Casimir elements for classical Lie algebras. and affine Kac Moody algebras

Casimir elements for classical Lie algebras and affine Kac Moody algebras Alexander Molev University of Sydney

Plan of lectures

Plan of lectures Casimir elements for the classical Lie algebras from the Schur Weyl duality.

Plan of lectures Casimir elements for the classical Lie algebras from the Schur Weyl duality. Affine Kac Moody algebras: center at the critical level.

Plan of lectures Casimir elements for the classical Lie algebras from the Schur Weyl duality. Affine Kac Moody algebras: center at the critical level. Affine Harish-Chandra isomorphism and classical W-algebras.

Symmetric group S m Let S m denote the group of permutations of the set {1,..., m}.

Symmetric group S m Let S m denote the group of permutations of the set {1,..., m}. Let s ab denote the transposition (a, b) for a < b.

Symmetric group S m Let S m denote the group of permutations of the set {1,..., m}. Let s ab denote the transposition (a, b) for a < b. The symmetrizer is the element h (m) = 1 s C[S m ]. m! s S m

Symmetric group S m Let S m denote the group of permutations of the set {1,..., m}. Let s ab denote the transposition (a, b) for a < b. The symmetrizer is the element h (m) = 1 s C[S m ]. m! s S m The anti-symmetrizer is the element a (m) = 1 sgn s s C[S m ]. m! s S m

Fusion formulas

Fusion formulas Theorem [Jucys 1966]. The following factorization formulas hold: h (m) = 1 m! 1 a<b m ( 1 + s ) ab, b a

Fusion formulas Theorem [Jucys 1966]. The following factorization formulas hold: h (m) = 1 m! 1 a<b m ( 1 + s ) ab, b a a (m) = 1 m! 1 a<b m ( 1 s ) ab, b a

Fusion formulas Theorem [Jucys 1966]. The following factorization formulas hold: h (m) = 1 m! 1 a<b m ( 1 + s ) ab, b a a (m) = 1 m! 1 a<b m ( 1 s ) ab, b a where both products are taken in the lexicographical order on the set of pairs (a, b).

Action of S m in tensors

Action of S m in tensors The symmetric group S m acts in the tensor space by the rule C N... C N }{{} m s a b P a b, 1 a < b m,

Action of S m in tensors The symmetric group S m acts in the tensor space by the rule C N... C N }{{} m s a b P a b, 1 a < b m, where P a b is the permutation operator N P a b = 1 (a 1) e i j 1 (b a 1) e j i 1 (m b) i,j=1

Action of S m in tensors The symmetric group S m acts in the tensor space by the rule C N... C N }{{} m s a b P a b, 1 a < b m, where P a b is the permutation operator P a b = N 1 (a 1) e i j 1 (b a 1) e j i 1 (m b) i,j=1 and e i j End C N are the matrix units.

The symmetrizer and anti-symmetrizer act as the operators H (m) = 1 m! 1 a<b m ( 1 + P ) ab b a

The symmetrizer and anti-symmetrizer act as the operators and H (m) = 1 m! A (m) = 1 m! 1 a<b m 1 a<b m ( 1 + P ) ab b a ( 1 P ) ab b a

The symmetrizer and anti-symmetrizer act as the operators and H (m) = 1 m! A (m) = 1 m! 1 a<b m 1 a<b m ( 1 + P ) ab b a ( 1 P ) ab b a which we regard as elements of the algebra End (C N ) m = End } C N. {{.. End C N }. m

Lie algebra gln

Lie algebra gl N We let E ij be the standard basis elements of gl N.

Lie algebra gl N We let E ij be the standard basis elements of gl N. The universal enveloping algebra U(gl N ) is the associative algebra generated by the N 2 elements E ij subject to the defining relations E ij E kl E kl E ij = δ kj E il δ il E kl. We will combine the generators into the matrix E = [E ij ] which will also be regarded as the element N E = e ij E ij End C N U(gl N ). i,j=1

Consider the algebra } End C N. {{.. End C N } U(gl N ) m

Consider the algebra } End C N. {{.. End C N } U(gl N ) m and for a = 1,..., m introduce its elements by E a = N 1 (a 1) e ij 1 (m a) E ij. i,j=1

Consider the algebra } End C N. {{.. End C N } U(gl N ) m and for a = 1,..., m introduce its elements by E a = N 1 (a 1) e ij 1 (m a) E ij. i,j=1 Note the property P a b E a = E b P a b.

Consider the algebra } End C N. {{.. End C N } U(gl N ) m and for a = 1,..., m introduce its elements by E a = N 1 (a 1) e ij 1 (m a) E ij. i,j=1 Note the property P a b E a = E b P a b. Key Lemma. The defining relations of U(gl N ) are equivalent to the single relation E 1 E 2 E 2 E 1 = (E 1 E 2 )P 12.

The trace is the linear map End C N C defined by tr : e ij δ ij.

The trace is the linear map End C N C defined by tr : e ij δ ij. The partial trace tr a acts on the a-th copy of End C N in End } C N. {{.. End C N } U(gl N ). m

The trace is the linear map End C N C defined by tr : e ij δ ij. The partial trace tr a acts on the a-th copy of End C N in End } C N. {{.. End C N } U(gl N ). m Theorem. For any s C[S m ] and u 1,..., u m C the element tr 1,...,m S (u 1 + E 1 )... (u m + E m ) belongs to the center Z(gl N ) of U(gl N ).

Proof. Consider the tensor product End C N End (C N ) m U(gl N ) with the copies of the algebra End C N labelled by 0, 1,..., m.

Proof. Consider the tensor product End C N End (C N ) m U(gl N ) with the copies of the algebra End C N labelled by 0, 1,..., m. We will show that [ E0, tr 1,...,m S (u 1 + E 1 )... (u m + E m ) ] = 0.

Proof. Consider the tensor product End C N End (C N ) m U(gl N ) with the copies of the algebra End C N labelled by 0, 1,..., m. We will show that [ E0, tr 1,...,m S (u 1 + E 1 )... (u m + E m ) ] = 0. By the Key Lemma, [E 0, u a + E a ] = P 0a (u a + E a ) (u a + E a )P 0a, where we used the relations P a b E b = E a P a b.

Hence [ E0, S (u 1 + E 1 )... (u m + E m ) ] m = S P 0a (u 1 + E 1 )... (u m + E m ) a=1 m S (u 1 + E 1 )... (u m + E m ) P 0 a, because E 0 S = SE 0 and P 0 a commutes with E b for b a. a=1

Hence [ E0, S (u 1 + E 1 )... (u m + E m ) ] m = S P 0a (u 1 + E 1 )... (u m + E m ) a=1 m S (u 1 + E 1 )... (u m + E m ) P 0 a, because E 0 S = SE 0 and P 0 a commutes with E b for b a. a=1 The sum of the permutation operators P 0a commutes with S (the Schur Weyl duality). Applying the trace tr 1,...,m and using its cyclic property we get 0.

