The MacMahon Master Theorem and

higher Sugawara operators

Alexander Molev

joint work with

Eric Ragoucy

University of Sydney

LAPTH, Annecy In the super case, zij are elements of a superalgebra and

¯ı¯+¯ık¯+¯k¯ [zij , zkl ] = [zkj , zil ](−1) for all i, j, k, l ∈ {1,..., m+n},

where ¯ı = 0 for i 6 m and ¯ı = 1 for i > m, deg zij = ¯ı + ¯.

Manin matrices

A matrix Z = [zij ] with entries in an algebra is a Manin matrix if

[zij , zkl ] = [zkj , zil ] for all i, j, k, l ∈ {1,..., m}. Manin matrices

A matrix Z = [zij ] with entries in an algebra is a Manin matrix if

[zij , zkl ] = [zkj , zil ] for all i, j, k, l ∈ {1,..., m}.

In the super case, zij are elements of a superalgebra and

¯ı¯+¯ık¯+¯k¯ [zij , zkl ] = [zkj , zil ](−1) for all i, j, k, l ∈ {1,..., m+n},

where ¯ı = 0 for i 6 m and ¯ı = 1 for i > m, deg zij = ¯ı + ¯. −1 −1 I Z = ∂u − E u , zij = ∂u δij − eij u , eij ∈ glm,

−∂u −∂u
I Z = e (u + E), zij = e u δij + eij .

−∂u I Z = e T (u), T (u) is the generator matrix

for the Yangian Y(glm|n).

Examples of Manin matrices.

I zij are elements of any (super)commutative (super)algebra. −∂u −∂u
I Z = e (u + E), zij = e u δij + eij .

−∂u I Z = e T (u), T (u) is the generator matrix

for the Yangian Y(glm|n).

Examples of Manin matrices.

I zij are elements of any (super)commutative (super)algebra.

−1 −1 I Z = ∂u − E u , zij = ∂u δij − eij u , eij ∈ glm, −∂u I Z = e T (u), T (u) is the generator matrix

for the Yangian Y(glm|n).

Examples of Manin matrices.

I zij are elements of any (super)commutative (super)algebra.

−1 −1 I Z = ∂u − E u , zij = ∂u δij − eij u , eij ∈ glm,

−∂u −∂u
I Z = e (u + E), zij = e u δij + eij . Examples of Manin matrices.

I zij are elements of any (super)commutative (super)algebra.

−1 −1 I Z = ∂u − E u , zij = ∂u δij − eij u , eij ∈ glm,

−∂u −∂u
I Z = e (u + E), zij = e u δij + eij .

−∂u I Z = e T (u), T (u) is the generator matrix

for the Yangian Y(glm|n). Regard the matrix Z = [ zij ] as the element m+n X ¯ı¯+¯ m|n Z = eij ⊗ zij (−1) ∈ End C ⊗ Mm|n. i,j=1

m|n Denote by Za the matrix Z in the a-th copy of End C in

m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k

MacMahon’s Master Theorem

The right quantum superalgebra Mm|n is generated by

elements zij , subject to the Manin matrix relations:

¯ı¯+¯ık¯+¯k¯ [zij , zkl ] = [zkj , zil ](−1) for all i, j, k, l ∈ {1,..., m+n}. m|n Denote by Za the matrix Z in the a-th copy of End C in

m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k

MacMahon’s Master Theorem

The right quantum superalgebra Mm|n is generated by

elements zij , subject to the Manin matrix relations:

¯ı¯+¯ık¯+¯k¯ [zij , zkl ] = [zkj , zil ](−1) for all i, j, k, l ∈ {1,..., m+n}.

Regard the matrix Z = [ zij ] as the element m+n X ¯ı¯+¯ m|n Z = eij ⊗ zij (−1) ∈ End C ⊗ Mm|n. i,j=1 MacMahon’s Master Theorem

The right quantum superalgebra Mm|n is generated by

elements zij , subject to the Manin matrix relations:

¯ı¯+¯ık¯+¯k¯ [zij , zkl ] = [zkj , zil ](−1) for all i, j, k, l ∈ {1,..., m+n}.

