Structure of classical (finite and affine) W-algebras

Daniele Valeri

Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste

Workshop on Geometric and Analytic Aspects of Integrable andnearly-Integrable Hamiltonian Systems, Universita di Milano Bicocca,

June 18-20, 2014

1 Overview on W-algebras

2 Structure of classical W-algebras

Basic physical theories

Classical Hamiltonian Mechanics: M a manifold (phase space) with a Poissonalgebra structure {· , ·} on C∞(M) (observables). An Hamiltonian equation is

du

dt= {h, u} , h ∈ C∞(M) is the Hamiltonian function .

Quantum Mechanics: V vector space (phase space) with some operators(they form an associative algebra) acting on it (observables). The Schroedingerequation is

dψ

dt= H(ψ) , H is the Hamiltonian operator .

Going from a finite number to an infinite number of degrees of freedomClassical Mechanics Classical Field Theory Poisson Vertex AlgebraQuantum Mechanics Quantum Field Theory Vertex Algebra

Basic physical theories

Classical Hamiltonian Mechanics: M a manifold (phase space) with a Poissonalgebra structure {· , ·} on C∞(M) (observables). An Hamiltonian equation is

du

dt= {h, u} , h ∈ C∞(M) is the Hamiltonian function .

Quantum Mechanics: V vector space (phase space) with some operators(they form an associative algebra) acting on it (observables). The Schroedingerequation is

dψ

dt= H(ψ) , H is the Hamiltonian operator .

Going from a finite number to an infinite number of degrees of freedomClassical Mechanics Classical Field Theory Poisson Vertex AlgebraQuantum Mechanics Quantum Field Theory Vertex Algebra

Basic physical theories

Classical Hamiltonian Mechanics: M a manifold (phase space) with a Poissonalgebra structure {· , ·} on C∞(M) (observables). An Hamiltonian equation is

du

dt= {h, u} , h ∈ C∞(M) is the Hamiltonian function .

Quantum Mechanics: V vector space (phase space) with some operators(they form an associative algebra) acting on it (observables). The Schroedingerequation is

dψ

dt= H(ψ) , H is the Hamiltonian operator .

Going from a finite number to an infinite number of degrees of freedomClassical Mechanics Classical Field Theory Poisson Vertex AlgebraQuantum Mechanics Quantum Field Theory Vertex Algebra

Basic physical theories

Classical Hamiltonian Mechanics: M a manifold (phase space) with a Poissonalgebra structure {· , ·} on C∞(M) (observables). An Hamiltonian equation is

du

dt= {h, u} , h ∈ C∞(M) is the Hamiltonian function .

Quantum Mechanics: V vector space (phase space) with some operators(they form an associative algebra) acting on it (observables). The Schroedingerequation is

dψ

dt= H(ψ) , H is the Hamiltonian operator .

Going from a finite number to an infinite number of degrees of freedomClassical Mechanics Classical Field Theory Poisson Vertex AlgebraQuantum Mechanics Quantum Field Theory Vertex Algebra

A nice picture

The basic physical theories...

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A nice picture

The basic physical theories...

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finite vs affine

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finite vs affine

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CM

quantization

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...and the corresponding algebraic structures

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A simple example

g simple finite dimensional Lie algebra, κ nondegenerate symmetric invariantbilinear form

V(g)

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S(g)

affiniz.

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U(g) is the universal enveloping algebra of g: if g =⊕`

i=1 Cui , then

U(g) = spanC{uk11 . . . uk`

` | k1, . . . , k` ∈ Z+} .

The associative product is given by the juxtaposition of monomials.

A simple example

g simple finite dimensional Lie algebra, κ nondegenerate symmetric invariantbilinear form

V(g)

Zhu

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V (g)cl.limitoo

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S(g)

affiniz.

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U(g) is the universal enveloping algebra of g: if g =⊕`

i=1 Cui , then

U(g) = spanC{uk11 . . . uk`

` | k1, . . . , k` ∈ Z+} .

The associative product is given by the juxtaposition of monomials.

A simple example

g simple finite dimensional Lie algebra, κ nondegenerate symmetric invariantbilinear form

V(g)

Zhu

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quantization

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V (g)cl.limitoo

Zhu

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S(g)

affiniz.

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quantization

77U(g)

cl.limitoo

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U(g) is the universal enveloping algebra of g: if g =⊕`

i=1 Cui , then

U(g) = spanC{uk11 . . . uk`

` | k1, . . . , k` ∈ Z+} .

The associative product is given by the juxtaposition of monomials.

A simple example

g simple finite dimensional Lie algebra, κ nondegenerate symmetric invariantbilinear form

V(g)

Zhu

��

quantization

''

V (g)cl.limitoo

Zhu

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S(g)

affiniz.

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quantization

77U(g)

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S(g) is the algebra of polynomials in the elements of g,

S(g) = C[u1, . . . , u`] ,

endowed with the Kirillov-Kostant Poisson bracket: for a, b ∈ g it is

{a, b} = [a, b] ,

and it is extended to S(g) using the Leibniz rule.

A simple example

g simple finite dimensional Lie algebra, κ nondegenerate symmetric invariantbilinear form

V(g)

Zhu

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quantization

''

V (g)cl.limitoo

Zhu

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S(g)

affiniz.