Example: Capelli determinant.

Example: Capelli determinant. Take m = N and introduce the Capelli determinant by C(u) = tr 1,...,N A (N) (u + E 1 )... (u + E N N + 1).

Example: Capelli determinant. Take m = N and introduce the Capelli determinant by C(u) = tr 1,...,N A (N) (u + E 1 )... (u + E N N + 1). Then C(u) coincides with the column-determinant u + E 11 E 12... E 1N E C(u) = cdet 21 u + E 22 1... E 2N.... E N1 E N2... u + E NN N + 1

Indeed, observe that by the Key Lemma ( 1 P ) ab (u + E a a + 1)(u + E b b + 1) b a ( = (u + E b b + 1)(u + E a a + 1) 1 P ) ab. b a

Indeed, observe that by the Key Lemma ( 1 P ) ab (u + E a a + 1)(u + E b b + 1) b a ( = (u + E b b + 1)(u + E a a + 1) 1 P ) ab. b a Hence, the fusion formula for A (N) gives A (N) (u+e 1 )... (u+e N N +1) = (u+e N N +1)... (u+e 1 ) A (N)

Example: Gelfand invariants.

Example: Gelfand invariants. Take s = (m m 1... 1) in the Theorem. Then S = P m 1 m... P 2 3 P 1 2.

Example: Gelfand invariants. Take s = (m m 1... 1) in the Theorem. Then S = P m 1 m... P 2 3 P 1 2. We get the Casimir elements (Gelfand invariants): tr 1,...,m S E 1... E m = tr E m.

Example: Gelfand invariants. Take s = (m m 1... 1) in the Theorem. Then S = P m 1 m... P 2 3 P 1 2. We get the Casimir elements (Gelfand invariants): tr 1,...,m S E 1... E m = tr E m. For instance, for m = 2 we get tr 1,2 P 12 E 1 E 2 = tr 1,2 E 2 P 12 E 2 = tr E 2 because tr 1 P 12 = 1.

The Newton identity Theorem [Perelomov Popov, 1966]. We have the identity ( 1) m tr E m C(u + 1) 1 + =. (u N + 1) m+1 C(u) m=0

The Newton identity Theorem [Perelomov Popov, 1966]. We have the identity ( 1) m tr E m C(u + 1) 1 + =. (u N + 1) m+1 C(u) Proof. Verify m=0 tr 1,...,N A (N) (u + E 1 )... (u + E N 1 N + 2)(u + E N + 1) = C(u + 1). Hence, C(u + 1) C(u) = N tr 1,...,N A (N) (u + E 1 )... (u + E N 1 N + 2) = N tr 1,...,N A (N) C(u) (u + E N N + 1) 1.

Harish-Chandra isomorphism

Harish-Chandra isomorphism Given an N-tuple of complex numbers λ = (λ 1,..., λ N ), the corresponding irreducible highest weight representation L(λ) of the Lie algebra gl N is generated by a nonzero vector ξ L(λ) (the highest vector) such that

Any element z Z(gl N ) acts in L(λ) by multiplying each vector by a scalar χ(z).

Any element z Z(gl N ) acts in L(λ) by multiplying each vector by a scalar χ(z). When regarded as a function of the highest weight, χ(z) is a symmetric polynomial in the variables l 1,..., l N, where l i = λ i i + 1. The mapping z χ(z) defines an algebra isomorphism χ : Z(gl N ) C[l 1,..., l N ] S N known as the Harish-Chandra isomorphism.

Consider the standard triangular decomposition gl N = n h n +.

Consider the standard triangular decomposition gl N = n h n +. Then χ can also be defined as the restriction to Z(gl N ) of the Harish-Chandra homomorphism U(gl N ) h U(h)

Consider the standard triangular decomposition gl N = n h n +. Then χ can also be defined as the restriction to Z(gl N ) of the Harish-Chandra homomorphism U(gl N ) h U(h) which is the projection of the h-centralizer U(gl N ) h with respect to the direct sum decomposition ) U(gl N ) h = U(h) (U(gl N ) h U(gl N )n +.

Example. Under the Harish-Chandra isomorphism we have χ : C(u) (u + l 1 )... (u + l N ), l i = E ii i + 1.

Example. Under the Harish-Chandra isomorphism we have χ : C(u) (u + l 1 )... (u + l N ), l i = E ii i + 1. This is immediate from the definition C(u) = σ S N sgn σ (u + E) σ(1)1... (u + E N + 1) σ(n)n. By the Newton formula, the Harish-Chandra images of the Gelfand invariants are found by 1 + m=0 ( 1) m χ(tr E m ) N (u N + 1) m+1 = i=1 u + l i + 1 u + l i.

Let g be a simple Lie algebra of rank n and g = n h n +.

Let g be a simple Lie algebra of rank n and g = n h n +. We have the direct sum decomposition ) U(g) h = U(h) (U(g) h U(g)n +

Let g be a simple Lie algebra of rank n and g = n h n +. We have the direct sum decomposition ) U(g) h = U(h) (U(g) h U(g)n + and the Harish-Chandra isomorphism χ : Z(g) U(h) W, with a shifted action of W. We have Z(g) = C[P 1,..., P n ], for certain algebraically independent invariants P 1,..., P n whose degrees d 1,..., d n are the exponents of g increased by 1.

Brauer algebra B m (ω)

Brauer algebra B m (ω) Multiplication of m-diagrams (m = 8): 1 2 3 4 5 6 7 8

Brauer algebra B m (ω) Multiplication of m-diagrams (m = 8): 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 = ω

The dimension of the Brauer algebra B m (ω) is (2m 1)!!.

The dimension of the Brauer algebra B m (ω) is (2m 1)!!. For 1 a < b m denote by s a b and g a b the diagrams 1 a b m and 1 a b m

The dimension of the Brauer algebra B m (ω) is (2m 1)!!. For 1 a < b m denote by s a b and g a b the diagrams 1 a b m and 1 a b m The symmetrizer in B m (ω) is the idempotent s (m) such that s a b s (m) = s (m) s a b = s (m) and g a b s (m) = s (m) g a b = 0.

Explicitly, s (m) = 1 m! m/2 r=0 ( 1) r ( ω/2 + m 2 r ) 1 d D (r) d, where D (r) B m (ω) denotes the set of diagrams which have exactly r horizontal edges in the top.

