A brief introduction to quantum groups
Pavel Etingof
MIT
May 5, 2020
1
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics.
Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications.
2
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications.
3
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theory
the Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications.
4
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands program
low-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications.
5
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands programlow-dimensional topology
category theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications.
6
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands programlow-dimensional topologycategory theory
enumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications.
7
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometry
quantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications.
8
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computation
algebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications.
9
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatorics
conformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications.
10
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theory
integrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications.
11
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systems
integrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications.
12
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probability
The goal of this talk is to review some of the main ideas andexamples of quantum groups and briefly describe some of theapplications.
13
Introduction
The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:
representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and
examples of quantum groups and briefly describe some of theapplications. 14
Hopf algebras
Classical physics:
X – a space of states;A = O(X ) – the algebra of observables, i.e., (say, complex)functions on X (a commutative, associative algebra).Example: X = R2. Then the coordinate x and momentum pare examples of observables (elements of A).
Quantum physics:A is deformed to a non-commutative (but still associative)algebra A~ with quantization parameter ~, e.g., theHeisenberg uncertainty relation [p, x ] = −i~.
15
Hopf algebras
Classical physics:X – a space of states;
A = O(X ) – the algebra of observables, i.e., (say, complex)functions on X (a commutative, associative algebra).Example: X = R2. Then the coordinate x and momentum pare examples of observables (elements of A).
Quantum physics:A is deformed to a non-commutative (but still associative)algebra A~ with quantization parameter ~, e.g., theHeisenberg uncertainty relation [p, x ] = −i~.
16
Hopf algebras
Classical physics:X – a space of states;A = O(X ) – the algebra of observables, i.e., (say, complex)functions on X (a commutative, associative algebra).
Example: X = R2. Then the coordinate x and momentum pare examples of observables (elements of A).
Quantum physics:A is deformed to a non-commutative (but still associative)algebra A~ with quantization parameter ~, e.g., theHeisenberg uncertainty relation [p, x ] = −i~.
17
Hopf algebras
Classical physics:X – a space of states;A = O(X ) – the algebra of observables, i.e., (say, complex)functions on X (a commutative, associative algebra).Example: X = R2. Then the coordinate x and momentum pare examples of observables (elements of A).
Quantum physics:A is deformed to a non-commutative (but still associative)algebra A~ with quantization parameter ~, e.g., theHeisenberg uncertainty relation [p, x ] = −i~.
18
Hopf algebras
Classical physics:X – a space of states;A = O(X ) – the algebra of observables, i.e., (say, complex)functions on X (a commutative, associative algebra).Example: X = R2. Then the coordinate x and momentum pare examples of observables (elements of A).
Quantum physics:
A is deformed to a non-commutative (but still associative)algebra A~ with quantization parameter ~, e.g., theHeisenberg uncertainty relation [p, x ] = −i~.
19
Hopf algebras
Classical physics:X – a space of states;A = O(X ) – the algebra of observables, i.e., (say, complex)functions on X (a commutative, associative algebra).Example: X = R2. Then the coordinate x and momentum pare examples of observables (elements of A).
Quantum physics:A is deformed to a non-commutative (but still associative)algebra A~ with quantization parameter ~,
e.g., theHeisenberg uncertainty relation [p, x ] = −i~.
20
Hopf algebras
Classical physics:X – a space of states;A = O(X ) – the algebra of observables, i.e., (say, complex)functions on X (a commutative, associative algebra).Example: X = R2. Then the coordinate x and momentum pare examples of observables (elements of A).
Quantum physics:A is deformed to a non-commutative (but still associative)algebra A~ with quantization parameter ~, e.g., theHeisenberg uncertainty relation [p, x ] = −i~.
21
Hopf algebras, ctd.
What if X = G is a group?
A group is equipped with an associative product, which has aunit and all elements are invertible:
m : G × G → G , m(x , y) = xy , x(yz) = (xy)z ,
∃ e : ∀g ∈ G : eg = ge = g ,
∀ g ∃ g−1 : gg−1 = g−1g = e.
Thus the algebra A = O(G ) of functions on a (say, finite)group has a natural structure of a coalgebra. Namely, decomposingO(G × G ) as O(G )⊗O(G ), one gets comultiplication, orcoproduct on A from the multiplication m in G :
∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).
Here we used Sweedler’s notation for the coproduct ∆f = f(1)⊗ f(2)where summation is implied, i.e., f(1) ⊗ f(2) is really
∑i f
i(1) ⊗ f i(2).
22
Hopf algebras, ctd.
What if X = G is a group?A group is equipped with an associative product, which has a
unit and all elements are invertible:
m : G × G → G , m(x , y) = xy , x(yz) = (xy)z ,
∃ e : ∀g ∈ G : eg = ge = g ,
∀ g ∃ g−1 : gg−1 = g−1g = e.
Thus the algebra A = O(G ) of functions on a (say, finite)group has a natural structure of a coalgebra. Namely, decomposingO(G × G ) as O(G )⊗O(G ), one gets comultiplication, orcoproduct on A from the multiplication m in G :
∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).
Here we used Sweedler’s notation for the coproduct ∆f = f(1)⊗ f(2)where summation is implied, i.e., f(1) ⊗ f(2) is really
∑i f
i(1) ⊗ f i(2).
23
Hopf algebras, ctd.
What if X = G is a group?A group is equipped with an associative product, which has a
unit and all elements are invertible:
m : G × G → G , m(x , y) = xy , x(yz) = (xy)z ,
∃ e : ∀g ∈ G : eg = ge = g ,
∀ g ∃ g−1 : gg−1 = g−1g = e.
Thus the algebra A = O(G ) of functions on a (say, finite)group has a natural structure of a coalgebra. Namely, decomposingO(G × G ) as O(G )⊗O(G ), one gets comultiplication, orcoproduct on A from the multiplication m in G :
∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).
Here we used Sweedler’s notation for the coproduct ∆f = f(1)⊗ f(2)where summation is implied, i.e., f(1) ⊗ f(2) is really
∑i f
i(1) ⊗ f i(2).
24
Hopf algebras, ctd.
What if X = G is a group?A group is equipped with an associative product, which has a
unit and all elements are invertible:
m : G × G → G , m(x , y) = xy , x(yz) = (xy)z ,
∃ e : ∀g ∈ G : eg = ge = g ,
∀ g ∃ g−1 : gg−1 = g−1g = e.
Thus the algebra A = O(G ) of functions on a (say, finite)group has a natural structure of a coalgebra.
Namely, decomposingO(G × G ) as O(G )⊗O(G ), one gets comultiplication, orcoproduct on A from the multiplication m in G :
∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).
Here we used Sweedler’s notation for the coproduct ∆f = f(1)⊗ f(2)where summation is implied, i.e., f(1) ⊗ f(2) is really
∑i f
i(1) ⊗ f i(2).
25
Hopf algebras, ctd.
What if X = G is a group?A group is equipped with an associative product, which has a
unit and all elements are invertible:
m : G × G → G , m(x , y) = xy , x(yz) = (xy)z ,
∃ e : ∀g ∈ G : eg = ge = g ,
∀ g ∃ g−1 : gg−1 = g−1g = e.
Thus the algebra A = O(G ) of functions on a (say, finite)group has a natural structure of a coalgebra. Namely, decomposingO(G × G ) as O(G )⊗O(G ), one gets comultiplication, orcoproduct on A from the multiplication m in G :
∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).
Here we used Sweedler’s notation for the coproduct ∆f = f(1)⊗ f(2)where summation is implied, i.e., f(1) ⊗ f(2) is really
∑i f
i(1) ⊗ f i(2).
26
Hopf algebras, ctd.
What if X = G is a group?A group is equipped with an associative product, which has a
unit and all elements are invertible:
m : G × G → G , m(x , y) = xy , x(yz) = (xy)z ,
∃ e : ∀g ∈ G : eg = ge = g ,
∀ g ∃ g−1 : gg−1 = g−1g = e.
Thus the algebra A = O(G ) of functions on a (say, finite)group has a natural structure of a coalgebra. Namely, decomposingO(G × G ) as O(G )⊗O(G ), one gets comultiplication, orcoproduct on A from the multiplication m in G :
∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).
Here we used Sweedler’s notation for the coproduct ∆f = f(1)⊗ f(2)where summation is implied, i.e., f(1) ⊗ f(2) is really
∑i f
i(1) ⊗ f i(2).
27
Hopf algebras, ctd.
What if X = G is a group?A group is equipped with an associative product, which has a
unit and all elements are invertible:
m : G × G → G , m(x , y) = xy , x(yz) = (xy)z ,
∃ e : ∀g ∈ G : eg = ge = g ,
∀ g ∃ g−1 : gg−1 = g−1g = e.
Thus the algebra A = O(G ) of functions on a (say, finite)group has a natural structure of a coalgebra. Namely, decomposingO(G × G ) as O(G )⊗O(G ), one gets comultiplication, orcoproduct on A from the multiplication m in G :
∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).
Here we used Sweedler’s notation for the coproduct ∆f = f(1)⊗ f(2)where summation is implied, i.e., f(1) ⊗ f(2) is really
∑i f
i(1) ⊗ f i(2).
28
Hopf algebras ctd.
It is clear that• ∆ is an algebra homomorphism.
The algebra A also has a natural counit and antipode obtainedfrom the unit and inversion in G :
ε : A→ C : ε(f ) = f (e),
S : A→ A : S(f )(x) = f (x−1).
The following properties could be easily checked:
∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),where µ : A⊗ A→ A is the multiplication.
Definition
A quantum group or Hopf algebra is a unital associative algebra A(not necessary commutative) which is equipped with ∆, ε,S andhas the properties listed above.
29
Hopf algebras ctd.
It is clear that• ∆ is an algebra homomorphism.The algebra A also has a natural counit and antipode obtained
from the unit and inversion in G :
ε : A→ C : ε(f ) = f (e),
S : A→ A : S(f )(x) = f (x−1).
The following properties could be easily checked:
∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),where µ : A⊗ A→ A is the multiplication.
Definition
A quantum group or Hopf algebra is a unital associative algebra A(not necessary commutative) which is equipped with ∆, ε,S andhas the properties listed above.
30
Hopf algebras ctd.
It is clear that• ∆ is an algebra homomorphism.The algebra A also has a natural counit and antipode obtained
from the unit and inversion in G :
ε : A→ C : ε(f ) = f (e),
S : A→ A : S(f )(x) = f (x−1).
The following properties could be easily checked:
∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),where µ : A⊗ A→ A is the multiplication.
Definition
A quantum group or Hopf algebra is a unital associative algebra A(not necessary commutative) which is equipped with ∆, ε,S andhas the properties listed above.
31
Hopf algebras ctd.
It is clear that• ∆ is an algebra homomorphism.The algebra A also has a natural counit and antipode obtained
from the unit and inversion in G :
ε : A→ C : ε(f ) = f (e),
S : A→ A : S(f )(x) = f (x−1).
The following properties could be easily checked:
∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;
(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),where µ : A⊗ A→ A is the multiplication.
Definition
A quantum group or Hopf algebra is a unital associative algebra A(not necessary commutative) which is equipped with ∆, ε,S andhas the properties listed above.
32
Hopf algebras ctd.
It is clear that• ∆ is an algebra homomorphism.The algebra A also has a natural counit and antipode obtained
from the unit and inversion in G :
ε : A→ C : ε(f ) = f (e),
S : A→ A : S(f )(x) = f (x−1).
The following properties could be easily checked:
∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;
µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),where µ : A⊗ A→ A is the multiplication.
Definition
A quantum group or Hopf algebra is a unital associative algebra A(not necessary commutative) which is equipped with ∆, ε,S andhas the properties listed above.
33
Hopf algebras ctd.
It is clear that• ∆ is an algebra homomorphism.The algebra A also has a natural counit and antipode obtained
from the unit and inversion in G :
ε : A→ C : ε(f ) = f (e),
S : A→ A : S(f )(x) = f (x−1).
The following properties could be easily checked:
∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),
where µ : A⊗ A→ A is the multiplication.
Definition
A quantum group or Hopf algebra is a unital associative algebra A(not necessary commutative) which is equipped with ∆, ε,S andhas the properties listed above.
34
Hopf algebras ctd.
It is clear that• ∆ is an algebra homomorphism.The algebra A also has a natural counit and antipode obtained
from the unit and inversion in G :
ε : A→ C : ε(f ) = f (e),
S : A→ A : S(f )(x) = f (x−1).
The following properties could be easily checked:
∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),where µ : A⊗ A→ A is the multiplication.
Definition
A quantum group or Hopf algebra is a unital associative algebra A(not necessary commutative) which is equipped with ∆, ε,S andhas the properties listed above.
35
Hopf algebras ctd.
It is clear that• ∆ is an algebra homomorphism.The algebra A also has a natural counit and antipode obtained
from the unit and inversion in G :
ε : A→ C : ε(f ) = f (e),
S : A→ A : S(f )(x) = f (x−1).
The following properties could be easily checked:
∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),where µ : A⊗ A→ A is the multiplication.
Definition
A quantum group or Hopf algebra is a unital associative algebra A(not necessary commutative) which is equipped with ∆, ε,S andhas the properties listed above.
36
Hopf algebras ctd.
Note that this definition makes sense over any field and evenover a commutative ring.
Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms. We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we canview the unit of A as a linear map ι : C→ A.
Proposition
1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).
37
Hopf algebras ctd.
Note that this definition makes sense over any field and evenover a commutative ring. Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms.
We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we canview the unit of A as a linear map ι : C→ A.
Proposition
1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).
38
Hopf algebras ctd.
Note that this definition makes sense over any field and evenover a commutative ring. Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms. We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u.
Finally, note that we canview the unit of A as a linear map ι : C→ A.
Proposition
1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).
39
Hopf algebras ctd.
Note that this definition makes sense over any field and evenover a commutative ring. Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms. We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we canview the unit of A as a linear map ι : C→ A.
Proposition
1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).
40
Hopf algebras ctd.
Note that this definition makes sense over any field and evenover a commutative ring. Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms. We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we canview the unit of A as a linear map ι : C→ A.
Proposition
1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.
2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).
41
Hopf algebras ctd.
Note that this definition makes sense over any field and evenover a commutative ring. Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms. We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we canview the unit of A as a linear map ι : C→ A.
Proposition
1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.
3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).
42
Hopf algebras ctd.
Note that this definition makes sense over any field and evenover a commutative ring. Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms. We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we canview the unit of A as a linear map ι : C→ A.
Proposition
1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).
4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).
43
Hopf algebras ctd.
Note that this definition makes sense over any field and evenover a commutative ring. Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms. We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we canview the unit of A as a linear map ι : C→ A.
Proposition
1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.
5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).
44
Hopf algebras ctd.
