Introduction to Hopf Algebras

Rob Ray

February 26, 2004

The last twenty years has seen a number of advances in

the area of Hopf algebras. Among these are the intro-

duction of quantum groups, and the unification of group

actions, actions of Lie Algebras, and graded algebras .

As an introduction to Hopf algebras, this talk will cover

basic definitions and some examples. We first briefly dis-

cuss the idea of a tensor product of vector spaces and

then move on to algebras, coalgebras, Hopf algebras and

finally quantum groups. In order to cover the amount of

material, many of the details will be omitted and there

will be no proofs, i.e., in order to present a view of the

forest, there will be no pictures of pine needles. How-

ever, those interested can find most of the material in

Montgomery[Mon91] and Kassel [Kas95]. If time per-

mits, some quantum groups will also be discussed.

1

Tensor Product

↓Algebras and Coalgebras

↓Bialgebras and Hopf Algebras

↓Quantum Groups

↓Examples

2

Tensor Products

Let V and W be any two vector spaces over a field F.

V ⊗ W can be thought of as the space of objects that

look like

v1 ⊗ w1 + v2 ⊗ w2 + · · · + vk ⊗ wk

where

vi ∈ V, wi ∈ W

and with the following bilinear relations

a1(v1 ⊗ w) + a2(v2 ⊗ w) = ((a1v1 + a2v2)⊗ w) (1)

and

a1(v ⊗ w1) + a2(v ⊗ w2) = (v ⊗ (a1w1 + a2w2)) (2)

where ai ∈ F, v, vi ∈ V and w, wi ∈ W .

3

Remark 0.1 If BV is a basis for V and BW is a basis

for W , then

BV⊗W = {e⊗ f |e ∈ BV and f ∈ BW}

is a basis for V ⊗W

4

Example 0.1 Consider the polynomials of one vari-

able over CC[x]

These polynomials are a vector space with basis

B = {1, x, x2, x3, x4, . . .}

So examples of objects in C[x]⊗ C[x] are

4x2 ⊗ 3x

x⊗ 3

(x2 − 4)⊗ (x3 − 2x2 + x)

x2 ⊗ x2 + x2 ⊗ 4x3 + (x− 2x2)⊗ x3

Note: We may use the bilinear relations 1 and 2 to do

the following:

x2 ⊗ x2 + x2 ⊗ 4x3 + (x− 2x2)⊗ x3)

= x2 ⊗ x2 + 4x2 ⊗ x3

+(x− 2x2)⊗ x3

= x2 ⊗ x2 + (4x2 + x− 2x2)⊗ x3

= x2 ⊗ x2 + (2x2 + x)⊗ x3

5

Algebras and Coalgebras

Definition 0.1 An algebra over a field k is a vector

space, A, together with two linear maps, a multiplica-

tion µ : A⊗ A → A, and a unit map η : k → A such

that the following diagrams commute:

A⊗ A A-µ

A⊗ A⊗ A A⊗ A-µ⊗ id

?

id⊗ µ

?

µ

and

A

@@

@@

@@

@R

k ⊗ A A⊗ A-η ⊗ id

A⊗ k�id⊗ η

?

µ

��

��

��

�

where the lower left and right maps are simply scalar

multiplication.

6

Example 0.2 Considering the example of C[x]⊗C[x],

we define µ as the standard polynomial multiplication.

E.g.

µ(x⊗ 3) = 3x

µ(4x2 ⊗ 3x) = 12x3

µ((x2 − 4)⊗ (x3 − 2x2 + x)

)= x5 − 2x4 − 3x3 + 8x2 − 4x

µ(x2 ⊗ x2 + x2 ⊗ 4x3 + (x− 2x2)⊗ x3

)= x4 + 4x5 + x4 − 2x5

= 2x4 + 2x5

7

Example 0.3 (A(X)) Now consider the polynomial

algebra

A(X) = C[x1,1, x1,2, x2,1, x2,2] (3)

As a vector space, it’s basis is

{xi1,1x

j1,2x

k2,1x

l2,2 : i, j, k, l ≥ 0 ∈ Z}

and examples of elements of A(X)

x1,2x2,2 − 3x1,1x2,2 + x2,2

x1,1x2,2 − x1,2x2,1

If we think of

X =

[x1,1 x1,2

x2,1 x2,2

]then we can think of the polynomials of A(X) as func-

tions from Mat(2, C) to C. They are in fact often

called the regular functions of Mat(2, C). Let f =

x1,1x2,2 − x1,2x2,1 then

f

([1 1

0 1

])= 1 · 1− 1 · 0 = 1

8

Definition 0.2 If A and B are algebras with respec-

tive multiplications µA and µB then

A linear map g : A → B is an algebra mor-

phism if g ◦ µA = µB ◦ (g ⊗ g) i.e. the following

diagram commutes.