Explicitly, s (m) = 1 m! m/2 r=0 ( 1) r ( ω/2 + m 2 r ) 1 d D (r) d, where D (r) B m (ω) denotes the set of diagrams which have exactly r horizontal edges in the top. Also, and s (m) = 1 a<b m ( g ) ab 1 h (m), ω + a + b 3

Explicitly, s (m) = 1 m! m/2 r=0 ( 1) r ( ω/2 + m 2 r ) 1 d D (r) d, where D (r) B m (ω) denotes the set of diagrams which have exactly r horizontal edges in the top. Also, s (m) = 1 a<b m ( g ) ab 1 h (m), ω + a + b 3 and s (m) = 1 m! 1 a<b m ( 1 + s ab b a g ) ab, ω/2 + b a 1 where the products are in the lexicographic order.

Brauer Schur Weyl duality

Brauer Schur Weyl duality There are commuting actions of the classical groups in types B, C or D and the Brauer algebra with a specialized parameter ω on the tensor product space C } N. {{.. C N }. m

Brauer Schur Weyl duality There are commuting actions of the classical groups in types B, C or D and the Brauer algebra with a specialized parameter ω on the tensor product space The dual pairs are C } N. {{.. C N }. m ( Bm (N), O N )

Action in tensors

Action in tensors In the case g = o N set ω = N. The generators of B m (N) act in the tensor space by the rule C N... C N }{{} m s a b P a b, g a b Q a b, 1 a < b m,

Action in tensors In the case g = o N set ω = N. The generators of B m (N) act in the tensor space by the rule C N... C N }{{} m s a b P a b, g a b Q a b, 1 a < b m, where i = N i + 1 and N Q a b = 1 (a 1) e i j 1 (b a 1) e i j 1 (m b). i,j=1

In the case g = sp N with N = 2n set ω = N. The generators of B m ( N) act in the tensor space (C N ) m by s a b P a b, g a b Q a b, 1 a < b m,

In the case g = sp N with N = 2n set ω = N. The generators of B m ( N) act in the tensor space (C N ) m by s a b P a b, g a b Q a b, 1 a < b m, with ε i = ε n+i = 1 for i = 1,..., n and N Q a b = ε i ε j 1 (a 1) e i j 1 (b a 1) e i j 1 (m b). i,j=1 In both cases denote by S (m) the image of the symmetrizer s (m) under the action in tensors, S (m) End } C N. {{.. End C N }. m

Explicitly, in the orthogonal case S (m) = 1 m! 1 a<b m ( 1 + P ab b a Q ) ab, N/2 + b a 1

Explicitly, in the orthogonal case S (m) = 1 m! 1 a<b m ( 1 + P ab b a Q ) ab, N/2 + b a 1 and in the symplectic case S (m) = 1 m! 1 a<b m ( 1 P ab b a Q ) ab. n b + a + 1

Explicitly, in the orthogonal case S (m) = 1 m! 1 a<b m ( 1 + P ab b a Q ) ab, N/2 + b a 1 and in the symplectic case S (m) = 1 m! 1 a<b m ( 1 P ab b a Q ) ab. n b + a + 1 Remark. S (n+1) = 0 for g = sp 2n.

Explicitly, in the orthogonal case S (m) = 1 m! 1 a<b m ( 1 + P ab b a Q ) ab, N/2 + b a 1 and in the symplectic case S (m) = 1 m! 1 a<b m ( 1 P ab b a Q ) ab. n b + a + 1 Remark. S (n+1) = 0 for g = sp 2n. Consider γ m ( 2n) S (m), γ m (ω) = ω + m 2 N for g = o N ω + 2m 2, ω = 2n for g = sp 2n.

Lie algebras on and spn

Lie algebras o N and sp N Let g = o N, sp N with N = 2n or N = 2n + 1.

Lie algebras o N and sp N Let g = o N, sp N with N = 2n or N = 2n + 1. Set F i j = E i j E j i or F i j = E i j ε i ε j E j i, respectively.

Lie algebras o N and sp N Let g = o N, sp N with N = 2n or N = 2n + 1. Set F i j = E i j E j i or F i j = E i j ε i ε j E j i, respectively. Introduce the N N matrix F = [F ij ] N F = e ij F i j End C N U(g). i,j=1

Theorem. For any s B m (ω) with ω = ±N and u 1,..., u m C the element tr 1,...,m S (u 1 + F 1 )... (u m + F m ) belongs to the center Z(g) of U(g).

Theorem. For any s B m (ω) with ω = ±N and u 1,..., u m C the element tr 1,...,m S (u 1 + F 1 )... (u m + F m ) belongs to the center Z(g) of U(g). In particular, there are analogues of the Capelli determinant and Gelfand invariants.

Proof of the theorem relies on the matrix form of the defining relations for U(g):

Proof of the theorem relies on the matrix form of the defining relations for U(g): F 1 F 2 F 2 F 1 = (P 12 Q 12 ) F 2 F 2 (P 12 Q 12 )

Proof of the theorem relies on the matrix form of the defining relations for U(g): F 1 F 2 F 2 F 1 = (P 12 Q 12 ) F 2 F 2 (P 12 Q 12 ) where both sides are regarded as elements of the algebra End C N End C N U(g) and N N F 1 = e ij 1 F ij, F 2 = 1 e ij F ij. i,j=1 i,j=1

Theorem. For g = o N the image of the Casimir element γ 2k (N) tr S (2k) (F 1 k)... (F 2k + k 1) under the Harish-Chandra isomorphism is given by

Theorem. For g = o N the image of the Casimir element γ 2k (N) tr S (2k) (F 1 k)... (F 2k + k 1) under the Harish-Chandra isomorphism is given by ( l 2 j1 (j 1 1/2) 2)... ( l 2 j k (j k + k 3/2) 2), 1 j 1 j k n where l i = F ii + n i + 1/2 in the case N = 2n + 1; and

Theorem. For g N = sp 2n the image of the Casimir element γ 2k ( 2n) tr S (2k) (F 1 + k)... (F 2k k + 1) under the Harish-Chandra isomorphism is given by

Theorem. For g N = sp 2n the image of the Casimir element γ 2k ( 2n) tr S (2k) (F 1 + k)... (F 2k k + 1) under the Harish-Chandra isomorphism is given by ( 1) k 1 j 1 < <j k n ( l 2 j1 j 2 ) ( 1... l 2 jk (j k k + 1) 2), where l i = F ii + n i + 1 for i = 1,..., n.

More constructions of Casimir elements for the Lie algebras gl N, o N and sp 2n are known.