Note that this definition makes sense over any field and evenover a commutative ring. Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms. We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we canview the unit of A as a linear map ι : C→ A.
Proposition
1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).
45
Examples of Hopf algebras
Example
1 A = O(G ), G is finite. Then A is commutative.
2 A = O(G ) (the algebra of regular functions), G is an affinealgebraic group, A is commutative.
3 A = CG - the group algebra,∆(g) = g ⊗ g , S(g) = g−1, ε(g) = 1, A is cocommutative:∆ = ∆op.
4 g a Lie algebra, A = U(g) (the universal enveloping algebra),∆(x) = x ⊗ 1 + 1⊗ x , S(x) = −x , ε(x) = 0 for x ∈ g, A iscocommutative.
46
Examples of Hopf algebras
Example
1 A = O(G ), G is finite. Then A is commutative.
2 A = O(G ) (the algebra of regular functions), G is an affinealgebraic group, A is commutative.
3 A = CG - the group algebra,∆(g) = g ⊗ g , S(g) = g−1, ε(g) = 1, A is cocommutative:∆ = ∆op.
4 g a Lie algebra, A = U(g) (the universal enveloping algebra),∆(x) = x ⊗ 1 + 1⊗ x , S(x) = −x , ε(x) = 0 for x ∈ g, A iscocommutative.
47
Quantum SL2
Let q ∈ C, q 6= 0,±1. The quantum group Uq(sl2) is generatedby e, f ,K±1 with relations
KeK−1 = q2e, KfK−1 = q−2f , ef − fe =K − K−1
q − q−1.
The coproduct and counit are defined by
∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,
ε(e) = 0, ε(f ) = 0, ε(K ) = 1.
The antipode can then be found from the Hopf algebra axioms:
µ ◦ (S ⊗ 1) ◦∆(e) = ε(e) ⇒ S(e) = −eK−1.Similarly one obtains S(f ) = −Kf , S(K ) = K−1.
This is a deformation of U(sl2) because if one sets K = qh andsends q → 1, one recovers the sl2 relations.
This example shows that S2 6= id in general: we haveS2(x) = KxK−1.
48
Quantum SL2
Let q ∈ C, q 6= 0,±1. The quantum group Uq(sl2) is generatedby e, f ,K±1 with relations
KeK−1 = q2e, KfK−1 = q−2f , ef − fe =K − K−1
q − q−1.
The coproduct and counit are defined by
∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,
ε(e) = 0, ε(f ) = 0, ε(K ) = 1.
The antipode can then be found from the Hopf algebra axioms:
µ ◦ (S ⊗ 1) ◦∆(e) = ε(e) ⇒ S(e) = −eK−1.Similarly one obtains S(f ) = −Kf , S(K ) = K−1.
This is a deformation of U(sl2) because if one sets K = qh andsends q → 1, one recovers the sl2 relations.
This example shows that S2 6= id in general: we haveS2(x) = KxK−1.
49
Quantum SL2
Let q ∈ C, q 6= 0,±1. The quantum group Uq(sl2) is generatedby e, f ,K±1 with relations
KeK−1 = q2e, KfK−1 = q−2f , ef − fe =K − K−1
q − q−1.
The coproduct and counit are defined by
∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,
ε(e) = 0, ε(f ) = 0, ε(K ) = 1.
The antipode can then be found from the Hopf algebra axioms:
µ ◦ (S ⊗ 1) ◦∆(e) = ε(e) ⇒ S(e) = −eK−1.Similarly one obtains S(f ) = −Kf , S(K ) = K−1.
This is a deformation of U(sl2) because if one sets K = qh andsends q → 1, one recovers the sl2 relations.
This example shows that S2 6= id in general: we haveS2(x) = KxK−1.
50
Quantum SL2
Let q ∈ C, q 6= 0,±1. The quantum group Uq(sl2) is generatedby e, f ,K±1 with relations
KeK−1 = q2e, KfK−1 = q−2f , ef − fe =K − K−1
q − q−1.
The coproduct and counit are defined by
∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,
ε(e) = 0, ε(f ) = 0, ε(K ) = 1.
The antipode can then be found from the Hopf algebra axioms:
µ ◦ (S ⊗ 1) ◦∆(e) = ε(e) ⇒ S(e) = −eK−1.
Similarly one obtains S(f ) = −Kf , S(K ) = K−1.This is a deformation of U(sl2) because if one sets K = qh and
sends q → 1, one recovers the sl2 relations.This example shows that S2 6= id in general: we have
S2(x) = KxK−1.
51
Quantum SL2
Let q ∈ C, q 6= 0,±1. The quantum group Uq(sl2) is generatedby e, f ,K±1 with relations
KeK−1 = q2e, KfK−1 = q−2f , ef − fe =K − K−1
q − q−1.
The coproduct and counit are defined by
∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,
ε(e) = 0, ε(f ) = 0, ε(K ) = 1.
The antipode can then be found from the Hopf algebra axioms:
µ ◦ (S ⊗ 1) ◦∆(e) = ε(e) ⇒ S(e) = −eK−1.Similarly one obtains S(f ) = −Kf , S(K ) = K−1.
This is a deformation of U(sl2) because if one sets K = qh andsends q → 1, one recovers the sl2 relations.
This example shows that S2 6= id in general: we haveS2(x) = KxK−1.
52
Quantum SL2
Let q ∈ C, q 6= 0,±1. The quantum group Uq(sl2) is generatedby e, f ,K±1 with relations
KeK−1 = q2e, KfK−1 = q−2f , ef − fe =K − K−1
q − q−1.
The coproduct and counit are defined by
∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,
ε(e) = 0, ε(f ) = 0, ε(K ) = 1.
The antipode can then be found from the Hopf algebra axioms:
µ ◦ (S ⊗ 1) ◦∆(e) = ε(e) ⇒ S(e) = −eK−1.Similarly one obtains S(f ) = −Kf , S(K ) = K−1.
This is a deformation of U(sl2) because if one sets K = qh andsends q → 1, one recovers the sl2 relations.
This example shows that S2 6= id in general: we haveS2(x) = KxK−1.
53
Representations of Uq(sl2)
Assume that q is not a root of unity.
Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.
Proposition
Finite dimensional representations of Uq(sl2) are semisimple.
So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.
Proposition
There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1, with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n} such that Kf jv = qn−2j f jv ,
ef jv = [j ]q[n − j + 1]qfj−1v , where [k]q := qk−q−k
q−q−1 .
54
Representations of Uq(sl2)
Assume that q is not a root of unity. Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.
Proposition
Finite dimensional representations of Uq(sl2) are semisimple.
So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.
Proposition
There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1, with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n} such that Kf jv = qn−2j f jv ,
ef jv = [j ]q[n − j + 1]qfj−1v , where [k]q := qk−q−k
q−q−1 .
55
Representations of Uq(sl2)
Assume that q is not a root of unity. Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.
Proposition
Finite dimensional representations of Uq(sl2) are semisimple.
So it remains to classify the irreducible f.d. representations V .
We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.
Proposition
There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1, with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n} such that Kf jv = qn−2j f jv ,
ef jv = [j ]q[n − j + 1]qfj−1v , where [k]q := qk−q−k
q−q−1 .
56
Representations of Uq(sl2)
Assume that q is not a root of unity. Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.
Proposition
Finite dimensional representations of Uq(sl2) are semisimple.
So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q.
E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.
Proposition
There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1, with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n} such that Kf jv = qn−2j f jv ,
ef jv = [j ]q[n − j + 1]qfj−1v , where [k]q := qk−q−k
q−q−1 .
57
Representations of Uq(sl2)
Assume that q is not a root of unity. Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.
Proposition
Finite dimensional representations of Uq(sl2) are semisimple.
So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I.
However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.
Proposition
There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1, with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n} such that Kf jv = qn−2j f jv ,
ef jv = [j ]q[n − j + 1]qfj−1v , where [k]q := qk−q−k
q−q−1 .
58
Representations of Uq(sl2)
Assume that q is not a root of unity. Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.
Proposition
Finite dimensional representations of Uq(sl2) are semisimple.
So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.
Thus it suffices to classify irreducibles of type I.
Proposition
There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1, with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n} such that Kf jv = qn−2j f jv ,
ef jv = [j ]q[n − j + 1]qfj−1v , where [k]q := qk−q−k
q−q−1 .
59
Representations of Uq(sl2)
Assume that q is not a root of unity. Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.
Proposition
Finite dimensional representations of Uq(sl2) are semisimple.
So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.
Proposition
There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1, with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n} such that Kf jv = qn−2j f jv ,
ef jv = [j ]q[n − j + 1]qfj−1v , where [k]q := qk−q−k
q−q−1 .
60
Representations of Uq(sl2)
Assume that q is not a root of unity. Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.
Proposition
Finite dimensional representations of Uq(sl2) are semisimple.
So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.
Proposition
There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1,
with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n} such that Kf jv = qn−2j f jv ,
ef jv = [j ]q[n − j + 1]qfj−1v , where [k]q := qk−q−k
q−q−1 .
61
Representations of Uq(sl2)
Assume that q is not a root of unity. Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.
Proposition
Finite dimensional representations of Uq(sl2) are semisimple.
So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.
Proposition
There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1, with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n}
such that Kf jv = qn−2j f jv ,
ef jv = [j ]q[n − j + 1]qfj−1v , where [k]q := qk−q−k
q−q−1 .
62
Representations of Uq(sl2)
Assume that q is not a root of unity. Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.
Proposition
Finite dimensional representations of Uq(sl2) are semisimple.
So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.
Proposition
There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1, with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n} such that Kf jv = qn−2j f jv ,
ef jv = [j ]q[n − j + 1]qfj−1v , where [k]q := qk−q−k
q−q−1 .
63
Tensor products of representations of Hopf algebras
For a group and a Lie algebra the representation categoriesRepG and Rep g are endowed with tensor products:
πV : G → AutV , πW : G → AutW ⇒
πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .
πV : g→ AutV , πW : g→ AutW ⇒
πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.
The formula for the tensor product of representations of a Hopfalgebra A is a straightforward generalization:
πV⊗W (x) = (πV ⊗ πW )(∆(x)) = πV (x(1))⊗ πW (x(2)) ∀x ∈ A.
So one can regard the category C = RepA of representations of Aas a category equipped with a tensor product bifunctor
⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .
64
Tensor products of representations of Hopf algebras
For a group and a Lie algebra the representation categoriesRepG and Rep g are endowed with tensor products:
πV : G → AutV , πW : G → AutW ⇒
πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .
πV : g→ AutV , πW : g→ AutW ⇒
πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.
The formula for the tensor product of representations of a Hopfalgebra A is a straightforward generalization:
πV⊗W (x) = (πV ⊗ πW )(∆(x)) = πV (x(1))⊗ πW (x(2)) ∀x ∈ A.
So one can regard the category C = RepA of representations of Aas a category equipped with a tensor product bifunctor
⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .
65
Tensor products of representations of Hopf algebras
For a group and a Lie algebra the representation categoriesRepG and Rep g are endowed with tensor products:
πV : G → AutV , πW : G → AutW ⇒
πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .
πV : g→ AutV , πW : g→ AutW ⇒
πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.
The formula for the tensor product of representations of a Hopfalgebra A is a straightforward generalization:
πV⊗W (x) = (πV ⊗ πW )(∆(x)) = πV (x(1))⊗ πW (x(2)) ∀x ∈ A.
So one can regard the category C = RepA of representations of Aas a category equipped with a tensor product bifunctor
⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .
66
Tensor products of representations of Hopf algebras
For a group and a Lie algebra the representation categoriesRepG and Rep g are endowed with tensor products:
πV : G → AutV , πW : G → AutW ⇒
πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .
πV : g→ AutV , πW : g→ AutW ⇒
πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.
The formula for the tensor product of representations of a Hopfalgebra A is a straightforward generalization:
πV⊗W (x) = (πV ⊗ πW )(∆(x)) = πV (x(1))⊗ πW (x(2)) ∀x ∈ A.
So one can regard the category C = RepA of representations of Aas a category equipped with a tensor product bifunctor
⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .
67
Tensor products of representations of Hopf algebras
For a group and a Lie algebra the representation categoriesRepG and Rep g are endowed with tensor products:
πV : G → AutV , πW : G → AutW ⇒
πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .
πV : g→ AutV , πW : g→ AutW ⇒
πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.
The formula for the tensor product of representations of a Hopfalgebra A is a straightforward generalization:
πV⊗W (x) = (πV ⊗ πW )(∆(x)) = πV (x(1))⊗ πW (x(2)) ∀x ∈ A.
So one can regard the category C = RepA of representations of Aas a category equipped with a tensor product bifunctor
⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .
68
Tensor products of representations of Hopf algebras
For a group and a Lie algebra the representation categoriesRepG and Rep g are endowed with tensor products:
πV : G → AutV , πW : G → AutW ⇒
πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .
πV : g→ AutV , πW : g→ AutW ⇒
πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.
The formula for the tensor product of representations of a Hopfalgebra A is a straightforward generalization:
πV⊗W (x) = (πV ⊗ πW )(∆(x)) = πV (x(1))⊗ πW (x(2)) ∀x ∈ A.
So one can regard the category C = RepA of representations of Aas a category equipped with a tensor product bifunctor
⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .
69
Tensor products of representations of Hopf algebras, ctd.
This product also has a unit (the trivial 1-dimensionalrepresentation):
1 = C : π1(a) = ε(a), 1⊗ X ∼= X ⊗ 1 ∼= X ∀X ∈ C.Finally, the tensor product is associative on isomorphism classes:
(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).
Thus C is a category with a unital tensor product associative up toan isomorphism.
However, this notion is not very useful; about such categoriesone can say very little, if anything at all. On the other hand, innatural examples a lot more structure is present, which is just alittle bit less obvious.
More precisely, a much better notion is obtained if, according tothe general yoga of category theory, we don’t just say simply that(X ⊗Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ), but make this isomorphism a part ofthe data and impose coherence conditions on this data.This leadsto the notion of a monoidal category.
70
Tensor products of representations of Hopf algebras, ctd.
This product also has a unit (the trivial 1-dimensionalrepresentation):
1 = C : π1(a) = ε(a), 1⊗ X ∼= X ⊗ 1 ∼= X ∀X ∈ C.
Finally, the tensor product is associative on isomorphism classes:
(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).
Thus C is a category with a unital tensor product associative up toan isomorphism.
However, this notion is not very useful; about such categoriesone can say very little, if anything at all. On the other hand, innatural examples a lot more structure is present, which is just alittle bit less obvious.
More precisely, a much better notion is obtained if, according tothe general yoga of category theory, we don’t just say simply that(X ⊗Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ), but make this isomorphism a part ofthe data and impose coherence conditions on this data.This leadsto the notion of a monoidal category.
71
Tensor products of representations of Hopf algebras, ctd.