B ⊗B B-µB

A⊗ A A-µA

?

g ⊗ g?

g

I.e. if a, b ∈ A then

g(ab) = g(a)g(b)

9

Definition 0.3 A coalgebra is a vector space C to-

gether with two linear maps, comultiplication ∆ : C →C ⊗ C and counit ε : C → k, such that the following

two diagrams commute.

C ⊗ C C ⊗ C ⊗ C-

id⊗∆

C C ⊗ C-∆

?

∆?

∆⊗ id

and

C ⊗ C

ε⊗ id

@@

@@

@@

@I

k ⊗ C C� 1⊗C ⊗ k-⊗1

?

∆ id⊗ ε

��

��

��

��

10

Example 0.4 (A(X)) We see that if we define the

comultiplication and counit maps on A(X) in the fol-

lowing manner, then A(X) is a coalgebra.

∆(xi,j) = xi,1 ⊗ x1,j + xi,2 ⊗ x2,j (4)

ε(xi,j) = δi,j =

{1, i = j

0, i 6= j

}(5)

We extend the action of ∆ to the rest of A(X) by

defining it to be an algebra morphism. I.e.

∆(xi,jxk,l) = ∆(xi,j)∆(xk,l)

E.g.

∆(x1,1) = x1,1 ⊗ x1,1 + x1,2 ⊗ x2,1

ε(x1,1) = 1

ε(x1,2) = 0

11

Example 0.5 (Coassociativity of ∆ on A(X))

(id⊗∆) ◦∆(x1,2) = (id⊗∆)(x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2)

= (x1,1 ⊗ (x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2)

+x1,2 ⊗ (x2,1 ⊗ x1,2 + x2,2 ⊗ x2,2))

= x1,1 ⊗ x1,1 ⊗ x1,2 + x1,1 ⊗ x1,2 ⊗ x2,2

+x1,2 ⊗ x2,1 ⊗ x1,2 + x1,2 ⊗ x2,2 ⊗ x2,2

and

(∆⊗ id) ◦∆(x1,2) = (∆⊗ id)(x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2)

= (x1,1 ⊗ x1,1 + x1,2 ⊗ x2,1)⊗ x1,2

+(x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2)⊗ x2,2

= x1,1 ⊗ x1,1 ⊗ x1,2 + x1,2 ⊗ x2,1 ⊗ x1,2

x1,1 ⊗ x1,2 ⊗ x2,2 + x1,2 ⊗ x2,2 ⊗ x2,2

Example 0.6 (Counit , ε, of A(X))

x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2

ε⊗ id

@@

@@

@@

@I

1⊗ x1,2 x1,2�1⊗ x1,2 ⊗ 1-⊗1

?

∆ id⊗ ε

��

��

��

��

12

Definition 0.4 Let C be any coalgebra, and let c ∈C. c is group-like if ∆(c) = c⊗ c and ε(c) = 1. The

set of group-like elements in C is denoted by G(C).

Example 0.7 In A(X) there is a special element

det = x1,1x2,2 − x1,2x2,1

It can be shown that det is group-like.