Regard the matrix Z = [ zij ] as the element m+n X ¯ı¯+¯ m|n Z = eij ⊗ zij (−1) ∈ End C ⊗ Mm|n. i,j=1

m|n Denote by Za the matrix Z in the a-th copy of End C in

m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k Let Hk and Ak denote the respective images in the algebra

m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k Set ∞ X Bos = 1 + str Hk Z1 ... Zk , k=1 ∞ X k Ferm = 1 + (−1) str Ak Z1 ... Zk , k=1 | taking supertrace with respect to all k copies of End Cm n.

Introduce the normalized symmetrizer and antisymmetrizer

1 X 1 X σ ∈ [S ], sgn σ · σ ∈ [S ]. k! C k k! C k σ∈Sk σ∈Sk Set ∞ X Bos = 1 + str Hk Z1 ... Zk , k=1 ∞ X k Ferm = 1 + (−1) str Ak Z1 ... Zk , k=1 | taking supertrace with respect to all k copies of End Cm n.

Introduce the normalized symmetrizer and antisymmetrizer

1 X 1 X σ ∈ [S ], sgn σ · σ ∈ [S ]. k! C k k! C k σ∈Sk σ∈Sk

Let Hk and Ak denote the respective images in the algebra

m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k Introduce the normalized symmetrizer and antisymmetrizer

1 X 1 X σ ∈ [S ], sgn σ · σ ∈ [S ]. k! C k k! C k σ∈Sk σ∈Sk

Let Hk and Ak denote the respective images in the algebra

m|n m|n End C ⊗ ... ⊗ End C ⊗Mm|n. | {z } k Set ∞ X Bos = 1 + str Hk Z1 ... Zk , k=1 ∞ X k Ferm = 1 + (−1) str Ak Z1 ... Zk , k=1 | taking supertrace with respect to all k copies of End Cm n. Remarks. Original proof in the even case n = 0:

I Garoufalidis, Lê and Zeilberger, 2006.

I Other proofs: Konvalinka and Pak, 2007, Foata and

Han, 2007, Hai and Lorenz, 2007.

I The proof in the super case relies on the matrix form

(1 − P12)[Z1, Z2] = 0

of the defining relations of Mm|n.

Theorem (MacMahon Master Theorem for Mm|n).

Bos × Ferm = 1. I Other proofs: Konvalinka and Pak, 2007, Foata and

Han, 2007, Hai and Lorenz, 2007.

I The proof in the super case relies on the matrix form

(1 − P12)[Z1, Z2] = 0

of the defining relations of Mm|n.

Theorem (MacMahon Master Theorem for Mm|n).

Bos × Ferm = 1.

Remarks. Original proof in the even case n = 0:

I Garoufalidis, Lê and Zeilberger, 2006. I The proof in the super case relies on the matrix form

(1 − P12)[Z1, Z2] = 0

of the defining relations of Mm|n.

Theorem (MacMahon Master Theorem for Mm|n).

Bos × Ferm = 1.

Remarks. Original proof in the even case n = 0:

I Garoufalidis, Lê and Zeilberger, 2006.

I Other proofs: Konvalinka and Pak, 2007, Foata and

Han, 2007, Hai and Lorenz, 2007. Theorem (MacMahon Master Theorem for Mm|n).

Bos × Ferm = 1.

Remarks. Original proof in the even case n = 0:

I Garoufalidis, Lê and Zeilberger, 2006.

I Other proofs: Konvalinka and Pak, 2007, Foata and

Han, 2007, Hai and Lorenz, 2007.