This product also has a unit (the trivial 1-dimensionalrepresentation):
1 = C : π1(a) = ε(a), 1⊗ X ∼= X ⊗ 1 ∼= X ∀X ∈ C.Finally, the tensor product is associative on isomorphism classes:
(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).
Thus C is a category with a unital tensor product associative up toan isomorphism.
However, this notion is not very useful; about such categoriesone can say very little, if anything at all. On the other hand, innatural examples a lot more structure is present, which is just alittle bit less obvious.
More precisely, a much better notion is obtained if, according tothe general yoga of category theory, we don’t just say simply that(X ⊗Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ), but make this isomorphism a part ofthe data and impose coherence conditions on this data.This leadsto the notion of a monoidal category.
72
Tensor products of representations of Hopf algebras, ctd.
This product also has a unit (the trivial 1-dimensionalrepresentation):
1 = C : π1(a) = ε(a), 1⊗ X ∼= X ⊗ 1 ∼= X ∀X ∈ C.Finally, the tensor product is associative on isomorphism classes:
(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).
Thus C is a category with a unital tensor product associative up toan isomorphism.
However, this notion is not very useful; about such categoriesone can say very little, if anything at all. On the other hand, innatural examples a lot more structure is present, which is just alittle bit less obvious.
More precisely, a much better notion is obtained if, according tothe general yoga of category theory, we don’t just say simply that(X ⊗Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ), but make this isomorphism a part ofthe data and impose coherence conditions on this data.This leadsto the notion of a monoidal category.
73
Tensor products of representations of Hopf algebras, ctd.
This product also has a unit (the trivial 1-dimensionalrepresentation):
1 = C : π1(a) = ε(a), 1⊗ X ∼= X ⊗ 1 ∼= X ∀X ∈ C.Finally, the tensor product is associative on isomorphism classes:
(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).
Thus C is a category with a unital tensor product associative up toan isomorphism.
However, this notion is not very useful; about such categoriesone can say very little, if anything at all. On the other hand, innatural examples a lot more structure is present, which is just alittle bit less obvious.
More precisely, a much better notion is obtained if, according tothe general yoga of category theory, we don’t just say simply that(X ⊗Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ), but make this isomorphism a part ofthe data and impose coherence conditions on this data.This leadsto the notion of a monoidal category.
74
Tensor products of representations of Hopf algebras, ctd.
This product also has a unit (the trivial 1-dimensionalrepresentation):
1 = C : π1(a) = ε(a), 1⊗ X ∼= X ⊗ 1 ∼= X ∀X ∈ C.Finally, the tensor product is associative on isomorphism classes:
(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).
Thus C is a category with a unital tensor product associative up toan isomorphism.
However, this notion is not very useful; about such categoriesone can say very little, if anything at all.
On the other hand, innatural examples a lot more structure is present, which is just alittle bit less obvious.
More precisely, a much better notion is obtained if, according tothe general yoga of category theory, we don’t just say simply that(X ⊗Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ), but make this isomorphism a part ofthe data and impose coherence conditions on this data.This leadsto the notion of a monoidal category.
75
Tensor products of representations of Hopf algebras, ctd.
This product also has a unit (the trivial 1-dimensionalrepresentation):
1 = C : π1(a) = ε(a), 1⊗ X ∼= X ⊗ 1 ∼= X ∀X ∈ C.Finally, the tensor product is associative on isomorphism classes:
(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).
Thus C is a category with a unital tensor product associative up toan isomorphism.
However, this notion is not very useful; about such categoriesone can say very little, if anything at all. On the other hand, innatural examples a lot more structure is present, which is just alittle bit less obvious.
More precisely, a much better notion is obtained if, according tothe general yoga of category theory, we don’t just say simply that(X ⊗Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ), but make this isomorphism a part ofthe data and impose coherence conditions on this data.This leadsto the notion of a monoidal category.
76
Tensor products of representations of Hopf algebras, ctd.
This product also has a unit (the trivial 1-dimensionalrepresentation):
1 = C : π1(a) = ε(a), 1⊗ X ∼= X ⊗ 1 ∼= X ∀X ∈ C.Finally, the tensor product is associative on isomorphism classes:
(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).
Thus C is a category with a unital tensor product associative up toan isomorphism.
However, this notion is not very useful; about such categoriesone can say very little, if anything at all. On the other hand, innatural examples a lot more structure is present, which is just alittle bit less obvious.
More precisely, a much better notion is obtained if, according tothe general yoga of category theory, we don’t just say simply that(X ⊗Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ), but make this isomorphism a part ofthe data and impose coherence conditions on this data.
This leadsto the notion of a monoidal category.
77
Tensor products of representations of Hopf algebras, ctd.
This product also has a unit (the trivial 1-dimensionalrepresentation):
1 = C : π1(a) = ε(a), 1⊗ X ∼= X ⊗ 1 ∼= X ∀X ∈ C.Finally, the tensor product is associative on isomorphism classes:
(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).
Thus C is a category with a unital tensor product associative up toan isomorphism.
However, this notion is not very useful; about such categoriesone can say very little, if anything at all. On the other hand, innatural examples a lot more structure is present, which is just alittle bit less obvious.
More precisely, a much better notion is obtained if, according tothe general yoga of category theory, we don’t just say simply that(X ⊗Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ), but make this isomorphism a part ofthe data and impose coherence conditions on this data.This leadsto the notion of a monoidal category.
78
Monoidal categories
Namely, we should equip C with an associativity isomorphism
αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )
functorial in X ,Y ,Z , which satisfies the pentagon identity
X (Y (ZT ))
(XY )(ZT )
((XY )Z )T (X (YZ ))T
X ((YZ )T )
where the arrows are induced by α and we have omitted the ⊗signs for brevity. The relation is that the diagram commutes.
79
Monoidal categories
Namely, we should equip C with an associativity isomorphism
αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )
functorial in X ,Y ,Z , which satisfies the pentagon identity
X (Y (ZT ))
(XY )(ZT )
((XY )Z )T (X (YZ ))T
X ((YZ )T )
where the arrows are induced by α and we have omitted the ⊗signs for brevity. The relation is that the diagram commutes.
80
Monoidal categories
Namely, we should equip C with an associativity isomorphism
αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )
functorial in X ,Y ,Z ,
which satisfies the pentagon identity
X (Y (ZT ))
(XY )(ZT )
((XY )Z )T (X (YZ ))T
X ((YZ )T )
where the arrows are induced by α and we have omitted the ⊗signs for brevity. The relation is that the diagram commutes.
81
Monoidal categories
Namely, we should equip C with an associativity isomorphism
αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )
functorial in X ,Y ,Z , which satisfies the pentagon identity
X (Y (ZT ))
(XY )(ZT )
((XY )Z )T (X (YZ ))T
X ((YZ )T )
where the arrows are induced by α and we have omitted the ⊗signs for brevity. The relation is that the diagram commutes.
82
Monoidal categories
Namely, we should equip C with an associativity isomorphism
αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )
functorial in X ,Y ,Z , which satisfies the pentagon identity
X (Y (ZT ))
(XY )(ZT )
((XY )Z )T (X (YZ ))T
X ((YZ )T )
where the arrows are induced by α and we have omitted the ⊗signs for brevity.
The relation is that the diagram commutes.
83
Monoidal categories
Namely, we should equip C with an associativity isomorphism
αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )
functorial in X ,Y ,Z , which satisfies the pentagon identity
X (Y (ZT ))
(XY )(ZT )
((XY )Z )T (X (YZ ))T
X ((YZ )T )
where the arrows are induced by α and we have omitted the ⊗signs for brevity. The relation is that the diagram commutes.
84
Monoidal categories, ctd.
We should also require the existence of a unit object 1
with anisomorphism
ι : 1⊗ 1 ∼= 1
such that the functors 1⊗ and ⊗1 are autoequivalences of C.
Definition
A category C with such structures and properties is called amonoidal category.
We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA. In fact, if dimA <∞ thenComodA = RepA∗. Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.
85
Monoidal categories, ctd.
We should also require the existence of a unit object 1 with anisomorphism
ι : 1⊗ 1 ∼= 1
such that the functors 1⊗ and ⊗1 are autoequivalences of C.
Definition
A category C with such structures and properties is called amonoidal category.
We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA. In fact, if dimA <∞ thenComodA = RepA∗. Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.
86
Monoidal categories, ctd.
We should also require the existence of a unit object 1 with anisomorphism
ι : 1⊗ 1 ∼= 1
such that the functors 1⊗ and ⊗1 are autoequivalences of C.
Definition
A category C with such structures and properties is called amonoidal category.
We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA. In fact, if dimA <∞ thenComodA = RepA∗. Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.
87
Monoidal categories, ctd.
We should also require the existence of a unit object 1 with anisomorphism
ι : 1⊗ 1 ∼= 1
such that the functors 1⊗ and ⊗1 are autoequivalences of C.
Definition
A category C with such structures and properties is called amonoidal category.
We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA. In fact, if dimA <∞ thenComodA = RepA∗. Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.
88
Monoidal categories, ctd.
We should also require the existence of a unit object 1 with anisomorphism
ι : 1⊗ 1 ∼= 1
such that the functors 1⊗ and ⊗1 are autoequivalences of C.
Definition
A category C with such structures and properties is called amonoidal category.
We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).
In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA. In fact, if dimA <∞ thenComodA = RepA∗. Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.
89
Monoidal categories, ctd.
We should also require the existence of a unit object 1 with anisomorphism
ι : 1⊗ 1 ∼= 1
such that the functors 1⊗ and ⊗1 are autoequivalences of C.
Definition
A category C with such structures and properties is called amonoidal category.
We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA.
In fact, if dimA <∞ thenComodA = RepA∗. Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.
90
Monoidal categories, ctd.
We should also require the existence of a unit object 1 with anisomorphism
ι : 1⊗ 1 ∼= 1
such that the functors 1⊗ and ⊗1 are autoequivalences of C.
Definition
A category C with such structures and properties is called amonoidal category.
We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA. In fact, if dimA <∞ thenComodA = RepA∗.
Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.
91
Monoidal categories, ctd.
We should also require the existence of a unit object 1 with anisomorphism
ι : 1⊗ 1 ∼= 1
such that the functors 1⊗ and ⊗1 are autoequivalences of C.
Definition
A category C with such structures and properties is called amonoidal category.
We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA. In fact, if dimA <∞ thenComodA = RepA∗. Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.
92
Duality in monoidal categories
Let us now discuss duality for representations of Hopf algebras,which generalizes duality for group and Lie algebrarepresentations.
The dual representations X ∗, ∗X ∈ RepA forX ∈ RepA are both the dual vector space to X with the actionsdefined as follows:
πX∗(a) = πX (S(a))∗ - left dual, π∗X (a) = πX (S−1(a))∗ - right dual.
For the pair X ,X ∗ there is the evaluation morphism
X ∗ ⊗ X → 1
(the usual pairing). For finite dimensional representations there isalso the coevaluation morphism
1→ X ⊗ X ∗.
and a pair of functorial isomorphisms
(∗X )∗ = X , ∗(X ∗) = X .
93
Duality in monoidal categories
Let us now discuss duality for representations of Hopf algebras,which generalizes duality for group and Lie algebrarepresentations.The dual representations X ∗, ∗X ∈ RepA forX ∈ RepA are both the dual vector space to X with the actionsdefined as follows:
πX∗(a) = πX (S(a))∗ - left dual, π∗X (a) = πX (S−1(a))∗ - right dual.
For the pair X ,X ∗ there is the evaluation morphism
X ∗ ⊗ X → 1
(the usual pairing). For finite dimensional representations there isalso the coevaluation morphism
1→ X ⊗ X ∗.
and a pair of functorial isomorphisms
(∗X )∗ = X , ∗(X ∗) = X .
94
Duality in monoidal categories
Let us now discuss duality for representations of Hopf algebras,which generalizes duality for group and Lie algebrarepresentations.The dual representations X ∗, ∗X ∈ RepA forX ∈ RepA are both the dual vector space to X with the actionsdefined as follows:
πX∗(a) = πX (S(a))∗ - left dual, π∗X (a) = πX (S−1(a))∗ - right dual.
For the pair X ,X ∗ there is the evaluation morphism
X ∗ ⊗ X → 1
(the usual pairing). For finite dimensional representations there isalso the coevaluation morphism
1→ X ⊗ X ∗.
and a pair of functorial isomorphisms
(∗X )∗ = X , ∗(X ∗) = X .
95
Duality in monoidal categories
Let us now discuss duality for representations of Hopf algebras,which generalizes duality for group and Lie algebrarepresentations.The dual representations X ∗, ∗X ∈ RepA forX ∈ RepA are both the dual vector space to X with the actionsdefined as follows:
πX∗(a) = πX (S(a))∗ - left dual, π∗X (a) = πX (S−1(a))∗ - right dual.
For the pair X ,X ∗ there is the evaluation morphism
X ∗ ⊗ X → 1
(the usual pairing).
For finite dimensional representations there isalso the coevaluation morphism
1→ X ⊗ X ∗.
and a pair of functorial isomorphisms
(∗X )∗ = X , ∗(X ∗) = X .
96
Duality in monoidal categories
Let us now discuss duality for representations of Hopf algebras,which generalizes duality for group and Lie algebrarepresentations.The dual representations X ∗, ∗X ∈ RepA forX ∈ RepA are both the dual vector space to X with the actionsdefined as follows:
πX∗(a) = πX (S(a))∗ - left dual, π∗X (a) = πX (S−1(a))∗ - right dual.
For the pair X ,X ∗ there is the evaluation morphism
X ∗ ⊗ X → 1
(the usual pairing). For finite dimensional representations there isalso the coevaluation morphism
1→ X ⊗ X ∗.
and a pair of functorial isomorphisms
(∗X )∗ = X , ∗(X ∗) = X .
97
Duality in monoidal categories
Let us now discuss duality for representations of Hopf algebras,which generalizes duality for group and Lie algebrarepresentations.The dual representations X ∗, ∗X ∈ RepA forX ∈ RepA are both the dual vector space to X with the actionsdefined as follows:
πX∗(a) = πX (S(a))∗ - left dual, π∗X (a) = πX (S−1(a))∗ - right dual.
For the pair X ,X ∗ there is the evaluation morphism
X ∗ ⊗ X → 1
(the usual pairing). For finite dimensional representations there isalso the coevaluation morphism
1→ X ⊗ X ∗.
and a pair of functorial isomorphisms
(∗X )∗ = X , ∗(X ∗) = X .
98
Duality in monoidal categories
Let us now discuss duality for representations of Hopf algebras,which generalizes duality for group and Lie algebrarepresentations.The dual representations X ∗, ∗X ∈ RepA forX ∈ RepA are both the dual vector space to X with the actionsdefined as follows:
πX∗(a) = πX (S(a))∗ - left dual, π∗X (a) = πX (S−1(a))∗ - right dual.