More constructions of Casimir elements for the Lie algebras gl N, o N and sp 2n are known. In particular, there is a linear basis of Z(gl N ) formed by the quantum immanants S λ with λ running over partitions with at most N parts (Okounkov Olshanski, 1996, 1998).

Affine Kac Moody algebras

Affine Kac Moody algebras Define an invariant bilinear form on a simple Lie algebra g, X, Y = 1 tr(ad X ad Y), 2h where h is the dual Coxeter number.

Affine Kac Moody algebras Define an invariant bilinear form on a simple Lie algebra g, X, Y = 1 tr(ad X ad Y), 2h where h is the dual Coxeter number. For the classical types, X, Y = const tr X Y, N for g = sl N, const = 1 h = N 2 for g = o N, const = 1 2 n + 1 for g = sp 2n, const = 1.

The affine Kac Moody algebra ĝ is the central extension ĝ = g[t, t 1 ] CK

The affine Kac Moody algebra ĝ is the central extension ĝ = g[t, t 1 ] CK with the commutation relations [ X[r], Y[s] ] = [X, Y][r + s] + r δr, s X, Y K, where X[r] = X t r for any X g and r Z.

The affine Kac Moody algebra ĝ is the central extension ĝ = g[t, t 1 ] CK with the commutation relations [ X[r], Y[s] ] = [X, Y][r + s] + r δr, s X, Y K, where X[r] = X t r for any X g and r Z. Problem: What are Casimir elements for ĝ?

The universal enveloping algebra at the critical level U h (ĝ) is the quotient of U(ĝ) by the ideal generated by K + h.

The universal enveloping algebra at the critical level U h (ĝ) is the quotient of U(ĝ) by the ideal generated by K + h. By [Kac 1974], the canonical quadratic Casimir element belongs to a completion Ũ h (ĝ) of U h (ĝ) with respect to the left ideals I m, m 0, generated by t m g[t].

Let Z(ĝ) be the center of the completed algebra Ũ h (ĝ).

Let Z(ĝ) be the center of the completed algebra Ũ h (ĝ). Known results: Algebraic structure of Z(ĝ).

Let Z(ĝ) be the center of the completed algebra Ũ h (ĝ). Known results: Algebraic structure of Z(ĝ). Explicit generators for classical types A, B, C, D.

Let Z(ĝ) be the center of the completed algebra Ũ h (ĝ). Known results: Algebraic structure of Z(ĝ). Explicit generators for classical types A, B, C, D. Questions: Extension to Lie superalgebras.

Let Z(ĝ) be the center of the completed algebra Ũ h (ĝ). Known results: Algebraic structure of Z(ĝ). Explicit generators for classical types A, B, C, D. Questions: Extension to Lie superalgebras. Extension to quantum affine algebras.

Example: g = gl N. Defining relations for U(ĝl N): E ij [r] E kl [s] E kl [s] E ij [r] ( = δ kj E i l [r + s] δ i l E kj [r + s] + rδ r, s δ kj δ i l δ ij δ ) kl K. N

Example: g = gl N. Defining relations for U(ĝl N): E ij [r] E kl [s] E kl [s] E ij [r] ( = δ kj E i l [r + s] δ i l E kj [r + s] + rδ r, s δ kj δ i l δ ij δ ) kl K. N The critical level is K = N.

Example: g = gl N. Defining relations for U(ĝl N): E ij [r] E kl [s] E kl [s] E ij [r] ( = δ kj E i l [r + s] δ i l E kj [r + s] + rδ r, s δ kj δ i l δ ij δ ) kl K. N The critical level is K = N. For all r Z the sums N E ii [r] i=1 are Casimir elements.

For r Z set C r = N ( i,j=1 s<0 E ij [s] E ji [r s] + s 0 ) E ji [r s] E ij [s].

For r Z set C r = N ( i,j=1 s<0 E ij [s] E ji [r s] + s 0 ) E ji [r s] E ij [s]. All C r are Casimir elements at the critical level, they belong to the completed universal enveloping algebra Ũ N (ĝl N).

Introduce the (formal) Laurent series E ij (z) = r Z E ij [r] z r 1

Introduce the (formal) Laurent series E ij (z) = r Z E ij [r] z r 1 and use the notation E ij (z) + = r<0 E ij [r] z r 1, E ij (z) = r 0 E ij [r] z r 1.

Introduce the (formal) Laurent series E ij (z) = r Z E ij [r] z r 1 and use the notation E ij (z) + = r<0 E ij [r] z r 1, E ij (z) = r 0 E ij [r] z r 1. Given two Laurent series a(z) and b(z), their normally ordered product is defined by : a(z)b(z) : = a(z) + b(z) + b(z) a(z).

Note C r z r 2 = r Z N i,j=1 (E ij (z) + E ji (z) + E ji (z)e ij (z) ).

Note C r z r 2 = r Z N i,j=1 (E ij (z) + E ji (z) + E ji (z)e ij (z) ). Hence, all coefficients of the series tr : E(z) 2 : = N i,j=1 : E ij (z)e ji (z) : are Casimir elements.

Similarly, all coefficients of the series tr : E(z) 3 : = N i,j,k=1 : E ij (z) E jk (z) E ki (z) : are Casimir elements, where the normal ordering is applied from right to left.

Similarly, all coefficients of the series tr : E(z) 3 : = N i,j,k=1 : E ij (z) E jk (z) E ki (z) : are Casimir elements, where the normal ordering is applied from right to left. However, the claim does not extend to tr : E(z) 4 :! Correction term: all coefficients of the series tr : E(z) 4 : tr : ( z E(z) ) 2 : are Casimir elements.

Invariants of the vacuum module

Invariants of the vacuum module The vacuum module at the critical level is the ĝ-module V(g) = U h (ĝ)/u h (ĝ)g[t].

Invariants of the vacuum module The vacuum module at the critical level is the ĝ-module V(g) = U h (ĝ)/u h (ĝ)g[t]. The Feigin Frenkel center z(ĝ) is the algebra of g[t]-invariants z(ĝ) = {v V(g) g[t]v = 0}. Note V(g) = U ( t 1 g[t 1 ] ) as a vector space.

Properties: The subalgebra z(ĝ) of U ( t 1 g[t 1 ] ) is commutative.

Properties: The subalgebra z(ĝ) of U ( t 1 g[t 1 ] ) is commutative. It is invariant with respect to the translation operator T defined as the derivation T = d/dt.

Properties: The subalgebra z(ĝ) of U ( t 1 g[t 1 ] ) is commutative. It is invariant with respect to the translation operator T defined as the derivation T = d/dt. Any element of z(ĝ) is called a Segal Sugawara vector.