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quantization

77U(g)

cl.limitoo

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S(g) is the algebra of polynomials in the elements of g,

S(g) = C[u1, . . . , u`] ,

endowed with the Kirillov-Kostant Poisson bracket: for a, b ∈ g it is

{a, b} = [a, b] ,

and it is extended to S(g) using the Leibniz rule.

A simple example

g simple finite dimensional Lie algebra, κ nondegenerate symmetric invariantbilinear form

V(g)

Zhu

��

quantization

''

V (g)cl.limitoo

Zhu

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S(g)

affiniz.

CC

quantization

77U(g)

cl.limitoo

affiniz.

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V (g) is the affine vertex algebra of g:

:(

I apologize but I do not know its definition!

A simple example

g simple finite dimensional Lie algebra, κ nondegenerate symmetric invariantbilinear form

V(g)

Zhu

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quantization

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V (g)cl.limitoo

Zhu

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S(g)

affiniz.

CC

quantization

77U(g)

cl.limitoo

affiniz.

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V (g) is the affine vertex algebra of g:

:(

I apologize but I do not know its definition!

A simple example

g simple finite dimensional Lie algebra, κ nondegenerate symmetric invariantbilinear form

V(g)

Zhu

��

quantization

''

V (g)cl.limitoo

Zhu

��

S(g)

affiniz.

CC

quantization

77U(g)

cl.limitoo

affiniz.

[[

V (g) is the affine vertex algebra of g:

:(

I apologize but I do not know its definition!

A simple example

g simple finite dimensional Lie algebra, κ nondegenerate symmetric invariantbilinear form

V(g)

Zhu

��

quantization

''

V (g)cl.limitoo

Zhu

��

S(g)

affiniz.

CC

quantization

77U(g)

cl.limitoo

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V(g) is the affine Poisson vertex algebra of g: it is the algebra of differentialpolynomials

V(g) = C[u(n)i | i = 1, . . . , `, n ∈ Z+] ,

with the following λ-bracket: for a, b ∈ g it is

{aλb} = [a, b] + κ(a | b)λ ,

and extended to V(g) using sesquilinearity and the Leibniz rule.

A simple example

g simple finite dimensional Lie algebra, κ nondegenerate symmetric invariantbilinear form

V(g)

Zhu

��

quantization

''

V (g)cl.limitoo

Zhu

��

S(g)

affiniz.

CC

quantization

77U(g)

cl.limitoo

affiniz.

[[

V(g) is the affine Poisson vertex algebra of g: it is the algebra of differentialpolynomials

V(g) = C[u(n)i | i = 1, . . . , `, n ∈ Z+] ,

with the following λ-bracket: for a, b ∈ g it is

{aλb} = [a, b] + κ(a | b)λ ,

and extended to V(g) using sesquilinearity and the Leibniz rule.

A simple example

g simple finite dimensional Lie algebra, κ nondegenerate symmetric invariantbilinear form

V(g)

Zhu

��

quantization

''

V (g)cl.limitoo

Zhu

��

S(g)

affiniz.

CC

quantization

77U(g)

cl.limitoo

affiniz.

[[

V(g) is the affine Poisson vertex algebra of g: it is the algebra of differentialpolynomials

V(g) = C[u(n)i | i = 1, . . . , `, n ∈ Z+] ,

with the following λ-bracket: for a, b ∈ g it is

{aλb} = [a, b] + κ(a | b)λ ,

and extended to V(g) using sesquilinearity and the Leibniz rule.

Poisson vertex algebras

Definition

A Poisson vertex algebra (PVA) is a differential algebra V (a commutativeassociative algebra with unity and a fixed derivation ∂) endowed with a C-linearmap {·λ·} : V ⊗ V −→ V[λ], called λ-bracket, such that, for a, b, c ∈ V, thefollowing properties hold:

sesquilinearity: {∂aλb} = −λ{aλb} and {aλ∂b} = (λ+ ∂){aλb};Leibniz rule: {aλbc} = {aλb}c + {aλc}b.

skewsymmetry and Jacobi identity.

For V(g) = C[u(n)i | i = 1, . . . , `, n ∈ Z+] the derivation ∂ is defined on

generators by (i = 1, . . . , `, n ∈ Z+):

∂u(n)i = u

(n+1)i .

An example of λ-bracket computation:

{u(2)1 λu

(1)2 u3} = {u(2)

1 λu(1)2 }u3 + {u(2)

1 λu3}u(1)2

= u3λ2(λ+ ∂){u1λu2}+ λ2{u1λu3}u(1)

2 .

Poisson vertex algebras

Definition

A Poisson vertex algebra (PVA) is a differential algebra V (a commutativeassociative algebra with unity and a fixed derivation ∂) endowed with a C-linearmap {·λ·} : V ⊗ V −→ V[λ], called λ-bracket, such that, for a, b, c ∈ V, thefollowing properties hold:

sesquilinearity: {∂aλb} = −λ{aλb} and {aλ∂b} = (λ+ ∂){aλb};Leibniz rule: {aλbc} = {aλb}c + {aλc}b.

skewsymmetry and Jacobi identity.

For V(g) = C[u(n)i | i = 1, . . . , `, n ∈ Z+] the derivation ∂ is defined on

generators by (i = 1, . . . , `, n ∈ Z+):

∂u(n)i = u

(n+1)i .