For the pair X ,X ∗ there is the evaluation morphism
X ∗ ⊗ X → 1
(the usual pairing). For finite dimensional representations there isalso the coevaluation morphism
1→ X ⊗ X ∗.
and a pair of functorial isomorphisms
(∗X )∗ = X , ∗(X ∗) = X .
99
Duality in monoidal categories
Let us now discuss duality for representations of Hopf algebras,which generalizes duality for group and Lie algebrarepresentations.The dual representations X ∗, ∗X ∈ RepA forX ∈ RepA are both the dual vector space to X with the actionsdefined as follows:
πX∗(a) = πX (S(a))∗ - left dual, π∗X (a) = πX (S−1(a))∗ - right dual.
For the pair X ,X ∗ there is the evaluation morphism
X ∗ ⊗ X → 1
(the usual pairing). For finite dimensional representations there isalso the coevaluation morphism
1→ X ⊗ X ∗.
and a pair of functorial isomorphisms
(∗X )∗ = X , ∗(X ∗) = X .
100
Duality in monoidal categories, ctd.
Let’s axiomatize this in the setting of monoidal categories.
Definition
An object Y of a monoidal category C is a left dual to X , denotedY = X ∗, if there exist evaluation and coevaluation morphisms
ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,
such that the following morphisms are the identities:
Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X
αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,
Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )
α−1YXY−−−−→ (Y ⊗ X )⊗ Y
ev⊗1−−−−→ Y .
If Y is a left dual to X then we have a functorial isomorphismHom(Z ,Y ) ∼= Hom(Z ⊗ X , 1). By the Yoneda lemma, this impliesthat the left dual, if exists, is unique up to a unique isomorphism.
101
Duality in monoidal categories, ctd.
Let’s axiomatize this in the setting of monoidal categories.
Definition
An object Y of a monoidal category C is a left dual to X , denotedY = X ∗, if there exist evaluation and coevaluation morphisms
ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,
such that the following morphisms are the identities:
Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X
αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,
Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )
α−1YXY−−−−→ (Y ⊗ X )⊗ Y
ev⊗1−−−−→ Y .
If Y is a left dual to X then we have a functorial isomorphismHom(Z ,Y ) ∼= Hom(Z ⊗ X , 1). By the Yoneda lemma, this impliesthat the left dual, if exists, is unique up to a unique isomorphism.
102
Duality in monoidal categories, ctd.
Let’s axiomatize this in the setting of monoidal categories.
Definition
An object Y of a monoidal category C is a left dual to X , denotedY = X ∗, if there exist evaluation and coevaluation morphisms
ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,
such that the following morphisms are the identities:
Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X
αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,
Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )
α−1YXY−−−−→ (Y ⊗ X )⊗ Y
ev⊗1−−−−→ Y .
If Y is a left dual to X then we have a functorial isomorphismHom(Z ,Y ) ∼= Hom(Z ⊗ X , 1). By the Yoneda lemma, this impliesthat the left dual, if exists, is unique up to a unique isomorphism.
103
Duality in monoidal categories, ctd.
Let’s axiomatize this in the setting of monoidal categories.
Definition
An object Y of a monoidal category C is a left dual to X , denotedY = X ∗, if there exist evaluation and coevaluation morphisms
ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,
such that the following morphisms are the identities:
Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X
αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,
Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )
α−1YXY−−−−→ (Y ⊗ X )⊗ Y
ev⊗1−−−−→ Y .
If Y is a left dual to X then we have a functorial isomorphismHom(Z ,Y ) ∼= Hom(Z ⊗ X , 1). By the Yoneda lemma, this impliesthat the left dual, if exists, is unique up to a unique isomorphism.
104
Duality in monoidal categories, ctd.
Let’s axiomatize this in the setting of monoidal categories.
Definition
An object Y of a monoidal category C is a left dual to X , denotedY = X ∗, if there exist evaluation and coevaluation morphisms
ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,
such that the following morphisms are the identities:
Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X
αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,
Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )
α−1YXY−−−−→ (Y ⊗ X )⊗ Y
ev⊗1−−−−→ Y .
If Y is a left dual to X then we have a functorial isomorphismHom(Z ,Y ) ∼= Hom(Z ⊗ X , 1). By the Yoneda lemma, this impliesthat the left dual, if exists, is unique up to a unique isomorphism.
105
Duality in monoidal categories, ctd.
Let’s axiomatize this in the setting of monoidal categories.
Definition
An object Y of a monoidal category C is a left dual to X , denotedY = X ∗, if there exist evaluation and coevaluation morphisms
ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,
such that the following morphisms are the identities:
Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X
αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,
Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )
α−1YXY−−−−→ (Y ⊗ X )⊗ Y
ev⊗1−−−−→ Y .
If Y is a left dual to X then we have a functorial isomorphismHom(Z ,Y ) ∼= Hom(Z ⊗ X , 1).
By the Yoneda lemma, this impliesthat the left dual, if exists, is unique up to a unique isomorphism.
106
Duality in monoidal categories, ctd.
Let’s axiomatize this in the setting of monoidal categories.
Definition
An object Y of a monoidal category C is a left dual to X , denotedY = X ∗, if there exist evaluation and coevaluation morphisms
ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,
such that the following morphisms are the identities:
Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X
αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,
Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )
α−1YXY−−−−→ (Y ⊗ X )⊗ Y
ev⊗1−−−−→ Y .
If Y is a left dual to X then we have a functorial isomorphismHom(Z ,Y ) ∼= Hom(Z ⊗ X , 1). By the Yoneda lemma, this impliesthat the left dual, if exists, is unique up to a unique isomorphism.
107
Duality in monoidal categories, ctd.
Definition
An object Z = ∗X is the right dual to X if X ∼= Z ∗.
As the left dual, the right dual is unique up to a uniqueisomorphism if exists.
Definition
An object X is rigid if it has both the left and the right dual. Acategory C is rigid if all its objects are rigid.
Thus, if A is a Hopf algebra then the category Repf A of finitedimensional representations of A is a rigid monoidal category.
108
Duality in monoidal categories, ctd.
Definition
An object Z = ∗X is the right dual to X if X ∼= Z ∗.
As the left dual, the right dual is unique up to a uniqueisomorphism if exists.
Definition
An object X is rigid if it has both the left and the right dual. Acategory C is rigid if all its objects are rigid.
Thus, if A is a Hopf algebra then the category Repf A of finitedimensional representations of A is a rigid monoidal category.
109
Duality in monoidal categories, ctd.
Definition
An object Z = ∗X is the right dual to X if X ∼= Z ∗.
As the left dual, the right dual is unique up to a uniqueisomorphism if exists.
Definition
An object X is rigid if it has both the left and the right dual.
Acategory C is rigid if all its objects are rigid.
Thus, if A is a Hopf algebra then the category Repf A of finitedimensional representations of A is a rigid monoidal category.
110
Duality in monoidal categories, ctd.
Definition
An object Z = ∗X is the right dual to X if X ∼= Z ∗.
As the left dual, the right dual is unique up to a uniqueisomorphism if exists.
Definition
An object X is rigid if it has both the left and the right dual. Acategory C is rigid if all its objects are rigid.
Thus, if A is a Hopf algebra then the category Repf A of finitedimensional representations of A is a rigid monoidal category.
111
Duality in monoidal categories, ctd.
Definition
An object Z = ∗X is the right dual to X if X ∼= Z ∗.
As the left dual, the right dual is unique up to a uniqueisomorphism if exists.
Definition
An object X is rigid if it has both the left and the right dual. Acategory C is rigid if all its objects are rigid.
Thus, if A is a Hopf algebra then the category Repf A of finitedimensional representations of A is a rigid monoidal category.
112
Examples of rigid monoidal categories
Example
Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G .
The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).
Example
The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).
113
Examples of rigid monoidal categories
Example
Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1.
ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).
Example
The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).
114
Examples of rigid monoidal categories
Example
Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category.
We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).
Example
The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).
115
Examples of rigid monoidal categories
Example
Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces).
This category makes sense for anygroup G (not necessarily finite).
Example
The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).
116
Examples of rigid monoidal categories
Example
Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).
Example
The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).
117
Examples of rigid monoidal categories
Example
Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).
Example
The previous example has the following twisted version.
Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).
118
Examples of rigid monoidal categories
Example
Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).
Example
The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.
Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).
119
Examples of rigid monoidal categories
Example
Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).
Example
The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G .
If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).
120
Examples of rigid monoidal categories
Example
Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).
Example
The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).
We will denote thiscategory Vec(G , α).
121
Examples of rigid monoidal categories
Example
Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).
Example
The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).
122
Monoidal functors
Let C,D be monoidal categories.
Definition
A functor F : C → D is a monoidal functor if F (1C) ∼= 1D and F isequipped with a functorial (in X ,Y ) isomorphismJX ,Y : F (X )⊗ F (Y )
∼−→ F (X ⊗ Y ) which makes the diagram
(F (X )⊗ F (Y ))⊗ F (Z )αD−−−−→ F (X )⊗ (F (Y )⊗ F (Z ))yJX ,Y⊗idZ
yidX⊗JY ,Z
F (X ⊗ Y )⊗ F (Z ) F (X )⊗ F (Y ⊗ Z )yJX⊗Y ,Z
yJX ,Y⊗Z
F ((X ⊗ Y )⊗ Z )F (αC)−−−−→ F (X ⊗ (Y ⊗ Z ))
commutative. A monoidal functor is an equivalence of monoidalcategories if it is an equivalence of categories.
123
Monoidal functors
Let C,D be monoidal categories.
Definition
A functor F : C → D is a monoidal functor if F (1C) ∼= 1D and F isequipped with a functorial (in X ,Y ) isomorphismJX ,Y : F (X )⊗ F (Y )
∼−→ F (X ⊗ Y )
which makes the diagram
(F (X )⊗ F (Y ))⊗ F (Z )αD−−−−→ F (X )⊗ (F (Y )⊗ F (Z ))yJX ,Y⊗idZ
yidX⊗JY ,Z
F (X ⊗ Y )⊗ F (Z ) F (X )⊗ F (Y ⊗ Z )yJX⊗Y ,Z
yJX ,Y⊗Z
F ((X ⊗ Y )⊗ Z )F (αC)−−−−→ F (X ⊗ (Y ⊗ Z ))
commutative. A monoidal functor is an equivalence of monoidalcategories if it is an equivalence of categories.
124
Monoidal functors
Let C,D be monoidal categories.
Definition
A functor F : C → D is a monoidal functor if F (1C) ∼= 1D and F isequipped with a functorial (in X ,Y ) isomorphismJX ,Y : F (X )⊗ F (Y )
∼−→ F (X ⊗ Y ) which makes the diagram
(F (X )⊗ F (Y ))⊗ F (Z )αD−−−−→ F (X )⊗ (F (Y )⊗ F (Z ))yJX ,Y⊗idZ
yidX⊗JY ,Z
F (X ⊗ Y )⊗ F (Z ) F (X )⊗ F (Y ⊗ Z )yJX⊗Y ,Z
yJX ,Y⊗Z
F ((X ⊗ Y )⊗ Z )F (αC)−−−−→ F (X ⊗ (Y ⊗ Z ))
commutative.
A monoidal functor is an equivalence of monoidalcategories if it is an equivalence of categories.
125
Monoidal functors
Let C,D be monoidal categories.
Definition
A functor F : C → D is a monoidal functor if F (1C) ∼= 1D and F isequipped with a functorial (in X ,Y ) isomorphismJX ,Y : F (X )⊗ F (Y )
∼−→ F (X ⊗ Y ) which makes the diagram
(F (X )⊗ F (Y ))⊗ F (Z )αD−−−−→ F (X )⊗ (F (Y )⊗ F (Z ))yJX ,Y⊗idZ
yidX⊗JY ,Z
F (X ⊗ Y )⊗ F (Z ) F (X )⊗ F (Y ⊗ Z )yJX⊗Y ,Z
yJX ,Y⊗Z
F ((X ⊗ Y )⊗ Z )F (αC)−−−−→ F (X ⊗ (Y ⊗ Z ))
commutative. A monoidal functor is an equivalence of monoidalcategories if it is an equivalence of categories.
126
Monoidal functors ctd.
The notion of monoidal equivalence is useful because monoidalcategories that are monoidally equivalent are “the same for allpractical purposes”.
Example
If α, β are two 3-cocycles on G then the identity functor
F = Id : Vec(G , α)→ Vec(G , β), g 7→ g
is monoidal with Jg ,h ∈ C∗ : g ⊗ h→ g ⊗ h iff dJ = α/β, where dis the differential in the standard complex of G with coefficients inC∗. In particular, F admits a monoidal structure if and only if thecohomology classes of α and β are the same.This shows thatVec(G , α) is equivalent to Vec(G ) iff α is trivial in H3(G ,C∗).
127
Monoidal functors ctd.
The notion of monoidal equivalence is useful because monoidalcategories that are monoidally equivalent are “the same for allpractical purposes”.
Example
If α, β are two 3-cocycles on G then the identity functor
F = Id : Vec(G , α)→ Vec(G , β), g 7→ g
is monoidal with Jg ,h ∈ C∗ : g ⊗ h→ g ⊗ h iff dJ = α/β, where dis the differential in the standard complex of G with coefficients inC∗. In particular, F admits a monoidal structure if and only if thecohomology classes of α and β are the same.This shows thatVec(G , α) is equivalent to Vec(G ) iff α is trivial in H3(G ,C∗).
128
Monoidal functors ctd.
The notion of monoidal equivalence is useful because monoidalcategories that are monoidally equivalent are “the same for allpractical purposes”.
Example
If α, β are two 3-cocycles on G then the identity functor
F = Id : Vec(G , α)→ Vec(G , β), g 7→ g
is monoidal with Jg ,h ∈ C∗ : g ⊗ h→ g ⊗ h iff dJ = α/β, where dis the differential in the standard complex of G with coefficients inC∗.
In particular, F admits a monoidal structure if and only if thecohomology classes of α and β are the same.This shows thatVec(G , α) is equivalent to Vec(G ) iff α is trivial in H3(G ,C∗).
129
Monoidal functors ctd.
The notion of monoidal equivalence is useful because monoidalcategories that are monoidally equivalent are “the same for allpractical purposes”.
Example
If α, β are two 3-cocycles on G then the identity functor
F = Id : Vec(G , α)→ Vec(G , β), g 7→ g
is monoidal with Jg ,h ∈ C∗ : g ⊗ h→ g ⊗ h iff dJ = α/β, where dis the differential in the standard complex of G with coefficients inC∗. In particular, F admits a monoidal structure if and only if thecohomology classes of α and β are the same.
This shows thatVec(G , α) is equivalent to Vec(G ) iff α is trivial in H3(G ,C∗).
130
Monoidal functors ctd.