Theorem (Feigin Frenkel, 1992, Frenkel, 2007). There exist Segal Sugawara vectors S 1,..., S n U ( t 1 g[t 1 ] ), n = rank g, such that

Theorem (Feigin Frenkel, 1992, Frenkel, 2007). There exist Segal Sugawara vectors S 1,..., S n U ( t 1 g[t 1 ] ), n = rank g, such that z(ĝ) = C[T k S l l = 1,..., n, k 0].

Theorem (Feigin Frenkel, 1992, Frenkel, 2007). There exist Segal Sugawara vectors S 1,..., S n U ( t 1 g[t 1 ] ), n = rank g, such that z(ĝ) = C[T k S l l = 1,..., n, k 0]. We call S 1,..., S n a complete set of Segal Sugawara vectors. Explicit constructions of such sets and a new proof of the theorem for the classical types A, B, C, D: [Chervov Talalaev, 2006, Chervov M., 2009, M. 2013].

Example: g = gl N.

Example: g = gl N. Set τ = d/dt and consider the N N matrix τ + E 11 [ 1] E 12 [ 1]... E 1N [ 1] E τ + E[ 1] = 21 [ 1] τ + E 22 [ 1]... E 2N [ 1]....... E N1 [ 1] E N2 [ 1]... τ + E NN [ 1]

The coefficients φ 1,..., φ N of the polynomial cdet ( τ + E[ 1] ) = τ N + φ 1 τ N 1 + + φ N 1 τ + φ N form a complete set of Segal Sugawara vectors.

The coefficients φ 1,..., φ N of the polynomial cdet ( τ + E[ 1] ) = τ N + φ 1 τ N 1 + + φ N 1 τ + φ N form a complete set of Segal Sugawara vectors. For N = 2 cdet ( τ + E[ 1] ) = ( τ + E 11 [ 1] )( τ + E 22 [ 1] ) E 21 [ 1]E 12 [ 1] = τ 2 + φ 1 τ + φ 2

To get another family of Segal Sugawara vectors, expand tr ( τ + E[ 1] ) m = θm 0 τ m + θ m1 τ m 1 + + θ m m

To get another family of Segal Sugawara vectors, expand tr ( τ + E[ 1] ) m = θm 0 τ m + θ m1 τ m 1 + + θ m m All coefficients θ m i belong to the Feigin Frenkel center z(ĝl N).

To get another family of Segal Sugawara vectors, expand tr ( τ + E[ 1] ) m = θm 0 τ m + θ m1 τ m 1 + + θ m m All coefficients θ m i belong to the Feigin Frenkel center z(ĝl N). The elements θ 1 1,..., θ N N form a complete set of Segal Sugawara vectors.

The corresponding central elements in Ũ N (ĝl N) are recovered by the state-field correspondence map Y which takes elements of the vacuum module V(gl N ) to Laurent series in z;

The corresponding central elements in Ũ N (ĝl N) are recovered by the state-field correspondence map Y which takes elements of the vacuum module V(gl N ) to Laurent series in z; its application to Segal Sugawara vectors yields Laurent series whose coefficients are Casimir elements.

Also, Y : E ij [ r 1] 1 r! r z E ij (z), r 0,

Also, Y : E ij [ r 1] 1 r! r z E ij (z), r 0, and Y : E ij [ 1] E kl [ 1] : E ij (z) E kl (z) :

Also, Y : E ij [ r 1] 1 r! r z E ij (z), r 0, and Y : E ij [ 1] E kl [ 1] : E ij (z) E kl (z) : We have Y : tr E[ 1] tr E(z) Y : tr E[ 1] 2 tr : E(z) 2 : Y : tr E[ 1] 3 tr : E(z) 3 : Y : tr E[ 1] 4 tr E[ 2] 2 tr : E(z) 4 : tr : ( z E(z) ) 2 :

Write tr : ( z + E(z) ) m : = θm0 (z) m z + + θ m m (z).

Write tr : ( z + E(z) ) m : = θm0 (z) m z + + θ m m (z). Theorem. The coefficients of the Laurent series θ 1 1 (z),..., θ N N (z) are topological generators of the center of Ũ N (ĝl N).

Write tr : ( z + E(z) ) m : = θm0 (z) m z + + θ m m (z). Theorem. The coefficients of the Laurent series θ 1 1 (z),..., θ N N (z) are topological generators of the center of Ũ N (ĝl N). Remark. The theorem holds in the same form for any complete set of Segal Sugawara vectors.

Proving the Feigin Frenkel theorem for the classical types:

Proving the Feigin Frenkel theorem for the classical types: Produce Segal Sugawara vectors S 1,..., S n explicitly.

Proving the Feigin Frenkel theorem for the classical types: Produce Segal Sugawara vectors S 1,..., S n explicitly. Show that all elements T k S l with l = 1,..., n and k 0 are algebraically independent and generate z(ĝ). Use the classical limit: gr U ( t 1 g[t 1 ] ) = S ( t 1 g[t 1 ] )

Let X 1,..., X d be a basis of g and let P = P(X 1,..., X d ) be a g-invariant in the symmetric algebra S(g).

Let X 1,..., X d be a basis of g and let P = P(X 1,..., X d ) be a g-invariant in the symmetric algebra S(g). Then each element P (r) = T r P ( X 1 [ 1],..., X d [ 1] ), r 0, is a g[t]-invariant in the symmetric algebra S ( t 1 g[t 1 ] ). Theorem (Raïs Tauvel 1992, Beilinson Drinfeld 1997). If P 1,..., P n are algebraically independent generators of S(g) g, then the elements P 1,(r),..., P n,(r) with r 0 are algebraically independent generators of S ( t 1 g[t 1 ] ) g[t].

Explicit generators of z(ĝ). Type A

Explicit generators of z(ĝ). Type A Set E ij [r] = E ij t r gl N [t, t 1 ]

Explicit generators of z(ĝ). Type A Set and E ij [r] = E ij t r gl N [t, t 1 ] N E[r] = e ij E ij [r] End C N U ( gl N [t, t 1 ] ). i,j=1

Explicit generators of z(ĝ). Type A Set E ij [r] = E ij t r gl N [t, t 1 ] and E[r] = N e ij E ij [r] End C N U ( gl N [t, t 1 ] ). i,j=1 Consider the algebra } End C N. {{.. End C N } U ( gl N [t, t 1 ] ) m

Explicit generators of z(ĝ). Type A Set E ij [r] = E ij t r gl N [t, t 1 ] and E[r] = N e ij E ij [r] End C N U ( gl N [t, t 1 ] ). i,j=1 Consider the algebra } End C N. {{.. End C N } U ( gl N [t, t 1 ] ) m and recall its elements H (m) and A (m).