An example of λ-bracket computation:

{u(2)1 λu

(1)2 u3} = {u(2)

1 λu(1)2 }u3 + {u(2)

1 λu3}u(1)2

= u3λ2(λ+ ∂){u1λu2}+ λ2{u1λu3}u(1)

2 .

Poisson vertex algebras

Definition

A Poisson vertex algebra (PVA) is a differential algebra V (a commutativeassociative algebra with unity and a fixed derivation ∂) endowed with a C-linearmap {·λ·} : V ⊗ V −→ V[λ], called λ-bracket, such that, for a, b, c ∈ V, thefollowing properties hold:

sesquilinearity: {∂aλb} = −λ{aλb} and {aλ∂b} = (λ+ ∂){aλb};Leibniz rule: {aλbc} = {aλb}c + {aλc}b.

skewsymmetry and Jacobi identity.

For V(g) = C[u(n)i | i = 1, . . . , `, n ∈ Z+] the derivation ∂ is defined on

generators by (i = 1, . . . , `, n ∈ Z+):

∂u(n)i = u

(n+1)i .

An example of λ-bracket computation:

{u(2)1 λu

(1)2 u3} = {u(2)

1 λu(1)2 }u3 + {u(2)

1 λu3}u(1)2

= u3λ2(λ+ ∂){u1λu2}+ λ2{u1λu3}u(1)

2 .

Poisson vertex algebras

Definition

A Poisson vertex algebra (PVA) is a differential algebra V (a commutativeassociative algebra with unity and a fixed derivation ∂) endowed with a C-linearmap {·λ·} : V ⊗ V −→ V[λ], called λ-bracket, such that, for a, b, c ∈ V, thefollowing properties hold:

sesquilinearity: {∂aλb} = −λ{aλb} and {aλ∂b} = (λ+ ∂){aλb};Leibniz rule: {aλbc} = {aλb}c + {aλc}b.

skewsymmetry and Jacobi identity.

For V(g) = C[u(n)i | i = 1, . . . , `, n ∈ Z+] the derivation ∂ is defined on

generators by (i = 1, . . . , `, n ∈ Z+):

∂u(n)i = u

(n+1)i .

An example of λ-bracket computation:

{u(2)1 λu

(1)2 u3} = {u(2)

1 λu(1)2 }u3 + {u(2)

1 λu3}u(1)2

= u3λ2(λ+ ∂){u1λu2}+ λ2{u1λu3}u(1)

2 .

Poisson vertex algebras

Definition

A Poisson vertex algebra (PVA) is a differential algebra V (a commutativeassociative algebra with unity and a fixed derivation ∂) endowed with a C-linearmap {·λ·} : V ⊗ V −→ V[λ], called λ-bracket, such that, for a, b, c ∈ V, thefollowing properties hold:

sesquilinearity: {∂aλb} = −λ{aλb} and {aλ∂b} = (λ+ ∂){aλb};Leibniz rule: {aλbc} = {aλb}c + {aλc}b.

skewsymmetry and Jacobi identity.

For V(g) = C[u(n)i | i = 1, . . . , `, n ∈ Z+] the derivation ∂ is defined on

generators by (i = 1, . . . , `, n ∈ Z+):

∂u(n)i = u

(n+1)i .

An example of λ-bracket computation:

{u(2)1 λu

(1)2 u3} = {u(2)

1 λu(1)2 }u3 + {u(2)

1 λu3}u(1)2

= u3λ2(λ+ ∂){u1λu2}+ λ2{u1λu3}u(1)

2 .

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

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quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

W-algebras were introduced separately and played important roles in differentareas of mathematics. Only later it became fully clear the relations betweenthem.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

W-algebras were introduced separately and played important roles in differentareas of mathematics. Only later it became fully clear the relations betweenthem.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

W-algebras were introduced separately and played important roles in differentareas of mathematics. Only later it became fully clear the relations betweenthem.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Finite W-algebras Wfin(g, f ): first constructed for a principal nilpotent elementby Kostant (1978) and Lynch (1979):

Wfin(g, fpr ) ' Z(U(g)) .

The general definition is given by Premet (2002): connection to representationtheory of simple finite dimensional Lie algebras, and to the theory of primitiveideals.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Finite W-algebras Wfin(g, f ): first constructed for a principal nilpotent elementby Kostant (1978) and Lynch (1979):

Wfin(g, fpr ) ' Z(U(g)) .

The general definition is given by Premet (2002): connection to representationtheory of simple finite dimensional Lie algebras, and to the theory of primitiveideals.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Finite W-algebras Wfin(g, f ): first constructed for a principal nilpotent elementby Kostant (1978) and Lynch (1979):

Wfin(g, fpr ) ' Z(U(g)) .