The notion of monoidal equivalence is useful because monoidalcategories that are monoidally equivalent are “the same for allpractical purposes”.
Example
If α, β are two 3-cocycles on G then the identity functor
F = Id : Vec(G , α)→ Vec(G , β), g 7→ g
is monoidal with Jg ,h ∈ C∗ : g ⊗ h→ g ⊗ h iff dJ = α/β, where dis the differential in the standard complex of G with coefficients inC∗. In particular, F admits a monoidal structure if and only if thecohomology classes of α and β are the same.This shows thatVec(G , α) is equivalent to Vec(G ) iff α is trivial in H3(G ,C∗).
131
The universal R-matrix of Uq(sl2)
If A is a Hopf algebra and X ,Y ∈ RepH then X ⊗Y � Y ⊗Xin general, as ∆ 6= ∆op (e.g. g ⊗ h � h ⊗ g for g , h ∈ Vec(G )).
However, sometimes ∆ 6= ∆op but still X ⊗ Y ∼= Y ⊗ X .
Example
For A = Uq(sl2) where qn 6= 1 define the universal R-matrix
R = qh⊗h2
∞∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k , where [k]q! = [1]q...[k]q.
This is an infinite series, but it makes sense as an operator onX ⊗ Y for any finite dimensional type I representations X ,Y ,
because the sum terminates. Here qh⊗h2 (x ⊗ y) = q
λµ2 x ⊗ y if x , y
have weights λ, µ, i.e., Kx = qλx ,Ky = qµy .
Theorem (Drinfeld)
The operator c = P ◦ R defines an isomorphism of representationsc : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)Ron X ⊗ Y for a ∈ Uq(sl2).
132
The universal R-matrix of Uq(sl2)
If A is a Hopf algebra and X ,Y ∈ RepH then X ⊗Y � Y ⊗Xin general, as ∆ 6= ∆op (e.g. g ⊗ h � h ⊗ g for g , h ∈ Vec(G )).However, sometimes ∆ 6= ∆op but still X ⊗ Y ∼= Y ⊗ X .
Example
For A = Uq(sl2) where qn 6= 1 define the universal R-matrix
R = qh⊗h2
∞∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k , where [k]q! = [1]q...[k]q.
This is an infinite series, but it makes sense as an operator onX ⊗ Y for any finite dimensional type I representations X ,Y ,
because the sum terminates. Here qh⊗h2 (x ⊗ y) = q
λµ2 x ⊗ y if x , y
have weights λ, µ, i.e., Kx = qλx ,Ky = qµy .
Theorem (Drinfeld)
The operator c = P ◦ R defines an isomorphism of representationsc : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)Ron X ⊗ Y for a ∈ Uq(sl2).
133
The universal R-matrix of Uq(sl2)
If A is a Hopf algebra and X ,Y ∈ RepH then X ⊗Y � Y ⊗Xin general, as ∆ 6= ∆op (e.g. g ⊗ h � h ⊗ g for g , h ∈ Vec(G )).However, sometimes ∆ 6= ∆op but still X ⊗ Y ∼= Y ⊗ X .
Example
For A = Uq(sl2) where qn 6= 1 define the universal R-matrix
R = qh⊗h2
∞∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k ,
where [k]q! = [1]q...[k]q.
This is an infinite series, but it makes sense as an operator onX ⊗ Y for any finite dimensional type I representations X ,Y ,
because the sum terminates. Here qh⊗h2 (x ⊗ y) = q
λµ2 x ⊗ y if x , y
have weights λ, µ, i.e., Kx = qλx ,Ky = qµy .
Theorem (Drinfeld)
The operator c = P ◦ R defines an isomorphism of representationsc : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)Ron X ⊗ Y for a ∈ Uq(sl2).
134
The universal R-matrix of Uq(sl2)
If A is a Hopf algebra and X ,Y ∈ RepH then X ⊗Y � Y ⊗Xin general, as ∆ 6= ∆op (e.g. g ⊗ h � h ⊗ g for g , h ∈ Vec(G )).However, sometimes ∆ 6= ∆op but still X ⊗ Y ∼= Y ⊗ X .
Example
For A = Uq(sl2) where qn 6= 1 define the universal R-matrix
R = qh⊗h2
∞∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k , where [k]q! = [1]q...[k]q.
This is an infinite series, but it makes sense as an operator onX ⊗ Y for any finite dimensional type I representations X ,Y ,
because the sum terminates. Here qh⊗h2 (x ⊗ y) = q
λµ2 x ⊗ y if x , y
have weights λ, µ, i.e., Kx = qλx ,Ky = qµy .
Theorem (Drinfeld)
The operator c = P ◦ R defines an isomorphism of representationsc : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)Ron X ⊗ Y for a ∈ Uq(sl2).
135
The universal R-matrix of Uq(sl2)
If A is a Hopf algebra and X ,Y ∈ RepH then X ⊗Y � Y ⊗Xin general, as ∆ 6= ∆op (e.g. g ⊗ h � h ⊗ g for g , h ∈ Vec(G )).However, sometimes ∆ 6= ∆op but still X ⊗ Y ∼= Y ⊗ X .
Example
For A = Uq(sl2) where qn 6= 1 define the universal R-matrix
R = qh⊗h2
∞∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k , where [k]q! = [1]q...[k]q.
This is an infinite series, but it makes sense as an operator onX ⊗ Y for any finite dimensional type I representations X ,Y ,
because the sum terminates.
Here qh⊗h2 (x ⊗ y) = q
λµ2 x ⊗ y if x , y
have weights λ, µ, i.e., Kx = qλx ,Ky = qµy .
Theorem (Drinfeld)
The operator c = P ◦ R defines an isomorphism of representationsc : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)Ron X ⊗ Y for a ∈ Uq(sl2).
136
The universal R-matrix of Uq(sl2)
If A is a Hopf algebra and X ,Y ∈ RepH then X ⊗Y � Y ⊗Xin general, as ∆ 6= ∆op (e.g. g ⊗ h � h ⊗ g for g , h ∈ Vec(G )).However, sometimes ∆ 6= ∆op but still X ⊗ Y ∼= Y ⊗ X .
Example
For A = Uq(sl2) where qn 6= 1 define the universal R-matrix
R = qh⊗h2
∞∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k , where [k]q! = [1]q...[k]q.
This is an infinite series, but it makes sense as an operator onX ⊗ Y for any finite dimensional type I representations X ,Y ,
because the sum terminates. Here qh⊗h2 (x ⊗ y) = q
λµ2 x ⊗ y if x , y
have weights λ, µ, i.e., Kx = qλx ,Ky = qµy .
Theorem (Drinfeld)
The operator c = P ◦ R defines an isomorphism of representationsc : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)Ron X ⊗ Y for a ∈ Uq(sl2).
137
The universal R-matrix of Uq(sl2)
If A is a Hopf algebra and X ,Y ∈ RepH then X ⊗Y � Y ⊗Xin general, as ∆ 6= ∆op (e.g. g ⊗ h � h ⊗ g for g , h ∈ Vec(G )).However, sometimes ∆ 6= ∆op but still X ⊗ Y ∼= Y ⊗ X .
Example
For A = Uq(sl2) where qn 6= 1 define the universal R-matrix
R = qh⊗h2
∞∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k , where [k]q! = [1]q...[k]q.
This is an infinite series, but it makes sense as an operator onX ⊗ Y for any finite dimensional type I representations X ,Y ,
because the sum terminates. Here qh⊗h2 (x ⊗ y) = q
λµ2 x ⊗ y if x , y
have weights λ, µ, i.e., Kx = qλx ,Ky = qµy .
Theorem (Drinfeld)
The operator c = P ◦ R defines an isomorphism of representationsc : X ⊗ Y → Y ⊗ X .
In other words, we have R∆(a) = ∆op(a)Ron X ⊗ Y for a ∈ Uq(sl2).
138
The universal R-matrix of Uq(sl2)
If A is a Hopf algebra and X ,Y ∈ RepH then X ⊗Y � Y ⊗Xin general, as ∆ 6= ∆op (e.g. g ⊗ h � h ⊗ g for g , h ∈ Vec(G )).However, sometimes ∆ 6= ∆op but still X ⊗ Y ∼= Y ⊗ X .
Example
For A = Uq(sl2) where qn 6= 1 define the universal R-matrix
R = qh⊗h2
∞∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k , where [k]q! = [1]q...[k]q.
This is an infinite series, but it makes sense as an operator onX ⊗ Y for any finite dimensional type I representations X ,Y ,
because the sum terminates. Here qh⊗h2 (x ⊗ y) = q
λµ2 x ⊗ y if x , y
have weights λ, µ, i.e., Kx = qλx ,Ky = qµy .
Theorem (Drinfeld)
The operator c = P ◦ R defines an isomorphism of representationsc : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)Ron X ⊗ Y for a ∈ Uq(sl2).
139
The universal R-matrix of Uq(sl2)
If A is a Hopf algebra and X ,Y ∈ RepH then X ⊗Y � Y ⊗Xin general, as ∆ 6= ∆op (e.g. g ⊗ h � h ⊗ g for g , h ∈ Vec(G )).However, sometimes ∆ 6= ∆op but still X ⊗ Y ∼= Y ⊗ X .
Example
For A = Uq(sl2) where qn 6= 1 define the universal R-matrix
R = qh⊗h2
∞∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k , where [k]q! = [1]q...[k]q.
This is an infinite series, but it makes sense as an operator onX ⊗ Y for any finite dimensional type I representations X ,Y ,
because the sum terminates. Here qh⊗h2 (x ⊗ y) = q
λµ2 x ⊗ y if x , y
have weights λ, µ, i.e., Kx = qλx ,Ky = qµy .
Theorem (Drinfeld)
The operator c = P ◦ R defines an isomorphism of representationsc : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)Ron X ⊗ Y for a ∈ Uq(sl2).
140
The Drinfeld center
A prototypical example of a monoidal category whereX ⊗ Y ∼= Y ⊗ X is the Drinfeld center of a monoidal category C.
Definition
The Drinfeld center Z(C) of C is the category of pairs (Y , ϕ)where Y ∈ C and ϕ : Y⊗?→?⊗ Y is a functorial isomorphismgiven by ϕX : Y ⊗ X
∼−→ X ⊗ Y ∀X ∈ C, satisfying the followingcommutative diagram:
Y ⊗ (X1 ⊗ X2)ϕX1⊗X2−−−−→ (X1 ⊗ X2)⊗ Yyα−1
YX1X2
xα−1X1X2Y
(Y ⊗ X1)⊗ X2 X1 ⊗ (X2 ⊗ Y )yϕX1⊗id
xid⊗ϕX2
(X1 ⊗ Y )⊗ X2
αX1YX2−−−−→ X1 ⊗ (Y ⊗ X2)
Morphisms of such pairs are morphisms in C which preserve ϕ.
141
The Drinfeld center
A prototypical example of a monoidal category whereX ⊗ Y ∼= Y ⊗ X is the Drinfeld center of a monoidal category C.
Definition
The Drinfeld center Z(C) of C is the category of pairs (Y , ϕ)where Y ∈ C and ϕ : Y⊗?→?⊗ Y is a functorial isomorphismgiven by ϕX : Y ⊗ X
∼−→ X ⊗ Y ∀X ∈ C, satisfying the followingcommutative diagram:
Y ⊗ (X1 ⊗ X2)ϕX1⊗X2−−−−→ (X1 ⊗ X2)⊗ Yyα−1
YX1X2
xα−1X1X2Y
(Y ⊗ X1)⊗ X2 X1 ⊗ (X2 ⊗ Y )yϕX1⊗id
xid⊗ϕX2
(X1 ⊗ Y )⊗ X2
αX1YX2−−−−→ X1 ⊗ (Y ⊗ X2)
Morphisms of such pairs are morphisms in C which preserve ϕ.
142
The Drinfeld center
A prototypical example of a monoidal category whereX ⊗ Y ∼= Y ⊗ X is the Drinfeld center of a monoidal category C.
Definition
The Drinfeld center Z(C) of C is the category of pairs (Y , ϕ)where Y ∈ C and ϕ : Y⊗?→?⊗ Y is a functorial isomorphismgiven by ϕX : Y ⊗ X
∼−→ X ⊗ Y ∀X ∈ C, satisfying the followingcommutative diagram:
Y ⊗ (X1 ⊗ X2)ϕX1⊗X2−−−−→ (X1 ⊗ X2)⊗ Yyα−1
YX1X2
xα−1X1X2Y
(Y ⊗ X1)⊗ X2 X1 ⊗ (X2 ⊗ Y )yϕX1⊗id
xid⊗ϕX2
(X1 ⊗ Y )⊗ X2
αX1YX2−−−−→ X1 ⊗ (Y ⊗ X2)
Morphisms of such pairs are morphisms in C which preserve ϕ.
143
The Drinfeld center
A prototypical example of a monoidal category whereX ⊗ Y ∼= Y ⊗ X is the Drinfeld center of a monoidal category C.
Definition
The Drinfeld center Z(C) of C is the category of pairs (Y , ϕ)where Y ∈ C and ϕ : Y⊗?→?⊗ Y is a functorial isomorphismgiven by ϕX : Y ⊗ X
∼−→ X ⊗ Y ∀X ∈ C, satisfying the followingcommutative diagram:
Y ⊗ (X1 ⊗ X2)ϕX1⊗X2−−−−→ (X1 ⊗ X2)⊗ Yyα−1
YX1X2
xα−1X1X2Y
(Y ⊗ X1)⊗ X2 X1 ⊗ (X2 ⊗ Y )yϕX1⊗id
xid⊗ϕX2
(X1 ⊗ Y )⊗ X2
αX1YX2−−−−→ X1 ⊗ (Y ⊗ X2)
Morphisms of such pairs are morphisms in C which preserve ϕ.144
The Drinfeld center, ctd.
The Drinfeld center Z(C) has a natural monoidal structuredefined by (Y , ϕ)⊗ (Z , ψ) = (Y ⊗ Z , η),
where (suppressing α)
ηX = (φX⊗idZ )◦(idY⊗ψX ) : Y⊗Z⊗X → Y⊗X⊗Z → X⊗Y⊗Z .
Note also that we have a monoidal forgetful functor Z (C)→ C,(Y , ϕ) 7→ Y . Moreover, if Y ,Z ∈ Z(C) then there are two ways
Y ⊗ ZcYZ ,c
−1ZY−−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ ,
cZY = ψY . This gives an action of the braid group Bn on V⊗n forV ∈ Z(C). Recall that Bn = π1(Cn\diagonals/Sn) =
〈s1, ..., sn−1 | si sj = sjsi if |i − j | ≥ 2, si si+1si = si+1si si+1〉
Proposition
There is an action of Bn on V⊗n is defined by
ρ : Bn → Aut(V⊗n), ρ(si ) = ci ,i+1 = (cV ,V )i ,i+1
145
The Drinfeld center, ctd.