I The proof in the super case relies on the matrix form

(1 − P12)[Z1, Z2] = 0

of the defining relations of Mm|n. Define the Berezinian Ber Z by

X Ber Z = sgn σ · zσ(1)1 ... zσ(m)m σ∈Sm

X 0 0 × sgn τ · zm+1,m+τ(1) ... zm+n,m+τ(n). τ∈Sn

In the supercommutative specialization,

" AB # Ber = det A · det(D − CA−1B)−1. CD

Noncommutative Berezinian

−1 0 Suppose a Manin matrix Z is invertible, Z = [zij ]. In the supercommutative specialization,

" AB # Ber = det A · det(D − CA−1B)−1. CD

Noncommutative Berezinian

−1 0 Suppose a Manin matrix Z is invertible, Z = [zij ]. Define the Berezinian Ber Z by

X Ber Z = sgn σ · zσ(1)1 ... zσ(m)m σ∈Sm

X 0 0 × sgn τ · zm+1,m+τ(1) ... zm+n,m+τ(n). τ∈Sn Noncommutative Berezinian

−1 0 Suppose a Manin matrix Z is invertible, Z = [zij ]. Define the Berezinian Ber Z by

X Ber Z = sgn σ · zσ(1)1 ... zσ(m)m σ∈Sm

X 0 0 × sgn τ · zm+1,m+τ(1) ... zm+n,m+τ(n). τ∈Sn

In the supercommutative specialization,

" AB # Ber = det A · det(D − CA−1B)−1. CD ∞  −1 X k Ber (1 − u Z ) = u str Hk Z1 ... Zk . k=0

Moreover, we have the noncommutative Newton identities:

∞  −1 X k k+1 Ber (1 + u Z ) ∂u Ber (1 + u Z ) = (−u) str Z . k=0

Theorem. If Z is a Manin matrix, then

∞ X k Ber (1 + u Z ) = u str Ak Z1 ... Zk , k=0 Moreover, we have the noncommutative Newton identities:

∞  −1 X k k+1 Ber (1 + u Z ) ∂u Ber (1 + u Z ) = (−u) str Z . k=0

Theorem. If Z is a Manin matrix, then

∞ X k Ber (1 + u Z ) = u str Ak Z1 ... Zk , k=0 ∞  −1 X k Ber (1 − u Z ) = u str Hk Z1 ... Zk . k=0 Theorem. If Z is a Manin matrix, then

∞ X k Ber (1 + u Z ) = u str Ak Z1 ... Zk , k=0 ∞  −1 X k Ber (1 − u Z ) = u str Hk Z1 ... Zk . k=0

Moreover, we have the noncommutative Newton identities:

∞  −1 X k k+1 Ber (1 + u Z ) ∂u Ber (1 + u Z ) = (−u) str Z . k=0 with the commutation relations

  (¯ı+¯)(k¯+¯l) eij [r], ekl [s] = δkj ei l [r + s] − δi l ekj [r + s](−1)

 δ δ ¯ + K δ δ (−1)¯ı − ij kl (−1)¯ı+k r δ , kj i l m − n r,−s

r the element K is even and central, and eij [r] := eij t .

In the case m = n the singular term is omitted.

Segal–Sugawara vectors

−1 Consider the affine Lie superalgebra glb m|n = glm|n[t, t ] ⊕ CK r the element K is even and central, and eij [r] := eij t .

In the case m = n the singular term is omitted.

Segal–Sugawara vectors

−1 Consider the affine Lie superalgebra glb m|n = glm|n[t, t ] ⊕ CK with the commutation relations

  (¯ı+¯)(k¯+¯l) eij [r], ekl [s] = δkj ei l [r + s] − δi l ekj [r + s](−1)

 δ δ ¯ + K δ δ (−1)¯ı − ij kl (−1)¯ı+k r δ , kj i l m − n r,−s In the case m = n the singular term is omitted.