Theorem. All coefficients of the polynomials in τ = d/d t

Theorem. All coefficients of the polynomials in τ = d/d t tr 1,...,m A (m)( ) ( ) τ + E[ 1] 1... τ + E[ 1]m = φ m 0 τ m + φ m1 τ m 1 + + φ m m, tr 1,...,m H (m)( ) ( ) τ + E[ 1] 1... τ + E[ 1]m = ψ m 0 τ m + ψ m1 τ m 1 + + ψ m m, and tr (τ + E[ 1]) m = θ m 0 τ m + θ m 1 τ m 1 + + θ m m

Proof. Use the matrix form of the defining relations of U N (ĝl N):

Proof. Use the matrix form of the defining relations of U N (ĝl N): For any r Z set N E[r] = e ij E ij [r] End C N U N (ĝl N). i,j=1

Proof. Use the matrix form of the defining relations of U N (ĝl N): For any r Z set N E[r] = e ij E ij [r] End C N U N (ĝl N). i,j=1 The defining relations can be written in the form E[r] 1 E[s] 2 E[s] 2 E[r] 1 = ( E[r + s] 1 E[r + s] 2 ) P12 + rδ r, s ( 1 N P12 ).

The required relations in the vacuum module are E[0] 0 tr 1,...,m A (m)( τ + E[ 1] 1 )... ( τ + E[ 1]m ) = 0

The required relations in the vacuum module are E[0] 0 tr 1,...,m A (m)( τ + E[ 1] 1 )... ( τ + E[ 1]m ) = 0 and E[1] 0 tr 1,...,m A (m)( τ + E[ 1] 1 )... ( τ + E[ 1]m ) = 0.

The required relations in the vacuum module are E[0] 0 tr 1,...,m A (m)( τ + E[ 1] 1 )... ( τ + E[ 1]m ) = 0 and E[1] 0 tr 1,...,m A (m)( τ + E[ 1] 1 )... ( τ + E[ 1]m ) = 0. The elements ψ m a and θ m a are expressed in terms of the φ m a through the MacMahon Master Theorem and the Newton identities, respectively.

The coefficients of the column-determinant are related to the φ m a through the relation cdet ( τ + E[ 1] ) = tr 1,...,N A (N) ( τ + E[ 1] 1 )... ( τ + E[ 1]N ).

The coefficients of the column-determinant are related to the φ m a through the relation cdet ( τ + E[ 1] ) = tr 1,...,N A (N) ( τ + E[ 1] 1 )... ( τ + E[ 1]N ). This follows from the property A (N) ( τ + E[ 1] 1 )... ( τ + E[ 1]N ) = A (N) ( τ + E[ 1] 1 )... ( τ + E[ 1]N ) A (N),

Types B, C and D

Types B, C and D Recall the symmetrizers associated with o N and sp 2n : S (m) = 1 m! 1 a<b m ( 1 + P ab b a Q ) ab, N/2 + b a 1

Types B, C and D Recall the symmetrizers associated with o N and sp 2n : and S (m) = 1 m! S (m) = 1 m! 1 a<b m 1 a<b m ( 1 + P ab b a Q ) ab, N/2 + b a 1 ( 1 P ab b a Q ) ab. n b + a + 1

Types B, C and D Recall the symmetrizers associated with o N and sp 2n : and S (m) = 1 m! S (m) = 1 m! 1 a<b m 1 a<b m ( 1 + P ab b a Q ) ab, N/2 + b a 1 ( 1 P ab b a Q ) ab. n b + a + 1 Also, γ m (ω) = ω + m 2 N for g = o N ω + 2m 2, ω = 2n for g = sp 2n.

Let g = o N, sp N with N = 2n or N = 2n + 1.

Let g = o N, sp N with N = 2n or N = 2n + 1. As before, F i j = E i j E j i or F i j = E i j ε i ε j E j i

Let g = o N, sp N with N = 2n or N = 2n + 1. As before, F i j = E i j E j i or F i j = E i j ε i ε j E j i and F i j [r] = F i j t r g[t, t 1 ].

Let g = o N, sp N with N = 2n or N = 2n + 1. As before, F i j = E i j E j i or F i j = E i j ε i ε j E j i and F i j [r] = F i j t r g[t, t 1 ]. Combine into a matrix N F[r] = e ij F i j [r] End C N U ( g[t, t 1 ] ). i,j=1

Theorem. All coefficients of the polynomial in τ = d/d t γ m (ω) tr 1,...,m S (m)( τ + F[ 1] 1 )... ( τ + F[ 1]m ) = φ m 0 τ m + φ m1 τ m 1 + + φ m m

Theorem. All coefficients of the polynomial in τ = d/d t γ m (ω) tr 1,...,m S (m)( τ + F[ 1] 1 )... ( τ + F[ 1]m ) = φ m 0 τ m + φ m1 τ m 1 + + φ m m belong to the Feigin Frenkel center z(ĝ).

Theorem. All coefficients of the polynomial in τ = d/d t γ m (ω) tr 1,...,m S (m)( τ + F[ 1] 1 )... ( τ + F[ 1]m ) = φ m 0 τ m + φ m1 τ m 1 + + φ m m belong to the Feigin Frenkel center z(ĝ). In addition, in the case g = o 2n, the Pfaffian Pf F[ 1] = 1 2 n n! belongs to z(ô 2n ). σ S 2n sgn σ F σ(1) σ(2) [ 1]... F σ(2n 1) σ(2n) [ 1]

Moreover, φ 2 2, φ 4 4,..., φ 2n 2n is a complete set of Segal Sugawara vectors for o 2n+1 and sp 2n, whereas

Moreover, φ 2 2, φ 4 4,..., φ 2n 2n is a complete set of Segal Sugawara vectors for o 2n+1 and sp 2n, whereas φ 2 2, φ 4 4,..., φ 2n 2 2n 2, φ n is a complete set of Segal Sugawara vectors for o 2n, where φ n = Pf F[ 1].

Affine Harish-Chandra isomorphism For a triangular decomposition g = n h n + consider the Harish-Chandra homomorphism U ( t 1 g[t 1 ] ) h U ( t 1 h[t 1 ] ),

Affine Harish-Chandra isomorphism For a triangular decomposition g = n h n + consider the Harish-Chandra homomorphism U ( t 1 g[t 1 ] ) h U ( t 1 h[t 1 ] ), the projection modulo the left ideal generated by t 1 n [t 1 ].