The Drinfeld center Z(C) has a natural monoidal structuredefined by (Y , ϕ)⊗ (Z , ψ) = (Y ⊗ Z , η), where (suppressing α)
ηX = (φX⊗idZ )◦(idY⊗ψX ) : Y⊗Z⊗X → Y⊗X⊗Z → X⊗Y⊗Z .
Note also that we have a monoidal forgetful functor Z (C)→ C,(Y , ϕ) 7→ Y . Moreover, if Y ,Z ∈ Z(C) then there are two ways
Y ⊗ ZcYZ ,c
−1ZY−−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ ,
cZY = ψY . This gives an action of the braid group Bn on V⊗n forV ∈ Z(C). Recall that Bn = π1(Cn\diagonals/Sn) =
〈s1, ..., sn−1 | si sj = sjsi if |i − j | ≥ 2, si si+1si = si+1si si+1〉
Proposition
There is an action of Bn on V⊗n is defined by
ρ : Bn → Aut(V⊗n), ρ(si ) = ci ,i+1 = (cV ,V )i ,i+1
146
The Drinfeld center, ctd.
The Drinfeld center Z(C) has a natural monoidal structuredefined by (Y , ϕ)⊗ (Z , ψ) = (Y ⊗ Z , η), where (suppressing α)
ηX = (φX⊗idZ )◦(idY⊗ψX ) : Y⊗Z⊗X → Y⊗X⊗Z → X⊗Y⊗Z .
Note also that we have a monoidal forgetful functor Z (C)→ C,(Y , ϕ) 7→ Y .
Moreover, if Y ,Z ∈ Z(C) then there are two ways
Y ⊗ ZcYZ ,c
−1ZY−−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ ,
cZY = ψY . This gives an action of the braid group Bn on V⊗n forV ∈ Z(C). Recall that Bn = π1(Cn\diagonals/Sn) =
〈s1, ..., sn−1 | si sj = sjsi if |i − j | ≥ 2, si si+1si = si+1si si+1〉
Proposition
There is an action of Bn on V⊗n is defined by
ρ : Bn → Aut(V⊗n), ρ(si ) = ci ,i+1 = (cV ,V )i ,i+1
147
The Drinfeld center, ctd.
The Drinfeld center Z(C) has a natural monoidal structuredefined by (Y , ϕ)⊗ (Z , ψ) = (Y ⊗ Z , η), where (suppressing α)
ηX = (φX⊗idZ )◦(idY⊗ψX ) : Y⊗Z⊗X → Y⊗X⊗Z → X⊗Y⊗Z .
Note also that we have a monoidal forgetful functor Z (C)→ C,(Y , ϕ) 7→ Y . Moreover, if Y ,Z ∈ Z(C) then there are two ways
Y ⊗ ZcYZ ,c
−1ZY−−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ ,
cZY = ψY .
This gives an action of the braid group Bn on V⊗n forV ∈ Z(C). Recall that Bn = π1(Cn\diagonals/Sn) =
〈s1, ..., sn−1 | si sj = sjsi if |i − j | ≥ 2, si si+1si = si+1si si+1〉
Proposition
There is an action of Bn on V⊗n is defined by
ρ : Bn → Aut(V⊗n), ρ(si ) = ci ,i+1 = (cV ,V )i ,i+1
148
The Drinfeld center, ctd.
The Drinfeld center Z(C) has a natural monoidal structuredefined by (Y , ϕ)⊗ (Z , ψ) = (Y ⊗ Z , η), where (suppressing α)
ηX = (φX⊗idZ )◦(idY⊗ψX ) : Y⊗Z⊗X → Y⊗X⊗Z → X⊗Y⊗Z .
Note also that we have a monoidal forgetful functor Z (C)→ C,(Y , ϕ) 7→ Y . Moreover, if Y ,Z ∈ Z(C) then there are two ways
Y ⊗ ZcYZ ,c
−1ZY−−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ ,
cZY = ψY . This gives an action of the braid group Bn on V⊗n forV ∈ Z(C).
Recall that Bn = π1(Cn\diagonals/Sn) =
〈s1, ..., sn−1 | si sj = sjsi if |i − j | ≥ 2, si si+1si = si+1si si+1〉
Proposition
There is an action of Bn on V⊗n is defined by
ρ : Bn → Aut(V⊗n), ρ(si ) = ci ,i+1 = (cV ,V )i ,i+1
149
The Drinfeld center, ctd.
The Drinfeld center Z(C) has a natural monoidal structuredefined by (Y , ϕ)⊗ (Z , ψ) = (Y ⊗ Z , η), where (suppressing α)
ηX = (φX⊗idZ )◦(idY⊗ψX ) : Y⊗Z⊗X → Y⊗X⊗Z → X⊗Y⊗Z .
Note also that we have a monoidal forgetful functor Z (C)→ C,(Y , ϕ) 7→ Y . Moreover, if Y ,Z ∈ Z(C) then there are two ways
Y ⊗ ZcYZ ,c
−1ZY−−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ ,
cZY = ψY . This gives an action of the braid group Bn on V⊗n forV ∈ Z(C). Recall that Bn = π1(Cn\diagonals/Sn) =
〈s1, ..., sn−1 | si sj = sjsi if |i − j | ≥ 2, si si+1si = si+1si si+1〉
Proposition
There is an action of Bn on V⊗n is defined by
ρ : Bn → Aut(V⊗n), ρ(si ) = ci ,i+1 = (cV ,V )i ,i+1
150
The Drinfeld center, ctd.
The Drinfeld center Z(C) has a natural monoidal structuredefined by (Y , ϕ)⊗ (Z , ψ) = (Y ⊗ Z , η), where (suppressing α)
ηX = (φX⊗idZ )◦(idY⊗ψX ) : Y⊗Z⊗X → Y⊗X⊗Z → X⊗Y⊗Z .
Note also that we have a monoidal forgetful functor Z (C)→ C,(Y , ϕ) 7→ Y . Moreover, if Y ,Z ∈ Z(C) then there are two ways
Y ⊗ ZcYZ ,c
−1ZY−−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ ,
cZY = ψY . This gives an action of the braid group Bn on V⊗n forV ∈ Z(C). Recall that Bn = π1(Cn\diagonals/Sn) =
〈s1, ..., sn−1 | si sj = sjsi if |i − j | ≥ 2, si si+1si = si+1si si+1〉
Proposition
There is an action of Bn on V⊗n is defined by
ρ : Bn → Aut(V⊗n), ρ(si ) = ci ,i+1 = (cV ,V )i ,i+1
151
The Drinfeld center, ctd.
The Drinfeld center Z(C) has a natural monoidal structuredefined by (Y , ϕ)⊗ (Z , ψ) = (Y ⊗ Z , η), where (suppressing α)
ηX = (φX⊗idZ )◦(idY⊗ψX ) : Y⊗Z⊗X → Y⊗X⊗Z → X⊗Y⊗Z .
Note also that we have a monoidal forgetful functor Z (C)→ C,(Y , ϕ) 7→ Y . Moreover, if Y ,Z ∈ Z(C) then there are two ways
Y ⊗ ZcYZ ,c
−1ZY−−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ ,
cZY = ψY . This gives an action of the braid group Bn on V⊗n forV ∈ Z(C). Recall that Bn = π1(Cn\diagonals/Sn) =
〈s1, ..., sn−1 | si sj = sjsi if |i − j | ≥ 2, si si+1si = si+1si si+1〉
Proposition
There is an action of Bn on V⊗n is defined by
ρ : Bn → Aut(V⊗n), ρ(si ) = ci ,i+1 = (cV ,V )i ,i+1
152
The Drinfeld center, ctd.
The Drinfeld center Z(C) has a natural monoidal structuredefined by (Y , ϕ)⊗ (Z , ψ) = (Y ⊗ Z , η), where (suppressing α)
ηX = (φX⊗idZ )◦(idY⊗ψX ) : Y⊗Z⊗X → Y⊗X⊗Z → X⊗Y⊗Z .
Note also that we have a monoidal forgetful functor Z (C)→ C,(Y , ϕ) 7→ Y . Moreover, if Y ,Z ∈ Z(C) then there are two ways
Y ⊗ ZcYZ ,c
−1ZY−−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ ,
cZY = ψY . This gives an action of the braid group Bn on V⊗n forV ∈ Z(C). Recall that Bn = π1(Cn\diagonals/Sn) =
〈s1, ..., sn−1 | si sj = sjsi if |i − j | ≥ 2, si si+1si = si+1si si+1〉
Proposition
There is an action of Bn on V⊗n is defined by
ρ : Bn → Aut(V⊗n), ρ(si ) = ci ,i+1 = (cV ,V )i ,i+1
153
The Drinfeld center, ctd.
Proof.
This follows from the hexagon relations for X ,Y ,Z ∈ Z(C):
X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z↘ ↓
Y ⊗ Z ⊗ X
X ⊗ Y ⊗ Z → X ⊗ Z ⊗ Y↘ ↓
Z ⊗ X ⊗ Y
where we suppress α and the maps are given by c .
They are call “hexagon relations” because they would havebeen hexagons had we not suppressed α.
This motivates the definition of a braided monoidal category.
154
The Drinfeld center, ctd.
Proof.
This follows from the hexagon relations for X ,Y ,Z ∈ Z(C):
X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z↘ ↓
Y ⊗ Z ⊗ X
X ⊗ Y ⊗ Z → X ⊗ Z ⊗ Y↘ ↓
Z ⊗ X ⊗ Y
where we suppress α and the maps are given by c .
They are call “hexagon relations” because they would havebeen hexagons had we not suppressed α.
This motivates the definition of a braided monoidal category.
155
The Drinfeld center, ctd.
Proof.
This follows from the hexagon relations for X ,Y ,Z ∈ Z(C):
X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z↘ ↓
Y ⊗ Z ⊗ X
X ⊗ Y ⊗ Z → X ⊗ Z ⊗ Y↘ ↓
Z ⊗ X ⊗ Y
where we suppress α and the maps are given by c .
They are call “hexagon relations” because they would havebeen hexagons had we not suppressed α.
This motivates the definition of a braided monoidal category.
156
The Drinfeld center, ctd.
Proof.
This follows from the hexagon relations for X ,Y ,Z ∈ Z(C):
X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z↘ ↓
Y ⊗ Z ⊗ X
X ⊗ Y ⊗ Z → X ⊗ Z ⊗ Y↘ ↓
Z ⊗ X ⊗ Y
where we suppress α and the maps are given by c .
They are call “hexagon relations” because they would havebeen hexagons had we not suppressed α.
This motivates the definition of a braided monoidal category.
157
Braided monoidal categories
Definition
A braided monoidal category is a monoidal category endowed witha functorial isomorphism c : ⊗ → ⊗op, cX ,Y : X ⊗ Y → Y ⊗ X
which satisfies the hexagon relations.
Thus we obtain
Theorem
The Drinfeld center Z(C) is a braided monoidal category.
Moreover, every braided monoidal category C is a braidedsubcategory of its Drinfeld center using the inclusion ι : C → Z(C)given by X 7→ (X , cX?). In this sense the Drinfeld center is aprototypical example of a braided category.
Theorem
The category of finite dimensional (type I) representations ofUq(sl2) is a braided monoidal category, with c = P ◦ R.
158
Braided monoidal categories
Definition
A braided monoidal category is a monoidal category endowed witha functorial isomorphism c : ⊗ → ⊗op, cX ,Y : X ⊗ Y → Y ⊗ Xwhich satisfies the hexagon relations.
Thus we obtain
Theorem
The Drinfeld center Z(C) is a braided monoidal category.
Moreover, every braided monoidal category C is a braidedsubcategory of its Drinfeld center using the inclusion ι : C → Z(C)given by X 7→ (X , cX?). In this sense the Drinfeld center is aprototypical example of a braided category.
Theorem
The category of finite dimensional (type I) representations ofUq(sl2) is a braided monoidal category, with c = P ◦ R.
159
Braided monoidal categories
Definition
A braided monoidal category is a monoidal category endowed witha functorial isomorphism c : ⊗ → ⊗op, cX ,Y : X ⊗ Y → Y ⊗ Xwhich satisfies the hexagon relations.
Thus we obtain
Theorem
The Drinfeld center Z(C) is a braided monoidal category.
Moreover, every braided monoidal category C is a braidedsubcategory of its Drinfeld center using the inclusion ι : C → Z(C)given by X 7→ (X , cX?). In this sense the Drinfeld center is aprototypical example of a braided category.
Theorem
The category of finite dimensional (type I) representations ofUq(sl2) is a braided monoidal category, with c = P ◦ R.
160
Braided monoidal categories
Definition
A braided monoidal category is a monoidal category endowed witha functorial isomorphism c : ⊗ → ⊗op, cX ,Y : X ⊗ Y → Y ⊗ Xwhich satisfies the hexagon relations.
Thus we obtain
Theorem
The Drinfeld center Z(C) is a braided monoidal category.
Moreover, every braided monoidal category C is a braidedsubcategory of its Drinfeld center using the inclusion ι : C → Z(C)given by X 7→ (X , cX?). In this sense the Drinfeld center is aprototypical example of a braided category.
Theorem
The category of finite dimensional (type I) representations ofUq(sl2) is a braided monoidal category, with c = P ◦ R.
161
Braided monoidal categories ctd.
Note that as a consequence R satisfies the QuantumYang-Baxter equation
R12R13R23 = R23R13R12
(this follows from the relation s1s2s1 = s2s1s2).
Remark
A braided category is called symmetric if cXY cYX = idX⊗Y for allX ,Y . For example, the categories RepG and Rep g aresymmetric. However, Z(C) is usually not symmetric, andRepf Uq(sl2) isn’t either.
We’ll explain that the theorem on Uq(sl2), and in fact theconstruction of R, are consequences of the theorem about theDrinfeld center. To this end, let us compute the Drinfeld center ofthe category RepA of representations of a Hopf algebra A.
162
Braided monoidal categories ctd.
Note that as a consequence R satisfies the QuantumYang-Baxter equation
R12R13R23 = R23R13R12
(this follows from the relation s1s2s1 = s2s1s2).
Remark
A braided category is called symmetric if cXY cYX = idX⊗Y for allX ,Y .
For example, the categories RepG and Rep g aresymmetric. However, Z(C) is usually not symmetric, andRepf Uq(sl2) isn’t either.
We’ll explain that the theorem on Uq(sl2), and in fact theconstruction of R, are consequences of the theorem about theDrinfeld center. To this end, let us compute the Drinfeld center ofthe category RepA of representations of a Hopf algebra A.
163
Braided monoidal categories ctd.
Note that as a consequence R satisfies the QuantumYang-Baxter equation
R12R13R23 = R23R13R12
(this follows from the relation s1s2s1 = s2s1s2).