Segal–Sugawara vectors

−1 Consider the affine Lie superalgebra glb m|n = glm|n[t, t ] ⊕ CK with the commutation relations

  (¯ı+¯)(k¯+¯l) eij [r], ekl [s] = δkj ei l [r + s] − δi l ekj [r + s](−1)

 δ δ ¯ + K δ δ (−1)¯ı − ij kl (−1)¯ı+k r δ , kj i l m − n r,−s

r the element K is even and central, and eij [r] := eij t . Segal–Sugawara vectors

−1 Consider the affine Lie superalgebra glb m|n = glm|n[t, t ] ⊕ CK with the commutation relations

  (¯ı+¯)(k¯+¯l) eij [r], ekl [s] = δkj ei l [r + s] − δi l ekj [r + s](−1)

 δ δ ¯ + K δ δ (−1)¯ı − ij kl (−1)¯ı+k r δ , kj i l m − n r,−s

r the element K is even and central, and eij [r] := eij t .

In the case m = n the singular term is omitted. Any element b of the glb m|n-module Vκ(glm|n) satisfying

glm|n[t] b = 0 is called a Segal–Sugawara vector.

The vacuum module Vκ(glm|n) has a vertex algebra structure.

Hence, the subspace z(glb m|n) spanned by the Segal–Sugawara vectors is a commutative associative algebra which can be

−1 −1 identified with a commutative subalgebra of U(t glm|n[t ]).

For any κ ∈ C the vacuum module Vκ(glm|n) of level κ is the quotient of U(glb m|n) by the left ideal generated by glm|n[t] and the element K − κ. The vacuum module Vκ(glm|n) has a vertex algebra structure.

Hence, the subspace z(glb m|n) spanned by the Segal–Sugawara vectors is a commutative associative algebra which can be

−1 −1 identified with a commutative subalgebra of U(t glm|n[t ]).

For any κ ∈ C the vacuum module Vκ(glm|n) of level κ is the quotient of U(glb m|n) by the left ideal generated by glm|n[t] and the element K − κ.

Any element b of the glb m|n-module Vκ(glm|n) satisfying glm|n[t] b = 0 is called a Segal–Sugawara vector. Hence, the subspace z(glb m|n) spanned by the Segal–Sugawara vectors is a commutative associative algebra which can be

−1 −1 identified with a commutative subalgebra of U(t glm|n[t ]).

For any κ ∈ C the vacuum module Vκ(glm|n) of level κ is the quotient of U(glb m|n) by the left ideal generated by glm|n[t] and the element K − κ.

Any element b of the glb m|n-module Vκ(glm|n) satisfying glm|n[t] b = 0 is called a Segal–Sugawara vector.

The vacuum module Vκ(glm|n) has a vertex algebra structure. For any κ ∈ C the vacuum module Vκ(glm|n) of level κ is the quotient of U(glb m|n) by the left ideal generated by glm|n[t] and the element K − κ.

Any element b of the glb m|n-module Vκ(glm|n) satisfying glm|n[t] b = 0 is called a Segal–Sugawara vector.

The vacuum module Vκ(glm|n) has a vertex algebra structure.

Hence, the subspace z(glb m|n) spanned by the Segal–Sugawara vectors is a commutative associative algebra which can be

−1 −1 identified with a commutative subalgebra of U(t glm|n[t ]). Consider the extended Lie superalgebra glb m|n ⊕ Cτ, where the element τ is even and

    τ, eij [r] = −r eij [r − 1], τ, K = 0.

Lemma. The matrix

 ¯ı τ + Eb [−1] = δij τ + eij [−1](−1)

with the entries in U(glb m|n ⊕ Cτ) is a Manin matrix.

The algebra z(glb m|n) of Segal–Sugawara vectors is nontrivial if and only if κ is the critical level, κ = n − m. Lemma. The matrix

 ¯ı τ + Eb [−1] = δij τ + eij [−1](−1)

with the entries in U(glb m|n ⊕ Cτ) is a Manin matrix.

The algebra z(glb m|n) of Segal–Sugawara vectors is nontrivial if and only if κ is the critical level, κ = n − m.