The general definition is given by Premet (2002): connection to representationtheory of simple finite dimensional Lie algebras, and to the theory of primitiveideals.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Classical finite W-algebras Wcl,fin(g, f ): introduced by Slodowy (1980) as aPoisson algebra of functions on the Slodowy slice with applications to thetheory of singularities of coadjoint orbits in mind.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Classical finite W-algebras Wcl,fin(g, f ): introduced by Slodowy (1980) as aPoisson algebra of functions on the Slodowy slice with applications to thetheory of singularities of coadjoint orbits in mind.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Classical finite W-algebras Wcl,fin(g, f ): introduced by Slodowy (1980) as aPoisson algebra of functions on the Slodowy slice with applications to thetheory of singularities of coadjoint orbits in mind.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Affine W-algebras Waff (g, f ): The first example is the ZamolodchikovW3-algebra (1985) (=Waff (sl3, fpr )). It is a “non-linear” infinite dimensionalLie algebra, extending the Virasoro algebra.Later, Feigin-Frenkel (1990) and Kac-Roan-Wakimoto (2003) provided ageneral construction via a quantization of the Drinfeld-Sokolov Hamiltonianreduction. Application to representation theory of superconformal algebras.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Affine W-algebras Waff (g, f ): The first example is the ZamolodchikovW3-algebra (1985) (=Waff (sl3, fpr )). It is a “non-linear” infinite dimensionalLie algebra, extending the Virasoro algebra.Later, Feigin-Frenkel (1990) and Kac-Roan-Wakimoto (2003) provided ageneral construction via a quantization of the Drinfeld-Sokolov Hamiltonianreduction. Application to representation theory of superconformal algebras.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Affine W-algebras Waff (g, f ): The first example is the ZamolodchikovW3-algebra (1985) (=Waff (sl3, fpr )). It is a “non-linear” infinite dimensionalLie algebra, extending the Virasoro algebra.Later, Feigin-Frenkel (1990) and Kac-Roan-Wakimoto (2003) provided ageneral construction via a quantization of the Drinfeld-Sokolov Hamiltonianreduction. Application to representation theory of superconformal algebras.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Classical affine W-algebras Wcl,affz (g, f ), z ∈ C: first introduced for principal

nilpotent f by Drinfeld-Sokolov (1985) as Poisson algebras of functions oversome infinite dimensional manifolds to study KdV-type integrable equations(Drinfeld-Sokolov hierarchies). In the 90’s were introduced generalized DShierarchies by deGroot, Delduc, Feher, Miramontes...Recently, with De Sole and Kac (2013), we generalized the construction toarbitrary nilpotent elements and formalized the approach to generalized DShierarchies using the theory of PVA.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Classical affine W-algebras Wcl,affz (g, f ), z ∈ C: first introduced for principal

nilpotent f by Drinfeld-Sokolov (1985) as Poisson algebras of functions oversome infinite dimensional manifolds to study KdV-type integrable equations(Drinfeld-Sokolov hierarchies). In the 90’s were introduced generalized DShierarchies by deGroot, Delduc, Feher, Miramontes...Recently, with De Sole and Kac (2013), we generalized the construction toarbitrary nilpotent elements and formalized the approach to generalized DShierarchies using the theory of PVA.

A more sophisticated example: W-algebras

g simple Lie algebra, f ∈ g nilpotent element

Wcl,affz (g, f )

Zhu

��

quantization**

Waff (g, f )cl.limitoo

Zhu

��

Wcl,fin(g, f )

affiniz.

CC

quantization

55Wfin(g, f )

cl.limitoo

affiniz.

[[

Classical affine W-algebras Wcl,affz (g, f ), z ∈ C: first introduced for principal

nilpotent f by Drinfeld-Sokolov (1985) as Poisson algebras of functions oversome infinite dimensional manifolds to study KdV-type integrable equations(Drinfeld-Sokolov hierarchies). In the 90’s were introduced generalized DShierarchies by deGroot, Delduc, Feher, Miramontes...Recently, with De Sole and Kac (2013), we generalized the construction toarbitrary nilpotent elements and formalized the approach to generalized DShierarchies using the theory of PVA.

The links among the four appearances of W-algebras are recent

Gan and Ginzburg (2002) constructed finite W-algebras as quantization ofclassical finite W-algebras:

Wfin(g, f )cl.limit // Wcl,fin(g, f ) .

The analogous result in the affine case

Waff (g, f )cl.limit // Wcl,aff (g, f )

has been proved by Suh (2013).De Sole-Kac (2006) and Arakawa (2007) proved that the (H-twisted) Zhualgebra of the affine W-algebra Waff (g , f ) is isomorphic to the finiteW-algebra Wfin(g , f ).

Waff (g, f )Zhu // Wfin(g, f ) .

Hence, their categories of irreducible representations are equivalent.It remains to understand the map

Wcl,affz (g, f )

Zhu // Wcl,fin(g, f ) .

1 Overview on W-algebras

2 Structure of classical W-algebras

Setup and notation

g simple finite dimensional Lie algebra, κ non degenerate symmetricbilinear form on it.

f ∈ g nilpotent element, there exists an sl2-triple {f , h = 2x , e} ⊂ g (ByJacobson-Morozov Theorem). We have

g =d⊕

k=−dk∈ 1

2Z

gk , gk = {a ∈ g | [x , a] = ka} .

g = gf ⊕ [e, g]. Denote ] : g→ gf , the projection map with kernel [e, g].

Fix a basis {qj}j∈Jf of gf . The dual basis w.r.t. κ, {qj}j∈Jf , is a basis ofge . We assume [x , qj ] = δ(j)qj (basis consisting of ad x-eigenvector).

Denote J = {(j , n) ∈ J f × Z+ | n = 0, . . . , 2δ(j)}. Then

{qjn = (ad f )nqj}(j,n)∈J and {qn

j = cj,n(ad e)nqj}(j,n)∈J

are dual (w.r.t. κ) basis of g. Note that J =∐

Jk , where {qjn}(j,n)∈J−k

is abasis of gk (hence, {qn

j }(j,n)∈Jkis the dual basis, w.r.t. κ, of g−k ).