∆(det) = det⊗ det

13

∆(det) = ∆(x1,1x2,2 − x1,2x2,1)

= ∆(x1,1)∆(x2,2)−∆(x1,2)∆(x2,1)

= (x1,1 ⊗ x1,1 + x1,2 ⊗ x2,1)(x2,1 ⊗ x1,2 + x2,2 ⊗ x2,2)

−(x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2)(x2,1 ⊗ x1,1 + x2,2 ⊗ x2,1)

= (x1,1 ⊗ x1,1)(x2,1 ⊗ x1,2) + (x1,1 ⊗ x1,1)(x2,2 ⊗ x2,2)

+(x1,2 ⊗ x2,1)(x2,1 ⊗ x1,2) + (x1,2 ⊗ x2,1)(x2,2 ⊗ x2,2)

−(x1,1 ⊗ x1,2)(x2,1 ⊗ x1,1)− (x1,1 ⊗ x1,2)(x2,2 ⊗ x2,1)

−(x1,2 ⊗ x2,2)(x2,1 ⊗ x1,1)− (x1,2 ⊗ x2,2)(x2,2 ⊗ x2,1)

= (x1,1x2,1 ⊗ x1,1x1,2) + (x1,1x2,2 ⊗ x1,1x2,2)

+(x1,2x2,1 ⊗ x2,1x1,2) + (x1,2x2,2 ⊗ x2,1x2,2)

−(x1,1x2,1 ⊗ x1,2x1,1)− (x1,1x2,2 ⊗ x1,2x2,1)

−(x1,2x2,1 ⊗ x2,2x1,1)− (x1,2x2,2 ⊗ x2,2x2,1)

= (x1,1x2,2 ⊗ x1,1x2,2)− (x1,1x2,2 ⊗ x1,2x2,1)

+(x1,2x2,1 ⊗ x2,1x1,2)− (x1,2x2,1 ⊗ x2,2x1,1)

= x1,1x2,2 ⊗ (x1,1x2,2 − x1,2x2,1)

+x1,2x2,1 ⊗ (x2,1x1,2 − x2,2x1,1)

= x1,1x2,2 ⊗ (x1,1x2,2 − x1,2x2,1)

−x1,2x2,1 ⊗ (x2,2x1,1 − x2,1x1,2)

= (x1,1x2,2 − x1,2x2,1)⊗ (x1,1x2,2 − x1,2x2,1)

= det⊗ det

14

Bialgebras and Hopf Algebras

Definition 0.5 Given a space B, B is a bialgebra

if (B, ∆, ε) is a coalgebra, (B, µ, η) is an algebra and

either of the following equivalent conditions is true:

1. ∆ and ε are algebra morphisms

2. µ and η are coalgebra morphisms

This bialgebra structure is often denoted by (B, µ, η, ∆, ε).

Remark 0.2 A(X) is a bialgebra.

15

Definition 0.6 (Convolution) Given an algebra (A, µ, η),

a coalgebra (C, ∆, ε) and two linear maps f, g : C →A then the convolution of f and g is the linear map

f ? g : C → A defined by

(f ? g)(c) = µ ◦ (f ⊗ g) ◦∆(c), c ∈ C

Definition 0.7 (Antipode and Hopf Algebra) Let

(H, µ, η, ∆, ε) be a bialgebra. An endomorphism S of

H is called an antipode for the bialgebra H if

idH ? S = S ? idH = η ◦ ε

A Hopf algebra is a bialgebra with an antipode.

16

Example 0.8 (A(G)) Unfortunately, A(X) has no an-

tipode, so we construct a new space.

A(G) = C[x1,1, x1,2, x2,1, x2,2, det−1

](6)

The comultiplication and counit are defined as with

A(X)

∆(xi,j) =

2∑k=1

xi,k ⊗ xk,j (7)

ε(xi,j) = δi,j (8)

Then

S(x1,1) = x2,2det−1

S(x1,2) = −x1,2det−1

S(x2,1) = −x2,1det−1

S(x2,2) = x1,1det−1

defines the antipode for A(G) and makes A(G) a Hopf

algebra.

17

Quantum Groups

Example 0.9 Let q ∈ C such that it is not a root of

unity, then define

Aq(X) = C[x1,1, x1,2, x2,1, x2,2] (9)

with added relations:

x1,1x1,2 = qx1,2x1,1

x2,1x2,2 = qx2,2x2,1

x1,1x2,1 = qx2,1x1,1

x1,2x2,2 = qx2,2x1,2

x1,2x2,1 = x2,1x1,2

x1,1x2,2 = x2,2x1,1 +

(q − 1

q

)x1,2x2,1

Now if we define ∆ and ε as we did for A(X), Aq(X)

becomes a bialgebra.