Consider the extended Lie superalgebra glb m|n ⊕ Cτ, where the element τ is even and

    τ, eij [r] = −r eij [r − 1], τ, K = 0. The algebra z(glb m|n) of Segal–Sugawara vectors is nontrivial if and only if κ is the critical level, κ = n − m.

Consider the extended Lie superalgebra glb m|n ⊕ Cτ, where the element τ is even and

    τ, eij [r] = −r eij [r − 1], τ, K = 0.

Lemma. The matrix

 ¯ı τ + Eb [−1] = δij τ + eij [−1](−1)

with the entries in U(glb m|n ⊕ Cτ) is a Manin matrix. Examples.

m+n X str(τ + Eb [−1]) = (m − n) τ + eii [−1], i=1 m+n 2 2 X str(τ + Eb [−1]) = (m − n) τ + 2 eii [−1] τ i=1 m+n m+n X k¯ X + eik [−1] eki [−1](−1) + eii [−2]. i,k=1 i=1

Theorem. For any k > 0 all coefficients skl in the expansion

k k k−1 str(τ + Eb [−1]) = sk 0 τ + sk1 τ + ··· + skk are Segal–Sugawara vectors. m+n 2 2 X str(τ + Eb [−1]) = (m − n) τ + 2 eii [−1] τ i=1 m+n m+n X k¯ X + eik [−1] eki [−1](−1) + eii [−2]. i,k=1 i=1

Theorem. For any k > 0 all coefficients skl in the expansion

k k k−1 str(τ + Eb [−1]) = sk 0 τ + sk1 τ + ··· + skk are Segal–Sugawara vectors.

Examples.

m+n X str(τ + Eb [−1]) = (m − n) τ + eii [−1], i=1 Theorem. For any k > 0 all coefficients skl in the expansion

k k k−1 str(τ + Eb [−1]) = sk 0 τ + sk1 τ + ··· + skk are Segal–Sugawara vectors.

Examples.

m+n X str(τ + Eb [−1]) = (m − n) τ + eii [−1], i=1 m+n 2 2 X str(τ + Eb [−1]) = (m − n) τ + 2 eii [−1] τ i=1 m+n m+n X k¯ X + eik [−1] eki [−1](−1) + eii [−2]. i,k=1 i=1 k k−1 str Ak T1 ... Tk = σk 0τ + σk1τ + ··· + σkk ,

k k−1 str Hk T1 ... Tk = hk 0τ + hk1τ + ··· + hkk ,

where T = τ + Eb [−1], are Segal–Sugawara vectors.

Moreover, bk l = σk l for all k and l.

−1 −1 Corollary. All the coefficients bk l , σk l , hk l ∈ U(t glm|n[t ]) in the expansions

∞ k
X X k k−l Ber 1 + u (τ + Eb [−1]) = bk l u τ , k=0 l=0 Moreover, bk l = σk l for all k and l.

−1 −1 Corollary. All the coefficients bk l , σk l , hk l ∈ U(t glm|n[t ]) in the expansions

∞ k
X X k k−l Ber 1 + u (τ + Eb [−1]) = bk l u τ , k=0 l=0

k k−1 str Ak T1 ... Tk = σk 0τ + σk1τ + ··· + σkk ,

k k−1 str Hk T1 ... Tk = hk 0τ + hk1τ + ··· + hkk , where T = τ + Eb [−1], are Segal–Sugawara vectors. −1 −1 Corollary. All the coefficients bk l , σk l , hk l ∈ U(t glm|n[t ]) in the expansions

∞ k
X X k k−l Ber 1 + u (τ + Eb [−1]) = bk l u τ , k=0 l=0

k k−1 str Ak T1 ... Tk = σk 0τ + σk1τ + ··· + σkk ,

k k−1 str Hk T1 ... Tk = hk 0τ + hk1τ + ··· + hkk , where T = τ + Eb [−1], are Segal–Sugawara vectors.

Moreover, bk l = σk l for all k and l. These central elements are called Sugawara operators.