Remark
A braided category is called symmetric if cXY cYX = idX⊗Y for allX ,Y . For example, the categories RepG and Rep g aresymmetric.
However, Z(C) is usually not symmetric, andRepf Uq(sl2) isn’t either.
We’ll explain that the theorem on Uq(sl2), and in fact theconstruction of R, are consequences of the theorem about theDrinfeld center. To this end, let us compute the Drinfeld center ofthe category RepA of representations of a Hopf algebra A.
164
Braided monoidal categories ctd.
Note that as a consequence R satisfies the QuantumYang-Baxter equation
R12R13R23 = R23R13R12
(this follows from the relation s1s2s1 = s2s1s2).
Remark
A braided category is called symmetric if cXY cYX = idX⊗Y for allX ,Y . For example, the categories RepG and Rep g aresymmetric. However, Z(C) is usually not symmetric, andRepf Uq(sl2) isn’t either.
We’ll explain that the theorem on Uq(sl2), and in fact theconstruction of R, are consequences of the theorem about theDrinfeld center.
To this end, let us compute the Drinfeld center ofthe category RepA of representations of a Hopf algebra A.
165
Braided monoidal categories ctd.
Note that as a consequence R satisfies the QuantumYang-Baxter equation
R12R13R23 = R23R13R12
(this follows from the relation s1s2s1 = s2s1s2).
Remark
A braided category is called symmetric if cXY cYX = idX⊗Y for allX ,Y . For example, the categories RepG and Rep g aresymmetric. However, Z(C) is usually not symmetric, andRepf Uq(sl2) isn’t either.
We’ll explain that the theorem on Uq(sl2), and in fact theconstruction of R, are consequences of the theorem about theDrinfeld center. To this end, let us compute the Drinfeld center ofthe category RepA of representations of a Hopf algebra A.
166
Yetter-Drinfeld modules
Let Y ∈ Z(RepA).
Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y . The compatibility condition between thiscomodule structure and the A-module structure on Y is
τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).
where y ∈ Y , τ(y) = y(1) ⊗ y(2) and
a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).
Definition
A Yetter-Drinfeld module over A is an A-module Y which is also anA-comodule with τ : Y → A⊗Y satisfying the above compatibilitycondition. The category of such modules is denoted YD(A).
Proposition
One has Z(RepA) ∼= YD(A).
167
Yetter-Drinfeld modules
Let Y ∈ Z(RepA). Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y .
The compatibility condition between thiscomodule structure and the A-module structure on Y is
τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).
where y ∈ Y , τ(y) = y(1) ⊗ y(2) and
a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).
Definition
A Yetter-Drinfeld module over A is an A-module Y which is also anA-comodule with τ : Y → A⊗Y satisfying the above compatibilitycondition. The category of such modules is denoted YD(A).
Proposition
One has Z(RepA) ∼= YD(A).
168
Yetter-Drinfeld modules
Let Y ∈ Z(RepA). Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y . The compatibility condition between thiscomodule structure and the A-module structure on Y is
τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).
where y ∈ Y , τ(y) = y(1) ⊗ y(2) and
a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).
Definition
A Yetter-Drinfeld module over A is an A-module Y which is also anA-comodule with τ : Y → A⊗Y satisfying the above compatibilitycondition. The category of such modules is denoted YD(A).
Proposition
One has Z(RepA) ∼= YD(A).
169
Yetter-Drinfeld modules
Let Y ∈ Z(RepA). Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y . The compatibility condition between thiscomodule structure and the A-module structure on Y is
τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).
where y ∈ Y , τ(y) = y(1) ⊗ y(2)
and
a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).
Definition
A Yetter-Drinfeld module over A is an A-module Y which is also anA-comodule with τ : Y → A⊗Y satisfying the above compatibilitycondition. The category of such modules is denoted YD(A).
Proposition
One has Z(RepA) ∼= YD(A).
170
Yetter-Drinfeld modules
Let Y ∈ Z(RepA). Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y . The compatibility condition between thiscomodule structure and the A-module structure on Y is
τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).
where y ∈ Y , τ(y) = y(1) ⊗ y(2) and
a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).
Definition
A Yetter-Drinfeld module over A is an A-module Y which is also anA-comodule with τ : Y → A⊗Y satisfying the above compatibilitycondition. The category of such modules is denoted YD(A).
Proposition
One has Z(RepA) ∼= YD(A).
171
Yetter-Drinfeld modules
Let Y ∈ Z(RepA). Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y . The compatibility condition between thiscomodule structure and the A-module structure on Y is
τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).
where y ∈ Y , τ(y) = y(1) ⊗ y(2) and
a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).
Definition
A Yetter-Drinfeld module over A is an A-module Y which is also anA-comodule with τ : Y → A⊗Y satisfying the above compatibilitycondition.
The category of such modules is denoted YD(A).
Proposition
One has Z(RepA) ∼= YD(A).
172
Yetter-Drinfeld modules
Let Y ∈ Z(RepA). Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y . The compatibility condition between thiscomodule structure and the A-module structure on Y is
τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).
where y ∈ Y , τ(y) = y(1) ⊗ y(2) and
a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).
Definition
A Yetter-Drinfeld module over A is an A-module Y which is also anA-comodule with τ : Y → A⊗Y satisfying the above compatibilitycondition. The category of such modules is denoted YD(A).
Proposition
One has Z(RepA) ∼= YD(A).
173
Yetter-Drinfeld modules
Let Y ∈ Z(RepA). Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y . The compatibility condition between thiscomodule structure and the A-module structure on Y is
τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).
where y ∈ Y , τ(y) = y(1) ⊗ y(2) and
a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).
Definition
A Yetter-Drinfeld module over A is an A-module Y which is also anA-comodule with τ : Y → A⊗Y satisfying the above compatibilitycondition. The category of such modules is denoted YD(A).
Proposition
One has Z(RepA) ∼= YD(A).
174
Yetter-Drinfeld modules
Let Y ∈ Z(RepA). Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y . The compatibility condition between thiscomodule structure and the A-module structure on Y is
τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).
where y ∈ Y , τ(y) = y(1) ⊗ y(2) and
a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).
Definition
A Yetter-Drinfeld module over A is an A-module Y which is also anA-comodule with τ : Y → A⊗Y satisfying the above compatibilitycondition. The category of such modules is denoted YD(A).
Proposition
One has Z(RepA) ∼= YD(A).
175
Quantum double
If A is finite dimensional, an A-comodule is the same as anA∗-module,
so the category YD(A) can be realized as the categoryof modules over some algebra generated by A and A∗ with somecommutation relation between them. Thus we get
Proposition
For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as avector space to A⊗ A∗ such that there is a monoidal equivalence
YD(A) ∼= RepD(A).
The Hopf algebra D(A) is called the (Drinfeld) quantum doubleof A. The algebras A and A∗op (with opposite coproduct) are Hopfsubalgebras of D(A), and the commutation relations between themhave the form ba = (a(1), b(1))(a(3), b(3))a(2)S
−1(b(2)),a ∈ A, b ∈ A∗.
176
Quantum double
If A is finite dimensional, an A-comodule is the same as anA∗-module, so the category YD(A) can be realized as the categoryof modules over some algebra generated by A and A∗ with somecommutation relation between them.
Thus we get
Proposition
For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as avector space to A⊗ A∗ such that there is a monoidal equivalence
YD(A) ∼= RepD(A).
The Hopf algebra D(A) is called the (Drinfeld) quantum doubleof A. The algebras A and A∗op (with opposite coproduct) are Hopfsubalgebras of D(A), and the commutation relations between themhave the form ba = (a(1), b(1))(a(3), b(3))a(2)S
−1(b(2)),a ∈ A, b ∈ A∗.
177
Quantum double
If A is finite dimensional, an A-comodule is the same as anA∗-module, so the category YD(A) can be realized as the categoryof modules over some algebra generated by A and A∗ with somecommutation relation between them. Thus we get
Proposition
For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as avector space to A⊗ A∗
such that there is a monoidal equivalence
YD(A) ∼= RepD(A).
The Hopf algebra D(A) is called the (Drinfeld) quantum doubleof A. The algebras A and A∗op (with opposite coproduct) are Hopfsubalgebras of D(A), and the commutation relations between themhave the form ba = (a(1), b(1))(a(3), b(3))a(2)S
−1(b(2)),a ∈ A, b ∈ A∗.
178
Quantum double
If A is finite dimensional, an A-comodule is the same as anA∗-module, so the category YD(A) can be realized as the categoryof modules over some algebra generated by A and A∗ with somecommutation relation between them. Thus we get
Proposition
For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as avector space to A⊗ A∗ such that there is a monoidal equivalence
YD(A) ∼= RepD(A).
The Hopf algebra D(A) is called the (Drinfeld) quantum doubleof A. The algebras A and A∗op (with opposite coproduct) are Hopfsubalgebras of D(A), and the commutation relations between themhave the form ba = (a(1), b(1))(a(3), b(3))a(2)S
−1(b(2)),a ∈ A, b ∈ A∗.
179
Quantum double
If A is finite dimensional, an A-comodule is the same as anA∗-module, so the category YD(A) can be realized as the categoryof modules over some algebra generated by A and A∗ with somecommutation relation between them. Thus we get
Proposition
For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as avector space to A⊗ A∗ such that there is a monoidal equivalence
YD(A) ∼= RepD(A).
The Hopf algebra D(A) is called the (Drinfeld) quantum doubleof A.
The algebras A and A∗op (with opposite coproduct) are Hopfsubalgebras of D(A), and the commutation relations between themhave the form ba = (a(1), b(1))(a(3), b(3))a(2)S
−1(b(2)),a ∈ A, b ∈ A∗.
180
Quantum double
If A is finite dimensional, an A-comodule is the same as anA∗-module, so the category YD(A) can be realized as the categoryof modules over some algebra generated by A and A∗ with somecommutation relation between them. Thus we get
Proposition
For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as avector space to A⊗ A∗ such that there is a monoidal equivalence
YD(A) ∼= RepD(A).
The Hopf algebra D(A) is called the (Drinfeld) quantum doubleof A. The algebras A and A∗op (with opposite coproduct) are Hopfsubalgebras of D(A), and the commutation relations between themhave the form ba = (a(1), b(1))(a(3), b(3))a(2)S
−1(b(2)),a ∈ A, b ∈ A∗.
181
Quantum double, ctd.
It is natural to ask how to express the braided structure ofYD(A) in terms of D(A).
This is addressed by the followingtheorem.
Theorem (Drinfeld)
The braiding of RepD(A) = YD(A) is given by the formulac = P ◦ R where R ∈ A⊗ A∗op ⊂ D(A)⊗ D(A) is given by theuniversal R-matrix
R =∑i
ai ⊗ a∗i ,
where ai is basis of A and a∗i the dual basis of A∗. In particular, Rsatisfies the quantum Yang-Baxter equation
R12R13R23 = R23R13R12 ∈ D(A)3.
Moreover, the commutation relation between A and A∗ is uniquelydetermined by this equation.
182
Quantum double, ctd.
It is natural to ask how to express the braided structure ofYD(A) in terms of D(A). This is addressed by the followingtheorem.
Theorem (Drinfeld)
The braiding of RepD(A) = YD(A) is given by the formulac = P ◦ R where R ∈ A⊗ A∗op ⊂ D(A)⊗ D(A) is given by theuniversal R-matrix
R =∑i
ai ⊗ a∗i ,
where ai is basis of A and a∗i the dual basis of A∗. In particular, Rsatisfies the quantum Yang-Baxter equation
R12R13R23 = R23R13R12 ∈ D(A)3.
Moreover, the commutation relation between A and A∗ is uniquelydetermined by this equation.
183
Quantum double, ctd.
It is natural to ask how to express the braided structure ofYD(A) in terms of D(A). This is addressed by the followingtheorem.
Theorem (Drinfeld)
The braiding of RepD(A) = YD(A) is given by the formulac = P ◦ R
where R ∈ A⊗ A∗op ⊂ D(A)⊗ D(A) is given by theuniversal R-matrix
R =∑i
ai ⊗ a∗i ,
where ai is basis of A and a∗i the dual basis of A∗. In particular, Rsatisfies the quantum Yang-Baxter equation
R12R13R23 = R23R13R12 ∈ D(A)3.
Moreover, the commutation relation between A and A∗ is uniquelydetermined by this equation.
184
Quantum double, ctd.
It is natural to ask how to express the braided structure ofYD(A) in terms of D(A). This is addressed by the followingtheorem.
Theorem (Drinfeld)
The braiding of RepD(A) = YD(A) is given by the formulac = P ◦ R where R ∈ A⊗ A∗op ⊂ D(A)⊗ D(A) is given by theuniversal R-matrix
R =∑i
ai ⊗ a∗i ,
where ai is basis of A and a∗i the dual basis of A∗. In particular, Rsatisfies the quantum Yang-Baxter equation
R12R13R23 = R23R13R12 ∈ D(A)3.
Moreover, the commutation relation between A and A∗ is uniquelydetermined by this equation.
185
Quantum double, ctd.
It is natural to ask how to express the braided structure ofYD(A) in terms of D(A). This is addressed by the followingtheorem.
Theorem (Drinfeld)
The braiding of RepD(A) = YD(A) is given by the formulac = P ◦ R where R ∈ A⊗ A∗op ⊂ D(A)⊗ D(A) is given by theuniversal R-matrix
R =∑i
ai ⊗ a∗i ,
where ai is basis of A and a∗i the dual basis of A∗.
In particular, Rsatisfies the quantum Yang-Baxter equation
R12R13R23 = R23R13R12 ∈ D(A)3.
Moreover, the commutation relation between A and A∗ is uniquelydetermined by this equation.
186
Quantum double, ctd.
It is natural to ask how to express the braided structure ofYD(A) in terms of D(A). This is addressed by the followingtheorem.
Theorem (Drinfeld)
The braiding of RepD(A) = YD(A) is given by the formulac = P ◦ R where R ∈ A⊗ A∗op ⊂ D(A)⊗ D(A) is given by theuniversal R-matrix
R =∑i
ai ⊗ a∗i ,
where ai is basis of A and a∗i the dual basis of A∗. In particular, Rsatisfies the quantum Yang-Baxter equation
R12R13R23 = R23R13R12 ∈ D(A)3.
Moreover, the commutation relation between A and A∗ is uniquelydetermined by this equation.
187
Quantum double, ctd.
It is natural to ask how to express the braided structure ofYD(A) in terms of D(A). This is addressed by the followingtheorem.
Theorem (Drinfeld)
The braiding of RepD(A) = YD(A) is given by the formulac = P ◦ R where R ∈ A⊗ A∗op ⊂ D(A)⊗ D(A) is given by theuniversal R-matrix
R =∑i
ai ⊗ a∗i ,
where ai is basis of A and a∗i the dual basis of A∗. In particular, Rsatisfies the quantum Yang-Baxter equation
R12R13R23 = R23R13R12 ∈ D(A)3.