Setup and notation

g simple finite dimensional Lie algebra, κ non degenerate symmetricbilinear form on it.

f ∈ g nilpotent element, there exists an sl2-triple {f , h = 2x , e} ⊂ g (ByJacobson-Morozov Theorem). We have

g =d⊕

k=−dk∈ 1

2Z

gk , gk = {a ∈ g | [x , a] = ka} .

g = gf ⊕ [e, g]. Denote ] : g→ gf , the projection map with kernel [e, g].

Fix a basis {qj}j∈Jf of gf . The dual basis w.r.t. κ, {qj}j∈Jf , is a basis ofge . We assume [x , qj ] = δ(j)qj (basis consisting of ad x-eigenvector).

Denote J = {(j , n) ∈ J f × Z+ | n = 0, . . . , 2δ(j)}. Then

{qjn = (ad f )nqj}(j,n)∈J and {qn

j = cj,n(ad e)nqj}(j,n)∈J

are dual (w.r.t. κ) basis of g. Note that J =∐

Jk , where {qjn}(j,n)∈J−k

is abasis of gk (hence, {qn

j }(j,n)∈Jkis the dual basis, w.r.t. κ, of g−k ).

Setup and notation

g simple finite dimensional Lie algebra, κ non degenerate symmetricbilinear form on it.

f ∈ g nilpotent element, there exists an sl2-triple {f , h = 2x , e} ⊂ g (ByJacobson-Morozov Theorem). We have

g =d⊕

k=−dk∈ 1

2Z

gk , gk = {a ∈ g | [x , a] = ka} .

g = gf ⊕ [e, g]. Denote ] : g→ gf , the projection map with kernel [e, g].

Fix a basis {qj}j∈Jf of gf . The dual basis w.r.t. κ, {qj}j∈Jf , is a basis ofge . We assume [x , qj ] = δ(j)qj (basis consisting of ad x-eigenvector).

Denote J = {(j , n) ∈ J f × Z+ | n = 0, . . . , 2δ(j)}. Then

{qjn = (ad f )nqj}(j,n)∈J and {qn

j = cj,n(ad e)nqj}(j,n)∈J

are dual (w.r.t. κ) basis of g. Note that J =∐

Jk , where {qjn}(j,n)∈J−k

is abasis of gk (hence, {qn

j }(j,n)∈Jkis the dual basis, w.r.t. κ, of g−k ).

Setup and notation

g simple finite dimensional Lie algebra, κ non degenerate symmetricbilinear form on it.

f ∈ g nilpotent element, there exists an sl2-triple {f , h = 2x , e} ⊂ g (ByJacobson-Morozov Theorem). We have

g =d⊕

k=−dk∈ 1

2Z

gk , gk = {a ∈ g | [x , a] = ka} .

g = gf ⊕ [e, g]. Denote ] : g→ gf , the projection map with kernel [e, g].

Fix a basis {qj}j∈Jf of gf . The dual basis w.r.t. κ, {qj}j∈Jf , is a basis ofge . We assume [x , qj ] = δ(j)qj (basis consisting of ad x-eigenvector).

Denote J = {(j , n) ∈ J f × Z+ | n = 0, . . . , 2δ(j)}. Then

{qjn = (ad f )nqj}(j,n)∈J and {qn

j = cj,n(ad e)nqj}(j,n)∈J

are dual (w.r.t. κ) basis of g. Note that J =∐

Jk , where {qjn}(j,n)∈J−k

is abasis of gk (hence, {qn

j }(j,n)∈Jkis the dual basis, w.r.t. κ, of g−k ).

Setup and notation

g simple finite dimensional Lie algebra, κ non degenerate symmetricbilinear form on it.

f ∈ g nilpotent element, there exists an sl2-triple {f , h = 2x , e} ⊂ g (ByJacobson-Morozov Theorem). We have

g =d⊕

k=−dk∈ 1

2Z

gk , gk = {a ∈ g | [x , a] = ka} .

g = gf ⊕ [e, g]. Denote ] : g→ gf , the projection map with kernel [e, g].

Fix a basis {qj}j∈Jf of gf . The dual basis w.r.t. κ, {qj}j∈Jf , is a basis ofge . We assume [x , qj ] = δ(j)qj (basis consisting of ad x-eigenvector).

Denote J = {(j , n) ∈ J f × Z+ | n = 0, . . . , 2δ(j)}. Then

{qjn = (ad f )nqj}(j,n)∈J and {qn

j = cj,n(ad e)nqj}(j,n)∈J

are dual (w.r.t. κ) basis of g. Note that J =∐

Jk , where {qjn}(j,n)∈J−k

is abasis of gk (hence, {qn

j }(j,n)∈Jkis the dual basis, w.r.t. κ, of g−k ).

Setup and notation

g simple finite dimensional Lie algebra, κ non degenerate symmetricbilinear form on it.

f ∈ g nilpotent element, there exists an sl2-triple {f , h = 2x , e} ⊂ g (ByJacobson-Morozov Theorem). We have

g =d⊕

k=−dk∈ 1

2Z

gk , gk = {a ∈ g | [x , a] = ka} .

g = gf ⊕ [e, g]. Denote ] : g→ gf , the projection map with kernel [e, g].