Under the state-field correspondence, τ 7→ ∂z ,

X −r−1 Y : eij [−1] 7→ eij (z) := eij [r]z . r∈Z

Sugawara operators

The application of the state-field correspondence map

−1 Y : Vn−m(glm|n) → End Vn−m(glm|n)[[z, z ]],

to the Segal–Sugawara vectors produces elements of the

center of the local completion Un−m(glb m|n)loc at the critical level. Under the state-field correspondence, τ 7→ ∂z ,

X −r−1 Y : eij [−1] 7→ eij (z) := eij [r]z . r∈Z

Sugawara operators

The application of the state-field correspondence map

−1 Y : Vn−m(glm|n) → End Vn−m(glm|n)[[z, z ]],

to the Segal–Sugawara vectors produces elements of the

center of the local completion Un−m(glb m|n)loc at the critical level.

These central elements are called Sugawara operators. X −r−1 Y : eij [−1] 7→ eij (z) := eij [r]z . r∈Z

Sugawara operators

The application of the state-field correspondence map

−1 Y : Vn−m(glm|n) → End Vn−m(glm|n)[[z, z ]],

to the Segal–Sugawara vectors produces elements of the

center of the local completion Un−m(glb m|n)loc at the critical level.

These central elements are called Sugawara operators.

Under the state-field correspondence, τ 7→ ∂z , Sugawara operators

The application of the state-field correspondence map

−1 Y : Vn−m(glm|n) → End Vn−m(glm|n)[[z, z ]],

to the Segal–Sugawara vectors produces elements of the

center of the local completion Un−m(glb m|n)loc at the critical level.

These central elements are called Sugawara operators.

Under the state-field correspondence, τ 7→ ∂z ,

X −r−1 Y : eij [−1] 7→ eij (z) := eij [r]z . r∈Z Corollary. All Fourier coefficients of the fields

skl (z), bkl (z), σkl (z) and hkl (z) defined by

k k k−1 : str T (z) : = sk 0(z) ∂z + sk1(z) ∂z + ··· + skk (z), ∞ k
X X k k−l : Ber 1 + u T (z) : = bkl (z) u ∂z , k=0 l=0

k k−1 : str Ak T1(z) ... Tk (z): = σk 0(z) ∂z + σk1(z) ∂z + ··· + σkk (z),

k k−1 : str Hk T1(z) ... Tk (z) : = hk 0(z) ∂z + hk1(z) ∂z + ··· + hkk (z)

are Sugawara operators for glb m|n. Moreover, σkl (z) = bkl (z).

¯ı Set Eb(z) = [eij (z)(−1) ] and T (z) = ∂z + Eb(z). ∞ k
X X k k−l : Ber 1 + u T (z) : = bkl (z) u ∂z , k=0 l=0

k k−1 : str Ak T1(z) ... Tk (z): = σk 0(z) ∂z + σk1(z) ∂z + ··· + σkk (z),

k k−1 : str Hk T1(z) ... Tk (z) : = hk 0(z) ∂z + hk1(z) ∂z + ··· + hkk (z)

are Sugawara operators for glb m|n. Moreover, σkl (z) = bkl (z).

¯ı Set Eb(z) = [eij (z)(−1) ] and T (z) = ∂z + Eb(z).

Corollary. All Fourier coefficients of the fields skl (z), bkl (z), σkl (z) and hkl (z) defined by

k k k−1 : str T (z) : = sk 0(z) ∂z + sk1(z) ∂z + ··· + skk (z), k k−1 : str Ak T1(z) ... Tk (z): = σk 0(z) ∂z + σk1(z) ∂z + ··· + σkk (z),

k k−1 : str Hk T1(z) ... Tk (z) : = hk 0(z) ∂z + hk1(z) ∂z + ··· + hkk (z)

are Sugawara operators for glb m|n. Moreover, σkl (z) = bkl (z).

¯ı Set Eb(z) = [eij (z)(−1) ] and T (z) = ∂z + Eb(z).