Moreover, the commutation relation between A and A∗ is uniquelydetermined by this equation.
188
Examples
Example
Let A = CG where G is a finite group.
Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action). We haveR =
∑g∈G g ⊗ δg , where δg (h) = δgh, g , h ∈ G .
Example
Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1
q−q−1 and coproduct given
by the above formulas and ∆(f ) = f ⊗ 1 + K−1 ⊗ f .
189
Examples
Example
Let A = CG where G is a finite group. Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action).
We haveR =
∑g∈G g ⊗ δg , where δg (h) = δgh, g , h ∈ G .
Example
Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1
q−q−1 and coproduct given
by the above formulas and ∆(f ) = f ⊗ 1 + K−1 ⊗ f .
190
Examples
Example
Let A = CG where G is a finite group. Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action). We haveR =
∑g∈G g ⊗ δg ,
where δg (h) = δgh, g , h ∈ G .
Example
Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1
q−q−1 and coproduct given
by the above formulas and ∆(f ) = f ⊗ 1 + K−1 ⊗ f .
191
Examples
Example
Let A = CG where G is a finite group. Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action). We haveR =
∑g∈G g ⊗ δg , where δg (h) = δgh, g , h ∈ G .
Example
Let ` be an odd positive integer, q a primitive root of unity of order`.
Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1
q−q−1 and coproduct given
by the above formulas and ∆(f ) = f ⊗ 1 + K−1 ⊗ f .
192
Examples
Example
Let A = CG where G is a finite group. Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action). We haveR =
∑g∈G g ⊗ δg , where δg (h) = δgh, g , h ∈ G .
Example
Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1,
andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1
q−q−1 and coproduct given
by the above formulas and ∆(f ) = f ⊗ 1 + K−1 ⊗ f .
193
Examples
Example
Let A = CG where G is a finite group. Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action). We haveR =
∑g∈G g ⊗ δg , where δg (h) = δgh, g , h ∈ G .
Example
Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K .
ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1
q−q−1 and coproduct given
by the above formulas and ∆(f ) = f ⊗ 1 + K−1 ⊗ f .
194
Examples
Example
Let A = CG where G is a finite group. Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action). We haveR =
∑g∈G g ⊗ δg , where δg (h) = δgh, g , h ∈ G .
Example
Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where
D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1
q−q−1 and coproduct given
by the above formulas and ∆(f ) = f ⊗ 1 + K−1 ⊗ f .
195
Examples
Example
Let A = CG where G is a finite group. Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action). We haveR =
∑g∈G g ⊗ δg , where δg (h) = δgh, g , h ∈ G .
Example
Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1
q−q−1
and coproduct given
by the above formulas and ∆(f ) = f ⊗ 1 + K−1 ⊗ f .
196
Examples
Example
Let A = CG where G is a finite group. Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action). We haveR =
∑g∈G g ⊗ δg , where δg (h) = δgh, g , h ∈ G .
Example
Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1
q−q−1 and coproduct given
by the above formulas and ∆(f ) = f ⊗ 1 + K−1 ⊗ f .
197
Small quantum group
Note that D0 is nothing but the quotient of Uq(sl2) by therelations E ` = F ` = K ` − 1 = 0.
It is called the small quantumgroup and was introduced by G. Lusztig (for any simple Liealgebra), and is denoted by uq(sl2). Note that by virtue of theconstruction D0 has a universal R-matrix, which can be computedto be
R = qh⊗h2
`−1∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k ,
i.e., it is just the truncation of the formula for Uq(sl2) for generalq. In fact, the punchline is that Uq(sl2) for general q can also beconstructed by an infinite dimensional version of the quantumdouble construction, which naturally produces both thecommutation relation between e and f and the R-matrix!
198
Small quantum group
Note that D0 is nothing but the quotient of Uq(sl2) by therelations E ` = F ` = K ` − 1 = 0. It is called the small quantumgroup and was introduced by G. Lusztig (for any simple Liealgebra), and is denoted by uq(sl2).
Note that by virtue of theconstruction D0 has a universal R-matrix, which can be computedto be
R = qh⊗h2
`−1∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k ,
i.e., it is just the truncation of the formula for Uq(sl2) for generalq. In fact, the punchline is that Uq(sl2) for general q can also beconstructed by an infinite dimensional version of the quantumdouble construction, which naturally produces both thecommutation relation between e and f and the R-matrix!
199
Small quantum group
Note that D0 is nothing but the quotient of Uq(sl2) by therelations E ` = F ` = K ` − 1 = 0. It is called the small quantumgroup and was introduced by G. Lusztig (for any simple Liealgebra), and is denoted by uq(sl2). Note that by virtue of theconstruction D0 has a universal R-matrix, which can be computedto be
R = qh⊗h2
`−1∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k ,
i.e., it is just the truncation of the formula for Uq(sl2) for generalq. In fact, the punchline is that Uq(sl2) for general q can also beconstructed by an infinite dimensional version of the quantumdouble construction, which naturally produces both thecommutation relation between e and f and the R-matrix!
200
Small quantum group
Note that D0 is nothing but the quotient of Uq(sl2) by therelations E ` = F ` = K ` − 1 = 0. It is called the small quantumgroup and was introduced by G. Lusztig (for any simple Liealgebra), and is denoted by uq(sl2). Note that by virtue of theconstruction D0 has a universal R-matrix, which can be computedto be
R = qh⊗h2
`−1∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k ,
i.e., it is just the truncation of the formula for Uq(sl2) for generalq. In fact, the punchline is that Uq(sl2) for general q can also beconstructed by an infinite dimensional version of the quantumdouble construction, which naturally produces both thecommutation relation between e and f and the R-matrix!
201
Small quantum group
Note that D0 is nothing but the quotient of Uq(sl2) by therelations E ` = F ` = K ` − 1 = 0. It is called the small quantumgroup and was introduced by G. Lusztig (for any simple Liealgebra), and is denoted by uq(sl2). Note that by virtue of theconstruction D0 has a universal R-matrix, which can be computedto be
R = qh⊗h2
`−1∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k ,
i.e., it is just the truncation of the formula for Uq(sl2) for generalq.
In fact, the punchline is that Uq(sl2) for general q can also beconstructed by an infinite dimensional version of the quantumdouble construction, which naturally produces both thecommutation relation between e and f and the R-matrix!
202
Small quantum group
Note that D0 is nothing but the quotient of Uq(sl2) by therelations E ` = F ` = K ` − 1 = 0. It is called the small quantumgroup and was introduced by G. Lusztig (for any simple Liealgebra), and is denoted by uq(sl2). Note that by virtue of theconstruction D0 has a universal R-matrix, which can be computedto be
R = qh⊗h2
`−1∑k=0
qk(k−1)
2(q − q−1)k
[k]q!ek ⊗ f k ,
i.e., it is just the truncation of the formula for Uq(sl2) for generalq. In fact, the punchline is that Uq(sl2) for general q can also beconstructed by an infinite dimensional version of the quantumdouble construction, which naturally produces both thecommutation relation between e and f and the R-matrix!
203
Higher rank quantum groups
Quantum groups attached to any simple Lie algebra can beconstructed similarly.
Given a Cartan matrix (aij) withsymmetrizing numbers di ∈ Z+ (i.e., diaij is symmetric), we startwith the Hopf algebra Uq(b+) (quantum Borel) generated byei ,K
±1i with relations
Kiej = qdiaij ejKi , [Ki ,Kj ] = 0,
with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei . Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi with commutation
relations [ei , fj ] = δijKi−K−1
i
qdi−q−di.
204
Higher rank quantum groups
Quantum groups attached to any simple Lie algebra can beconstructed similarly. Given a Cartan matrix (aij) withsymmetrizing numbers di ∈ Z+ (i.e., diaij is symmetric), we startwith the Hopf algebra Uq(b+) (quantum Borel) generated byei ,K
±1i with relations
Kiej = qdiaij ejKi , [Ki ,Kj ] = 0,
with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei . Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi with commutation
relations [ei , fj ] = δijKi−K−1
i
qdi−q−di.
205
Higher rank quantum groups
Quantum groups attached to any simple Lie algebra can beconstructed similarly. Given a Cartan matrix (aij) withsymmetrizing numbers di ∈ Z+ (i.e., diaij is symmetric), we startwith the Hopf algebra Uq(b+) (quantum Borel) generated byei ,K
±1i with relations
Kiej = qdiaij ejKi , [Ki ,Kj ] = 0,
with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei . Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi with commutation
relations [ei , fj ] = δijKi−K−1
i
qdi−q−di.
206
Higher rank quantum groups
Quantum groups attached to any simple Lie algebra can beconstructed similarly. Given a Cartan matrix (aij) withsymmetrizing numbers di ∈ Z+ (i.e., diaij is symmetric), we startwith the Hopf algebra Uq(b+) (quantum Borel) generated byei ,K
±1i with relations
Kiej = qdiaij ejKi , [Ki ,Kj ] = 0,
with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei
and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei . Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi with commutation
relations [ei , fj ] = δijKi−K−1
i
qdi−q−di.
207
Higher rank quantum groups
Quantum groups attached to any simple Lie algebra can beconstructed similarly. Given a Cartan matrix (aij) withsymmetrizing numbers di ∈ Z+ (i.e., diaij is symmetric), we startwith the Hopf algebra Uq(b+) (quantum Borel) generated byei ,K
±1i with relations
Kiej = qdiaij ejKi , [Ki ,Kj ] = 0,
with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei .
Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi with commutation
relations [ei , fj ] = δijKi−K−1
i
qdi−q−di.
208
Higher rank quantum groups
Quantum groups attached to any simple Lie algebra can beconstructed similarly. Given a Cartan matrix (aij) withsymmetrizing numbers di ∈ Z+ (i.e., diaij is symmetric), we startwith the Hopf algebra Uq(b+) (quantum Borel) generated byei ,K
±1i with relations
Kiej = qdiaij ejKi , [Ki ,Kj ] = 0,
with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei . Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.
This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi with commutation
relations [ei , fj ] = δijKi−K−1
i
qdi−q−di.
209
Higher rank quantum groups
Quantum groups attached to any simple Lie algebra can beconstructed similarly. Given a Cartan matrix (aij) withsymmetrizing numbers di ∈ Z+ (i.e., diaij is symmetric), we startwith the Hopf algebra Uq(b+) (quantum Borel) generated byei ,K
±1i with relations
Kiej = qdiaij ejKi , [Ki ,Kj ] = 0,
with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei . Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi
with commutation
relations [ei , fj ] = δijKi−K−1
i
qdi−q−di.
210
Higher rank quantum groups
Quantum groups attached to any simple Lie algebra can beconstructed similarly. Given a Cartan matrix (aij) withsymmetrizing numbers di ∈ Z+ (i.e., diaij is symmetric), we startwith the Hopf algebra Uq(b+) (quantum Borel) generated byei ,K
±1i with relations
Kiej = qdiaij ejKi , [Ki ,Kj ] = 0,
with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei . Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi with commutation
relations [ei , fj ] = δijKi−K−1
i
qdi−q−di.
211
Jones polynomial
These commutation relations, as well as the R-matrix (which isnow much more complicated) are produced automatically by thedouble construction. In fact, this works more generally, for anysymmetrizable Kac-Moody algebra.
Example
In conclusion let us point out a connection to the Jonespolynomial.
It is well known that any knot K can be obtained byclosing up a braid b. On the other hand, as we have learned, bacts as an operator on the space V⊗n, where V = V1 is the2-dimensional representation of the quantum sl2.
Theorem
The trace of b · K in V⊗n (called the quantum trace of b) is theJones polynomial of K (up to normalization).
For Vi for i > 1 one gets the colored Jones polynomial, and forother Lie algebras – more complex invariants of knots called theReshetikhin-Turaev invariants.
212
Jones polynomial
These commutation relations, as well as the R-matrix (which isnow much more complicated) are produced automatically by thedouble construction. In fact, this works more generally, for anysymmetrizable Kac-Moody algebra.
Example
In conclusion let us point out a connection to the Jonespolynomial. It is well known that any knot K can be obtained byclosing up a braid b.
On the other hand, as we have learned, bacts as an operator on the space V⊗n, where V = V1 is the2-dimensional representation of the quantum sl2.
Theorem
The trace of b · K in V⊗n (called the quantum trace of b) is theJones polynomial of K (up to normalization).
For Vi for i > 1 one gets the colored Jones polynomial, and forother Lie algebras – more complex invariants of knots called theReshetikhin-Turaev invariants.
213
Jones polynomial
These commutation relations, as well as the R-matrix (which isnow much more complicated) are produced automatically by thedouble construction. In fact, this works more generally, for anysymmetrizable Kac-Moody algebra.
Example
In conclusion let us point out a connection to the Jonespolynomial. It is well known that any knot K can be obtained byclosing up a braid b. On the other hand, as we have learned, bacts as an operator on the space V⊗n, where V = V1 is the2-dimensional representation of the quantum sl2.
Theorem
The trace of b · K in V⊗n (called the quantum trace of b) is theJones polynomial of K (up to normalization).
For Vi for i > 1 one gets the colored Jones polynomial, and forother Lie algebras – more complex invariants of knots called theReshetikhin-Turaev invariants.
214
Jones polynomial
These commutation relations, as well as the R-matrix (which isnow much more complicated) are produced automatically by thedouble construction. In fact, this works more generally, for anysymmetrizable Kac-Moody algebra.
Example
In conclusion let us point out a connection to the Jonespolynomial. It is well known that any knot K can be obtained byclosing up a braid b. On the other hand, as we have learned, bacts as an operator on the space V⊗n, where V = V1 is the2-dimensional representation of the quantum sl2.
Theorem
The trace of b · K in V⊗n (called the quantum trace of b) is theJones polynomial of K (up to normalization).
For Vi for i > 1 one gets the colored Jones polynomial, and forother Lie algebras – more complex invariants of knots called theReshetikhin-Turaev invariants.
215
Jones polynomial
These commutation relations, as well as the R-matrix (which isnow much more complicated) are produced automatically by thedouble construction. In fact, this works more generally, for anysymmetrizable Kac-Moody algebra.
Example
In conclusion let us point out a connection to the Jonespolynomial. It is well known that any knot K can be obtained byclosing up a braid b. On the other hand, as we have learned, bacts as an operator on the space V⊗n, where V = V1 is the2-dimensional representation of the quantum sl2.
Theorem
The trace of b · K in V⊗n (called the quantum trace of b) is theJones polynomial of K (up to normalization).
For Vi for i > 1 one gets the colored Jones polynomial, and forother Lie algebras – more complex invariants of knots called theReshetikhin-Turaev invariants. 216
Thank you!
217