The restriction to z(ĝ) yields the Harish-Chandra isomorphism f : z(ĝ) W( L g),

The restriction to z(ĝ) yields the Harish-Chandra isomorphism f : z(ĝ) W( L g), where W( L g) is the classical W-algebra associated with the Langlands dual Lie algebra L g [Feigin and Frenkel, 1992].

Example g = gl N. Set µ i [r] = E i i [r]. We have f : cdet ( τ + E[ 1] ) ( τ + µ N [ 1] )... ( τ + µ 1 [ 1] ).

Example g = gl N. Set µ i [r] = E i i [r]. We have f : cdet ( τ + E[ 1] ) ( τ + µ N [ 1] )... ( τ + µ 1 [ 1] ). Define the elements E 1,..., E N by the Miura transformation ( τ + µn [ 1] )... ( τ + µ 1 [ 1] ) = τ N + E 1 τ N 1 + + E N. Explicitly, E m = e m ( T + µ1 [ 1],..., T + µ N [ 1] ) 1 is the noncommutative elementary symmetric function, e m (x 1,..., x p ) = i 1 > >i m x i1... x im.

If N = 2 then E 1 = µ 1 [ 1] + µ 2 [ 1], E 2 = µ 1 [ 1] µ 2 [ 1] + µ 1 [ 2].

If N = 2 then E 1 = µ 1 [ 1] + µ 2 [ 1], E 2 = µ 1 [ 1] µ 2 [ 1] + µ 1 [ 2]. If N = 3 then E 1 = µ 1 [ 1] + µ 2 [ 1] + µ 3 [ 1], E 2 = µ 1 [ 1] µ 2 [ 1] + µ 1 [ 1] µ 3 [ 1] + µ 2 [ 1] µ 3 [ 1] + 2 µ 1 [ 2] + µ 2 [ 2], E 3 = µ 1 [ 1] µ 2 [ 1] µ 3 [ 1] + µ 1 [ 2] µ 2 [ 1] + µ 1 [ 2] µ 3 [ 1] + µ 1 [ 1] µ 2 [ 2] + 2 µ 1 [ 3].

Then W(gl N ) = C[T k E 1,..., T k E N k 0].

Then W(gl N ) = C[T k E 1,..., T k E N k 0]. Also, W(gl N ) = C[T k H 1,..., T k H N k 0],

Then W(gl N ) = C[T k E 1,..., T k E N k 0]. Also, W(gl N ) = C[T k H 1,..., T k H N k 0], where ( H m = h m T + µ1 [ 1],..., T + µ N [ 1] ) 1 is the noncommutative complete symmetric function, h m (x 1,..., x p ) = i 1 i m x i1... x im.

Example g = o N. Set µ i [r] = F i i [r].

Example g = o N. Set µ i [r] = F i i [r]. The Harish-Chandra image of the polynomial γ m (N) tr S (m)( τ + F[ 1] 1 )... ( τ + F[ 1]m ) equals

Example g = o N. Set µ i [r] = F i i [r]. The Harish-Chandra image of the polynomial γ m (N) tr S (m)( τ + F[ 1] 1 )... ( τ + F[ 1]m ) equals h m ( τ + µ1 [ 1],..., τ + µ n [ 1], τ µ n [ 1],..., τ µ 1 [ 1] )

Example g = o N. Set µ i [r] = F i i [r]. The Harish-Chandra image of the polynomial γ m (N) tr S (m)( τ + F[ 1] 1 )... ( τ + F[ 1]m ) equals h m ( τ + µ1 [ 1],..., τ + µ n [ 1], τ µ n [ 1],..., τ µ 1 [ 1] ) for N = 2n + 1.

For the Lie algebra g = o 2n the image is 1 2 h ( m τ + µ1 [ 1],..., τ + µ n 1 [ 1], τ µ n [ 1],..., τ µ 1 [ 1] ) + 1 2 h m( τ + µ1 [ 1],..., τ + µ n [ 1], τ µ n 1 [ 1],..., τ µ 1 [ 1] ).

For the Lie algebra g = o 2n the image is 1 2 h ( m τ + µ1 [ 1],..., τ + µ n 1 [ 1], τ µ n [ 1],..., τ µ 1 [ 1] ) + 1 2 h m( τ + µ1 [ 1],..., τ + µ n [ 1], τ µ n 1 [ 1],..., τ µ 1 [ 1] ). The Harish-Chandra image of the Pfaffian Pf F[ 1] = 1 2 n n! is found by σ S 2n sgn σ F σ(1) σ(2) [ 1]... F σ(2n 1) σ(2n) [ 1]

Example g = sp 2n. Set µ i [r] = F i i [r].

Example g = sp 2n. Set µ i [r] = F i i [r]. The Harish-Chandra image of the polynomial γ m ( 2n) tr S (m)( τ + F[ 1] 1 )... ( τ + F[ 1]m ) with 1 m 2n + 1 equals

Example g = sp 2n. Set µ i [r] = F i i [r]. The Harish-Chandra image of the polynomial γ m ( 2n) tr S (m)( ) ( ) τ + F[ 1] 1... τ + F[ 1]m with 1 m 2n + 1 equals ( e m τ + µ1 [ 1],..., τ + µ n [ 1], τ, τ µ n [ 1],..., τ µ 1 [ 1] ). Miura transformation for o 2n+1 [Drinfeld Sokolov 1985]: ( τ µ1 [ 1] )... ( τ µ n [ 1] ) τ ( τ + µ n [ 1] )... ( τ + µ 1 [ 1] ) = τ 2n+1 + E 2 τ 2n 1 + E 3 τ 2n 2 + + E 2n+1.

Classical W-algebras

Classical W-algebras Let µ 1,..., µ n be a basis of the Cartan subalgebra h of g.

Classical W-algebras Let µ 1,..., µ n be a basis of the Cartan subalgebra h of g. Set µ i [r] = µ i t r and identify U ( t 1 h[t 1 ] ) = C [ µ 1 [r],..., µ n [r] r < 0 ] =: P n.

Classical W-algebras Let µ 1,..., µ n be a basis of the Cartan subalgebra h of g. Set µ i [r] = µ i t r and identify U ( t 1 h[t 1 ] ) = C [ µ 1 [r],..., µ n [r] r < 0 ] =: P n. The classical W-algebra W(g) is defined by W(g) = {P P n V i P = 0, i = 1,..., n},

Example. For W(gl N ) the operators V 1,..., V N 1 are

Example. For W(gl N ) the operators V 1,..., V N 1 are V i = ( V i (r) µ i [ r 1] ), µ i+1 [ r 1] r=0

Example. For W(gl N ) the operators V 1,..., V N 1 are V i = ( V i (r) µ i [ r 1] ), µ i+1 [ r 1] r=0 V i (r) z r = exp r=0 m=1 µ i [ m] µ i+1 [ m] m z m.