Fix a basis {qj}j∈Jf of gf . The dual basis w.r.t. κ, {qj}j∈Jf , is a basis ofge . We assume [x , qj ] = δ(j)qj (basis consisting of ad x-eigenvector).

Denote J = {(j , n) ∈ J f × Z+ | n = 0, . . . , 2δ(j)}. Then

{qjn = (ad f )nqj}(j,n)∈J and {qn

j = cj,n(ad e)nqj}(j,n)∈J

are dual (w.r.t. κ) basis of g. Note that J =∐

Jk , where {qjn}(j,n)∈J−k

is abasis of gk (hence, {qn

j }(j,n)∈Jkis the dual basis, w.r.t. κ, of g−k ).

Structure of classical finite W-algebras

Theorem (De Sole-Kac-V,2014)

(a) As a vector space, the classical finite W-algebra is isomorphic to thealgebra of polynomials in the elements of gf :

Wcl,fin(g, f ) ' S(gf ) = C[qj | j ∈ J f ] .

(b) The Poisson structure on Wcl,fin(g, f ) is given on generators by (a, b ∈ gf ):

{a, b} = [a, b]+∞∑

t=1

∑j1,...,jt∈Jf

2δ(j1)−1∑n1=0

· · ·2δ(jt )−1∑

nt =0

[a, qj1n1

]][qn1+1j1

, qj2n2

]] . . . [qnt +1jt

, b]] .

Note: when f is principal nilpotent, we have {a, b} = 0 for every a, b ∈ gf .

Structure of classical finite W-algebras

Theorem (De Sole-Kac-V,2014)

(a) As a vector space, the classical finite W-algebra is isomorphic to thealgebra of polynomials in the elements of gf :

Wcl,fin(g, f ) ' S(gf ) = C[qj | j ∈ J f ] .

(b) The Poisson structure on Wcl,fin(g, f ) is given on generators by (a, b ∈ gf ):

{a, b} = [a, b]+∞∑

t=1

∑j1,...,jt∈Jf

2δ(j1)−1∑n1=0

· · ·2δ(jt )−1∑

nt =0

[a, qj1n1

]][qn1+1j1

, qj2n2

]] . . . [qnt +1jt

, b]] .

Note: when f is principal nilpotent, we have {a, b} = 0 for every a, b ∈ gf .

Structure of classical finite W-algebras

Theorem (De Sole-Kac-V,2014)

(a) As a vector space, the classical finite W-algebra is isomorphic to thealgebra of polynomials in the elements of gf :

Wcl,fin(g, f ) ' S(gf ) = C[qj | j ∈ J f ] .

(b) The Poisson structure on Wcl,fin(g, f ) is given on generators by (a, b ∈ gf ):

{a, b} = [a, b]+∞∑

t=1

∑j1,...,jt∈Jf

2δ(j1)−1∑n1=0

· · ·2δ(jt )−1∑

nt =0

[a, qj1n1

]][qn1+1j1

, qj2n2

]] . . . [qnt +1jt

, b]] .

Note: when f is principal nilpotent, we have {a, b} = 0 for every a, b ∈ gf .

Structure of classical affine W-algebras

Theorem (De Sole-Kac-V,2014)

(a) As a vector space, the classical affine W-algebra is isomorphic to thealgebra of differential polynomials in the elements of gf :

Wcl,affz (g, f ) ' S(C[∂]gf ) = C[q

(n)j | j ∈ J f , n ∈ Z+] .

(b) The Poisson vertex algebra structure on Wcl,affz (g, f ) is given on generators

by (a, b ∈ gf ):

{aλb}z = [a, b] + κ(a|b)λ+ zκ(s|[a, b])

−∞∑

t=1

∑−h+1≤kt≺···≺k1≤k

∑(~j,~n)∈J−~k

([b, qj1

n1]] − κ(b|qj1

n1)(λ+ ∂) + zκ(s|[b, qj1

n1]))

([qn1+1

j1, qj2

n2]] − κ(qn1+1

j1|qj2

n2)(λ+ ∂) + zκ(s|[qn1+1

j1, qj2

n2])). . .

. . .([q

nt−1+1

jt−1, qjt

nt]] − κ(q

nt−1+1

jt−1|qjt

nt)(λ+ ∂) + zκ(s|[qnt−1+1

jt−1, qjt

nt]))(

[qnt +1jt

, a]] − κ(qnt +1jt|a)λ+ zκ(s|[qnt +1

jt, a])

).

where s ∈ gd , and h ≺ k if and only if h ≤ k − 1.

Structure of classical affine W-algebras

Theorem (De Sole-Kac-V,2014)

(a) As a vector space, the classical affine W-algebra is isomorphic to thealgebra of differential polynomials in the elements of gf :

Wcl,affz (g, f ) ' S(C[∂]gf ) = C[q

(n)j | j ∈ J f , n ∈ Z+] .

(b) The Poisson vertex algebra structure on Wcl,affz (g, f ) is given on generators

by (a, b ∈ gf ):

{aλb}z = [a, b] + κ(a|b)λ+ zκ(s|[a, b])

−∞∑

t=1

∑−h+1≤kt≺···≺k1≤k

∑(~j,~n)∈J−~k

([b, qj1

n1]] − κ(b|qj1

n1)(λ+ ∂) + zκ(s|[b, qj1

n1]))

([qn1+1

j1, qj2

n2]] − κ(qn1+1

j1|qj2

n2)(λ+ ∂) + zκ(s|[qn1+1

j1, qj2

n2])). . .

