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

Transcription

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

2 Plan of lectures

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

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

5 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.

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

7 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.

8 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

9 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

10 Fusion formulas

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

12 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

13 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).

14 Action of S m in tensors

15 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,

16 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

17 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.

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

19 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

20 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

21 Lie algebra gln

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

23 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.

24 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

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

26 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

27 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.

28 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.

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

30 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

31 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 ).

32 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.

33 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.

34 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.

35 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

36 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.

37 Example: Capelli determinant.

38 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).

39 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 E 1N E C(u) = cdet 21 u + E E 2N.... E N1 E N2... u + E NN N + 1

40 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 E 1N E C(u) = cdet 21 u + E E 2N.... E N1 E N2... u + E NN N + 1 All coefficients of the polynomial C(u) are Casimir elements.

41 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

42 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)

43 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) and that this equals A (N) C(u).

44 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) and that this equals A (N) C(u). It remains to note that tr 1,...,N A (N) = 1.

45 Example: Gelfand invariants.

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

47 Example: Gelfand invariants. Take s = (m m ) 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.

48 Example: Gelfand invariants. Take s = (m m ) 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.

49 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

50 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).

51 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.

52 Harish-Chandra isomorphism

53 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

54 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 E ij ξ = 0 for 1 i < j N, and E ii ξ = λ i ξ for 1 i N.

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

56 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.

57 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.

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

59 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)

60 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 +.

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

62 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.

63 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.

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

65 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 +

66 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.

67 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.

68 Brauer algebra B m (ω)

69 Brauer algebra B m (ω) Multiplication of m-diagrams (m = 8):

70 Brauer algebra B m (ω) Multiplication of m-diagrams (m = 8):

71 Brauer algebra B m (ω) Multiplication of m-diagrams (m = 8):

72 Brauer algebra B m (ω) Multiplication of m-diagrams (m = 8): = ω

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

74 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

75 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.

76 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.

77 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

78 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.

79 Brauer Schur Weyl duality

80 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

81 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 )

82 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 ) and ( Bm ( N), Sp N ) with N = 2n.

83 Action in tensors

84 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,

85 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

86 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,

87 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

88 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

89 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

90 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

91 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.

92 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.

93 Lie algebras on and spn

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

95 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.

96 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

97 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).

98 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.

99 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. A version of the Newton identity also holds.

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

101 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 )

102 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

103 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

104 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

105 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 ( l 2 j1 (j 1 1) 2)... ( l 2 j k (j k + k 2) 2), 1 j 1 j k n where l i = F ii + n i in the case N = 2n.

106 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

107 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.

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

109 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).

110 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). The Harish-Chandra images χ(s λ ) are the shifted Schur polynomials.

111 Affine Kac Moody algebras

112 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.

113 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.

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

115 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.

116 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?

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

118 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].

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

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

121 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.

122 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.

123 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.

124 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

125 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.

126 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.

127 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].

128 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).

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

130 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.

131 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).

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

133 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.

134 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.

135 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 :!

136 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.

137 Invariants of the vacuum module

138 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].

139 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}.

140 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.

141 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. Hence, z(ĝ) is a subalgebra of U ( t 1 g[t 1 ] ).

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

143 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.

144 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.

145 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

146 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].

147 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.

148 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].

149 Example: g = gl N.

150 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]

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

152 The coefficients φ 1,..., φ N of the polynomial cdet ( τ + E[ 1] ) = τ N + φ 1 τ N 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

153 The coefficients φ 1,..., φ N of the polynomial cdet ( τ + E[ 1] ) = τ N + φ 1 τ N 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 with φ 1 = E 11 [ 1] + E 22 [ 1], φ 2 = E 11 [ 1] E 22 [ 1] E 21 [ 1] E 12 [ 1] + E 22 [ 2].

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

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

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

157 To get another family of Segal Sugawara vectors, expand tr ( τ + E[ 1] ) m = θm 0 τ m + θ m1 τ m 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 following are Segal Sugawara vectors for gl N : tr E[ 1], tr E[ 1] 2, tr E[ 1] 3, tr E[ 1] 4 tr E[ 2] 2.

158 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;

159 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.

160 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. By definition, Y : E ij [ 1] E ij (z) = r Z E ij [r] z r 1.

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

162 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) :

163 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 :

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

165 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).

166 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.

167 Proving the Feigin Frenkel theorem for the classical types:

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

169 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).

170 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 ] )

171 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 ] ) which yields a g[t]-module structure on the symmetric algebra S ( t 1 g[t 1 ] ) = S ( g[t, t 1 ]/g[t] ).

172 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).

173 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 ] ).

174 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].

175 Explicit generators of z(ĝ). Type A

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

177 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

178 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

179 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).

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

181 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 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 ψ m m, and tr (τ + E[ 1]) m = θ m 0 τ m + θ m 1 τ m θ m m

182 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 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 ψ m m, and tr (τ + E[ 1]) m = θ m 0 τ m + θ m 1 τ m θ m m belong to the Feigin Frenkel center z(ĝl N).

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

184 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

185 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 ).

186 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

187 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.

188 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.

189 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 ).

190 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),

191 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), implied by the fact that τ + E[ 1] is a Manin matrix.

192 Types B, C and D

193 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

194 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

195 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.

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

197 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

198 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 ].

199 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

200 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 φ m m

201 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 φ m m belong to the Feigin Frenkel center z(ĝ).

202 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 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]

203 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

204 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].

205 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 ] ),

206 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 ].

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

208 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].

209 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] ).

210 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 E N.

211 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 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.

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

213 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].

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

215 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],

216 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.

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

218 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

219 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] )

220 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.

221 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] ) h m( τ + µ1 [ 1],..., τ + µ n [ 1], τ µ n 1 [ 1],..., τ µ 1 [ 1] ).

222 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] ) 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]

223 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] ) 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] Pf F[ 1] ( µ 1 [ 1] T )... ( µ n [ 1] T ) 1.

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

225 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

226 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] ).

227 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 E 2n+1.

228 Classical W-algebras

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

230 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.

231 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},

232 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}, the V i are the screening operators.

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

234 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

235 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.

236 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.

237 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,

238 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,

239 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

240 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.

241 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,

242 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, equipped with the derivation, (X (r) i ) = X (r+1) i for all i = 1,..., d and r 0.

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

244 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,

245 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},

246 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},

247 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}, and the Leibniz rule (a, b, c V): {a λ bc} = {a λ b}c + {a λ c}b.

248 Hamiltonian reduction

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

250 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.

251 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.

252 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 defined by W(g) = {P V(p) ρ{x λ P} = 0 for all X n + }.

253 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).

254 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).

255 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). De Sole, Kac and Valeri, ; Drinfeld and Sokolov, 1985.

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

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

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

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

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

261 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.

262 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.

263 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 E E N N 1.

264 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 E E N N 1. We will work with the algebra V(p) C[ ], E (r) ij E (r) ij = E (r+1) ij.

265 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 E E N N 1. We will work with the algebra V(p) C[ ], E (r) ij E (r) ij = E (r+1) ij. The invariant symmetric bilinear form on gl N is defined by X, Y = tr XY, X, Y gl N.

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

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

268 Chevalley-type theorem

269 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.

270 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.

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

272 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