Corollary. All Fourier coefficients of the fields skl (z), bkl (z), σkl (z) and hkl (z) defined by

k k k−1 : str T (z) : = sk 0(z) ∂z + sk1(z) ∂z + ··· + skk (z), ∞ k
X X k k−l : Ber 1 + u T (z) : = bkl (z) u ∂z , k=0 l=0 Moreover, σkl (z) = bkl (z).

¯ı Set Eb(z) = [eij (z)(−1) ] and T (z) = ∂z + Eb(z).

Corollary. All Fourier coefficients of the fields skl (z), bkl (z), σkl (z) and hkl (z) defined by

k k k−1 : str T (z) : = sk 0(z) ∂z + sk1(z) ∂z + ··· + skk (z), ∞ k
X X k k−l : Ber 1 + u T (z) : = bkl (z) u ∂z , k=0 l=0

k k−1 : str Ak T1(z) ... Tk (z): = σk 0(z) ∂z + σk1(z) ∂z + ··· + σkk (z),

k k−1 : str Hk T1(z) ... Tk (z) : = hk 0(z) ∂z + hk1(z) ∂z + ··· + hkk (z) are Sugawara operators for glb m|n. ¯ı Set Eb(z) = [eij (z)(−1) ] and T (z) = ∂z + Eb(z).

Corollary. All Fourier coefficients of the fields skl (z), bkl (z), σkl (z) and hkl (z) defined by

k k k−1 : str T (z) : = sk 0(z) ∂z + sk1(z) ∂z + ··· + skk (z), ∞ k
X X k k−l : Ber 1 + u T (z) : = bkl (z) u ∂z , k=0 l=0

k k−1 : str Ak T1(z) ... Tk (z): = σk 0(z) ∂z + σk1(z) ∂z + ··· + σkk (z),

k k−1 : str Hk T1(z) ... Tk (z) : = hk 0(z) ∂z + hk1(z) ∂z + ··· + hkk (z) are Sugawara operators for glb m|n. Moreover, σkl (z) = bkl (z). ¯ı L(z) = ∂z − Eb(z)−, Eb(z)− = [(−1) eij (z)−].

Corollary. All coefficients of the series Skl (z) and Bkl (z) in

k k k−1 str L(z) = Sk 0(z) ∂z + Sk1(z) ∂z + ··· + Skk (z), ∞ k
X X k k−l Ber 1 + u L(z) = Bkl (z) u ∂z , k=0 l=0

generate a commutative subalgebra of U(glm|n[t]).

Gaudin Hamiltonians

∞ X −r−1 Set eij (z)− = eij [r]z , r=0 Corollary. All coefficients of the series Skl (z) and Bkl (z) in

k k k−1 str L(z) = Sk 0(z) ∂z + Sk1(z) ∂z + ··· + Skk (z), ∞ k
X X k k−l Ber 1 + u L(z) = Bkl (z) u ∂z , k=0 l=0

generate a commutative subalgebra of U(glm|n[t]).

Gaudin Hamiltonians

∞ X −r−1 Set eij (z)− = eij [r]z , r=0

¯ı L(z) = ∂z − Eb(z)−, Eb(z)− = [(−1) eij (z)−]. Gaudin Hamiltonians

∞ X −r−1 Set eij (z)− = eij [r]z , r=0

¯ı L(z) = ∂z − Eb(z)−, Eb(z)− = [(−1) eij (z)−].

Corollary. All coefficients of the series Skl (z) and Bkl (z) in

k k k−1 str L(z) = Sk 0(z) ∂z + Sk1(z) ∂z + ··· + Skk (z), ∞ k
X X k k−l Ber 1 + u L(z) = Bkl (z) u ∂z , k=0 l=0

generate a commutative subalgebra of U(glm|n[t]).   ∂z − e (z)− −e (z)− ... −e (z)−  11 12 1m       −e21(z)− ∂z − e22(z)− ... −e2m(z)−  cdet   .  . . . .   . . .. .      −em1(z)− −em2(z)− . . . ∂z − emm(z)−

This yields the commutative subalgebras of U(glm[t]) first discovered by Talalaev, 2006.