. . .([q

nt−1+1

jt−1, qjt

nt]] − κ(q

nt−1+1

jt−1|qjt

nt)(λ+ ∂) + zκ(s|[qnt−1+1

jt−1, qjt

nt]))(

[qnt +1jt

, a]] − κ(qnt +1jt|a)λ+ zκ(s|[qnt +1

jt, a])

).

where s ∈ gd , and h ≺ k if and only if h ≤ k − 1.

Structure of classical affine W-algebras

Theorem (De Sole-Kac-V,2014)

(a) As a vector space, the classical affine W-algebra is isomorphic to thealgebra of differential polynomials in the elements of gf :

Wcl,affz (g, f ) ' S(C[∂]gf ) = C[q

(n)j | j ∈ J f , n ∈ Z+] .

(b) The Poisson vertex algebra structure on Wcl,affz (g, f ) is given on generators

by (a, b ∈ gf ):

{aλb}z = [a, b] + κ(a|b)λ+ zκ(s|[a, b])

−∞∑

t=1

∑−h+1≤kt≺···≺k1≤k

∑(~j,~n)∈J−~k

([b, qj1

n1]] − κ(b|qj1

n1)(λ+ ∂) + zκ(s|[b, qj1

n1]))

([qn1+1

j1, qj2

n2]] − κ(qn1+1

j1|qj2

n2)(λ+ ∂) + zκ(s|[qn1+1

j1, qj2

n2])). . .

. . .([q

nt−1+1

jt−1, qjt

nt]] − κ(q

nt−1+1

jt−1|qjt

nt)(λ+ ∂) + zκ(s|[qnt−1+1

jt−1, qjt

nt]))(

[qnt +1jt

, a]] − κ(qnt +1jt|a)λ+ zκ(s|[qnt +1

jt, a])

).

where s ∈ gd , and h ≺ k if and only if h ≤ k − 1.

Note: the RHS above is linear in z!

The links among the four appearances of W-algebras: final step

Corollary

For every z ∈ C, the (H-twisted) Zhu algebra of the Poisson vertex algebraWcl,aff

z (g, f ) is isomorphic to the Poisson algebra Wfin(g, f ):

Wcl,affz (g, f )

Zhu // Wcl,fin(g, f ) .

Generalized DS hierarchies

Using the Lenard-Magri scheme of integrability we have the following result:

Theorem (De Sole-Kac-V,2013)

If f + s ∈ g is semisimple, we can construct an integrable hierarchy ofbi-Hamiltonian equations associated to Wcl,aff

z (g, f ).

For f = fpr principal nilpotent, choose s = eθ highest root vector. Thenfpr + eθ is semisimple DS hierarchies.

Different choices of f generalized DS hierarchies.

In particular, for f minimal and short, with De Sole and Kac (2013), weclassified all the pairs (f , s) such that f + s is semisimple and computed thefirst non trivial equations of the generalized DS hierarchies.After performing a Dirac reduction procedure we provide a bi-Hamiltonianstructure (one of the two is non local!) for:f minimal unknown to me integrable hierarchies.f short integrable hierarchies associated to Jordan algebras, studied bySvinolupov (1991)

Generalized DS hierarchies

Using the Lenard-Magri scheme of integrability we have the following result:

Theorem (De Sole-Kac-V,2013)

If f + s ∈ g is semisimple, we can construct an integrable hierarchy ofbi-Hamiltonian equations associated to Wcl,aff

z (g, f ).

For f = fpr principal nilpotent, choose s = eθ highest root vector. Thenfpr + eθ is semisimple DS hierarchies.

Different choices of f generalized DS hierarchies.

In particular, for f minimal and short, with De Sole and Kac (2013), weclassified all the pairs (f , s) such that f + s is semisimple and computed thefirst non trivial equations of the generalized DS hierarchies.After performing a Dirac reduction procedure we provide a bi-Hamiltonianstructure (one of the two is non local!) for:f minimal unknown to me integrable hierarchies.f short integrable hierarchies associated to Jordan algebras, studied bySvinolupov (1991)

Generalized DS hierarchies

Using the Lenard-Magri scheme of integrability we have the following result:

Theorem (De Sole-Kac-V,2013)

If f + s ∈ g is semisimple, we can construct an integrable hierarchy ofbi-Hamiltonian equations associated to Wcl,aff

z (g, f ).

For f = fpr principal nilpotent, choose s = eθ highest root vector. Thenfpr + eθ is semisimple DS hierarchies.

Different choices of f generalized DS hierarchies.

In particular, for f minimal and short, with De Sole and Kac (2013), weclassified all the pairs (f , s) such that f + s is semisimple and computed thefirst non trivial equations of the generalized DS hierarchies.After performing a Dirac reduction procedure we provide a bi-Hamiltonianstructure (one of the two is non local!) for:f minimal unknown to me integrable hierarchies.f short integrable hierarchies associated to Jordan algebras, studied bySvinolupov (1991)

Generalized DS hierarchies

Using the Lenard-Magri scheme of integrability we have the following result:

Theorem (De Sole-Kac-V,2013)

If f + s ∈ g is semisimple, we can construct an integrable hierarchy ofbi-Hamiltonian equations associated to Wcl,aff

z (g, f ).