18

Definition 0.8 We define the quantum determinant

by

detq = x1,1x2,2 − qx1,2x2,1 ∈ Aq(X) (10)

19

Example 0.10 (Aq(Gl(n)), a quantum Hopf algebra)

Aq(Gl(n)) = C[x1,1, x1,2, x2,1, x2,2, detq

−1]

(11)

The comultiplication and counit are defined as with

A(X) Then

S(x1,1) = det−1q x2,2

S(x1,2) = −qdet−1q x1,2

S(x2,1) = −1

qdet−1

q x2,1

S(x2,2) = det−1q x1,1

defines an antipode.

20

More Examples

Example 0.11 (Divided Powers) If we let C = C[t]

be the polynomials of one variable over C, then we can

define a comultiplication and counit by

∆(tn) =∑

p+q=n

tp ⊗ tq

ε(tn) = δn0

We can demonstrate the coassociativity (for n = 2)

with the following calculations:

(id⊗∆) ◦∆(t2) = (id⊗∆)(t2 ⊗ 1 + t⊗ t + 1⊗ t2)

= t2 ⊗ 1⊗ 1 + t⊗ 1⊗ t + t⊗ t⊗ 1

+1⊗ t2 ⊗ 1 + 1⊗ t⊗ t + 1⊗ 1⊗ t2

= (∆⊗ id)(t2 ⊗ 1 + t⊗ t + 1⊗ t2)

= (∆⊗ id) ◦∆(t2)

To show that ε(tn) = δn0 defines a counit map we

check that (id⊗ ε) ◦∆(tn) = tn ⊗ 1 and that (ε⊗ id) ◦∆(tn) = 1⊗ tn

(id⊗ ε) ◦∆(tn) = (id⊗ ε)

( ∑p+q=n

tp ⊗ tq

)= tn ⊗ 1

Similarly, (ε⊗ id) ◦∆(tn) = 1⊗ tn.

21

Example 0.12 If we let G be a group then B = CG,

the associated group algebra, becomes a bialgebra with

the following defined maps

∆(g) = g ⊗ g, ∀g ∈ G

ε(g) = 1, ∀g ∈ G

and a Hopf algeba with the antipode defined by

S(g) = g−1

Remark 0.3 The set of group-like elements of this

Hopf algebra is the original group G.

22

Example 0.13 (U(sl(2))) Consider the universal en-

veloping algebra of sl(2), U(sl(2)). One can think of

U(sl(2)) as the polynomial algebra of three generators

e, f , and h, with the added relations

[e, f ] = h, [h, e] = 2x, [h, f ] = −2y (12)

Also, note that the set {eif jhk : i, j, k ≥ 0 ∈ Z+} is a

basis of U(sl(2)) as a result of the Poincare-Birkhoff-

Witt theorem [Kas95]. If we define the comultipli-

cation and counit maps on U(sl(2)) in the following

manner, then it has a bialgebra structure.

∆(x) = x⊗ 1 + 1⊗ x, ε(x) = 0, ∀x ∈ sl(2) (13)

S(x) = −x (14)

Infact, the enveloping algebra of any Lie algebra is a Hopf

algebra with the above definitions (13 and 14).

23

Example 0.14 (Quantum Plane) Choose q to be

an invertible element in C and define

Cq [x, y] = C {x, y|xy = qyx} (15)

With comultiplication and counits defined as

∆(x) = x⊗ x, ∆(y) = y ⊗ 1 + 1⊗ y (16)

and

ε(x) = 1, ε(y) = 0 (17)

Cq [x, y] is a bialgebra.

24

Example 0.15 A four dimensional noncommutative,

noncocomutative Hopf algebra [Mon91].

H4 ={1, g, x, gx|g2 = 1, x2 = 0, xg = −gx

}(18)

with structure maps

∆g = g ⊗ g

∆x = 1⊗ x + x⊗ 1

ε(g) = 1

ε(x) = 0

S(g) = g = g−1

S(x) = −gx

25

References

[Kas95] Christian Kassel. Quantum Groups. Springer-

Verlag New York, Inc., 1995.

[Mon91] Susan Montgomery. Hopf algebras and their ac-

tions on rings. In CBMS Regional Conference

Series in Mathematics, volume 82. Conference

Board of the Mathematical Sciences, American

Mathematical Society, 1991.

26