Example. For W(gl N ) the operators V 1,..., V N 1 are V i = ( V i (r) µ i [ r 1] ), µ i+1 [ r 1] r=0 V i (r) z r = exp r=0 m=1 µ i [ m] µ i+1 [ m] m z m. One verifies directly that V i ( τ + µn [ 1] )... ( τ + µ 1 [ 1] ) = 0.

Equivalently, V i : µ i (z) exp (µi (z) µ i+1 (z) ) dz, V i : µ i+1 (z) exp (µi (z) µ i+1 (z) ) dz,

Equivalently, V i : µ i (z) exp (µi (z) µ i+1 (z) ) dz, V i : µ i+1 (z) exp (µi (z) µ i+1 (z) ) dz, and V i : µ j (z) 0 for j i, i + 1,

Equivalently, V i : µ i (z) exp (µi (z) µ i+1 (z) ) dz, V i : µ i+1 (z) exp (µi (z) µ i+1 (z) ) dz, and V i : µ j (z) 0 for j i, i + 1, where µ i (z) = µ i [ r 1] z r, i = 1,..., N. r=0

Affine Poisson vertex algebra V(g) Let g be a simple Lie algebra over C and let X 1,..., X d be a basis of g.

Affine Poisson vertex algebra V(g) Let g be a simple Lie algebra over C and let X 1,..., X d be a basis of g. Consider the differential algebra V = V(g), V = C[X (r) 1,..., X(r) d r = 0, 1, 2,... ] with X (0) i = X i,

Introduce the λ-bracket on V as a linear map V V C[λ] V, a b {a λ b}.

Introduce the λ-bracket on V as a linear map V V C[λ] V, a b {a λ b}. By definition, it is given by {X λ Y} = [X, Y] + X, Y λ for X, Y g,

Introduce the λ-bracket on V as a linear map V V C[λ] V, a b {a λ b}. By definition, it is given by {X λ Y} = [X, Y] + X, Y λ for X, Y g, and extended to V by sesquilinearity (a, b V): { a λ b} = λ {a λ b}, skewsymmetry {a λ b} = {b λ a},

Hamiltonian reduction

Hamiltonian reduction For a triangular decomposition g = n h n + set p = n h and define the projection π p : g p.

Hamiltonian reduction For a triangular decomposition g = n h n + set p = n h and define the projection π p : g p. Let f n be a principal nilpotent in g.

Hamiltonian reduction For a triangular decomposition g = n h n + set p = n h and define the projection π p : g p. Let f n be a principal nilpotent in g. Define the differential algebra homomorphism ρ : V V(p), ρ(x) = π p (X) + f, X, X g.

The classical W-algebra W(g) is a Poisson vertex algebra equipped with the λ-bracket {a λ b} ρ = ρ{a λ b}, a, b W(g).

The classical W-algebra W(g) is a Poisson vertex algebra equipped with the λ-bracket {a λ b} ρ = ρ{a λ b}, a, b W(g). Motivation: Hamiltonian equations u t = {H λ u} λ=0 for u = u(t) W(g) with the Hamiltonian H W(g).

Example. Let g = sl 2 with the basis e, f, h.

Example. Let g = sl 2 with the basis e, f, h. W(sl 2 ) = C[u, u, u,... ], u = h 2 4 + h 2 + f V(p).

Example. Let g = sl 2 with the basis e, f, h. W(sl 2 ) = C[u, u, u,... ], u = h 2 4 + h 2 + f V(p). The λ-bracket (of Virasoro Magri) on W(sl 2 ) is given by {u λ u} = (2λ + )u λ3 2.

Example. Let g = sl 2 with the basis e, f, h. W(sl 2 ) = C[u, u, u,... ], u = h 2 4 + h 2 + f V(p). The λ-bracket (of Virasoro Magri) on W(sl 2 ) is given by {u λ u} = (2λ + )u λ3 2. The Hamiltonian equation with H = u2 is equivalent to 2 the KdV equation u t = 3uu 1 2 u.

Another λ-bracket (of Gardner Faddeev Zakharov) on W(sl 2 ) is given by {u λ u} 0 = λ.

Another λ-bracket (of Gardner Faddeev Zakharov) on W(sl 2 ) is given by {u λ u} 0 = λ. The Hamiltonian equation with K = 1 2 u3 1 4 u u is also equivalent to the KdV equation.

Generators of W(gl N ) Consider gl N = span of {E ij i, j = 1,..., N}. Here p is the subalgebra of lower triangular matrices.

Generators of W(gl N ) Consider gl N = span of {E ij i, j = 1,..., N}. Here p is the subalgebra of lower triangular matrices. Set f = E 2 1 + E 3 2 + + E N N 1.

Generators of W(gl N ) Consider gl N = span of {E ij i, j = 1,..., N}. Here p is the subalgebra of lower triangular matrices. Set f = E 2 1 + E 3 2 + + E N N 1. We will work with the algebra V(p) C[ ], E (r) ij E (r) ij = E (r+1) ij.

Expand the determinant with entries in V(p) C[ ], + E 11 1 0 0... 0 E 21 + E 22 1 0... 0 det.................. E N 1 1 E N 1 2 E N 1 3...... 1 E N 1 E N 2 E N 3...... + E N N = N + w 1 N 1 + + w N.

Expand the determinant with entries in V(p) C[ ], + E 11 1 0 0... 0 E 21 + E 22 1 0... 0 det.................. E N 1 1 E N 1 2 E N 1 3...... 1 E N 1 E N 2 E N 3...... + E N N = N + w 1 N 1 + + w N. Theorem. All elements w 1,..., w N belong to W(gl N ). Moreover, W(gl N ) = C[w (r) 1,..., w(r) N r 0].

Chevalley-type theorem

Chevalley-type theorem Let φ : V(p) V(h) denote the homomorphism of differential algebras defined on the generators as the projection p h with the kernel n.

Chevalley-type theorem Let φ : V(p) V(h) denote the homomorphism of differential algebras defined on the generators as the projection p h with the kernel n. The restriction of φ to W(g) is injective. The embedding φ : W(g) V(h) is often called the Miura transformation.

Theorem. The restriction of the homomorphism φ to the classical W-algebra W(g) yields an isomorphism φ : W(g) W(g),

Theorem. The restriction of the homomorphism φ to the classical W-algebra W(g) yields an isomorphism φ : W(g) W(g), where W(g) is the subalgebra of V(h) which consists of the elements annihilated by all screening operators V i, n W(g) = ker V i. i=1