For f = fpr principal nilpotent, choose s = eθ highest root vector. Thenfpr + eθ is semisimple DS hierarchies.

Different choices of f generalized DS hierarchies.

In particular, for f minimal and short, with De Sole and Kac (2013), weclassified all the pairs (f , s) such that f + s is semisimple and computed thefirst non trivial equations of the generalized DS hierarchies.After performing a Dirac reduction procedure we provide a bi-Hamiltonianstructure (one of the two is non local!) for:f minimal unknown to me integrable hierarchies.f short integrable hierarchies associated to Jordan algebras, studied bySvinolupov (1991)

Generalized DS hierarchies

Using the Lenard-Magri scheme of integrability we have the following result:

Theorem (De Sole-Kac-V,2013)

If f + s ∈ g is semisimple, we can construct an integrable hierarchy ofbi-Hamiltonian equations associated to Wcl,aff

z (g, f ).

For f = fpr principal nilpotent, choose s = eθ highest root vector. Thenfpr + eθ is semisimple DS hierarchies.

Different choices of f generalized DS hierarchies.

In particular, for f minimal and short, with De Sole and Kac (2013), weclassified all the pairs (f , s) such that f + s is semisimple and computed thefirst non trivial equations of the generalized DS hierarchies.After performing a Dirac reduction procedure we provide a bi-Hamiltonianstructure (one of the two is non local!) for:f minimal unknown to me integrable hierarchies.f short integrable hierarchies associated to Jordan algebras, studied bySvinolupov (1991)

Generalized DS hierarchies

Using the Lenard-Magri scheme of integrability we have the following result:

Theorem (De Sole-Kac-V,2013)

If f + s ∈ g is semisimple, we can construct an integrable hierarchy ofbi-Hamiltonian equations associated to Wcl,aff

z (g, f ).

For f = fpr principal nilpotent, choose s = eθ highest root vector. Thenfpr + eθ is semisimple DS hierarchies.

Different choices of f generalized DS hierarchies.

In particular, for f minimal and short, with De Sole and Kac (2013), weclassified all the pairs (f , s) such that f + s is semisimple and computed thefirst non trivial equations of the generalized DS hierarchies.After performing a Dirac reduction procedure we provide a bi-Hamiltonianstructure (one of the two is non local!) for:f minimal unknown to me integrable hierarchies.f short integrable hierarchies associated to Jordan algebras, studied bySvinolupov (1991)

May a formula be important?

Classify all the pairs (f , s) such that f + s is semisimple and compute thecorresponding generalized DS hierarchies.

Try to apply directly the Lenard-Magri scheme of integrability and check ifit is possible to construct generalized DS hierarchies for every Wcl,aff (g, f ).

Understand the relations between generalized DS hierarchies andFrobenius manifolds (this has be done in the principal nilpotent case byDubrovin-Liu-Zhang, 2008), and with the Kac-Wakimoto hierarchies (thishas been done in the principal nilpotent case by Wu, 2012).

Find an explicit description of the quantum part of the diagram.

May a formula be important?

Classify all the pairs (f , s) such that f + s is semisimple and compute thecorresponding generalized DS hierarchies.

Try to apply directly the Lenard-Magri scheme of integrability and check ifit is possible to construct generalized DS hierarchies for every Wcl,aff (g, f ).

Understand the relations between generalized DS hierarchies andFrobenius manifolds (this has be done in the principal nilpotent case byDubrovin-Liu-Zhang, 2008), and with the Kac-Wakimoto hierarchies (thishas been done in the principal nilpotent case by Wu, 2012).

Find an explicit description of the quantum part of the diagram.

May a formula be important?

Classify all the pairs (f , s) such that f + s is semisimple and compute thecorresponding generalized DS hierarchies.

Try to apply directly the Lenard-Magri scheme of integrability and check ifit is possible to construct generalized DS hierarchies for every Wcl,aff (g, f ).

Understand the relations between generalized DS hierarchies andFrobenius manifolds (this has be done in the principal nilpotent case byDubrovin-Liu-Zhang, 2008), and with the Kac-Wakimoto hierarchies (thishas been done in the principal nilpotent case by Wu, 2012).

Find an explicit description of the quantum part of the diagram.

May a formula be important?

Classify all the pairs (f , s) such that f + s is semisimple and compute thecorresponding generalized DS hierarchies.

Try to apply directly the Lenard-Magri scheme of integrability and check ifit is possible to construct generalized DS hierarchies for every Wcl,aff (g, f ).

Understand the relations between generalized DS hierarchies andFrobenius manifolds (this has be done in the principal nilpotent case byDubrovin-Liu-Zhang, 2008), and with the Kac-Wakimoto hierarchies (thishas been done in the principal nilpotent case by Wu, 2012).

Find an explicit description of the quantum part of the diagram.

May a formula be important?

Classify all the pairs (f , s) such that f + s is semisimple and compute thecorresponding generalized DS hierarchies.

Try to apply directly the Lenard-Magri scheme of integrability and check ifit is possible to construct generalized DS hierarchies for every Wcl,aff (g, f ).

Understand the relations between generalized DS hierarchies andFrobenius manifolds (this has be done in the principal nilpotent case byDubrovin-Liu-Zhang, 2008), and with the Kac-Wakimoto hierarchies (thishas been done in the principal nilpotent case by Wu, 2012).

Find an explicit description of the quantum part of the diagram.

Thank you!