Example. In the case n = 0 the Berezinian turns into the column determinant This yields the commutative subalgebras of U(glm[t]) first discovered by Talalaev, 2006.

Example. In the case n = 0 the Berezinian turns into the column determinant

  ∂z − e (z)− −e (z)− ... −e (z)−  11 12 1m       −e21(z)− ∂z − e22(z)− ... −e2m(z)−  cdet   .  . . . .   . . .. .      −em1(z)− −em2(z)− . . . ∂z − emm(z)− Example. In the case n = 0 the Berezinian turns into the column determinant

  ∂z − e (z)− −e (z)− ... −e (z)−  11 12 1m       −e21(z)− ∂z − e22(z)− ... −e2m(z)−  cdet   .  . . . .   . . .. .      −em1(z)− −em2(z)− . . . ∂z − emm(z)−

This yields the commutative subalgebras of U(glm[t]) first discovered by Talalaev, 2006. The tensor product

M(1) ⊗ ... ⊗ M(k)

becomes a glm|n[t]-module, where the images of the matrix

elements of the matrix L(z) = ∂z − Eb(z)− are found by

k e(r) ¯ı X ij `ij (z) = δij ∂z − (−1) , z − ar r=1

(r) (r) and eij denotes the image of eij in the glm|n-module M .

(1) (k) Consider finite-dimensional glm|n-modules M ,..., M and let a1,..., ak be complex parameters. (1) (k) Consider finite-dimensional glm|n-modules M ,..., M and let a1,..., ak be complex parameters. The tensor product

M(1) ⊗ ... ⊗ M(k)

becomes a glm|n[t]-module, where the images of the matrix elements of the matrix L(z) = ∂z − Eb(z)− are found by

k e(r) ¯ı X ij `ij (z) = δij ∂z − (−1) , z − ar r=1

(r) (r) and eij denotes the image of eij in the glm|n-module M . Corollary.

Higher Gaudin Hamiltonians associated with glm|n are provided by the coefficients of the Berezinian Ber 1 + u L(z)

and the supertrace

k k k−1 str L(z) = Sk 0(z) ∂z + Sk1(z) ∂z + ··· + Skk (z).

Set L(z) = [`ij (z)]. and the supertrace

k k k−1 str L(z) = Sk 0(z) ∂z + Sk1(z) ∂z + ··· + Skk (z).

Set L(z) = [`ij (z)].

Corollary.

Higher Gaudin Hamiltonians associated with glm|n are provided by the coefficients of the Berezinian Ber 1 + u L(z)
Set L(z) = [`ij (z)].

Corollary.

Higher Gaudin Hamiltonians associated with glm|n are provided by the coefficients of the Berezinian Ber 1 + u L(z)
and the supertrace

k k k−1 str L(z) = Sk 0(z) ∂z + Sk1(z) ∂z + ··· + Skk (z). where the ai are assumed to be distinct,

(r) X 1 X (r) (s) ¯ H = eij eji (−1) ar − as s6=r i,j

and ∆(r) denotes the eigenvalue of the Casimir element

P ¯ P (r) eij eji (−1) + eii of glm|n in the representation M .

Example.

The quadratic Gaudin Hamiltonian H(z) = S22(z) is given by

k k X H(r) X ∆(r) H( ) = + , z 2 2 z − ar (z − ar ) r=1 r=1 and ∆(r) denotes the eigenvalue of the Casimir element

P ¯ P (r) eij eji (−1) + eii of glm|n in the representation M .

Example.

The quadratic Gaudin Hamiltonian H(z) = S22(z) is given by

k k X H(r) X ∆(r) H( ) = + , z 2 2 z − ar (z − ar ) r=1 r=1 where the ai are assumed to be distinct,

(r) X 1 X (r) (s) ¯ H = eij eji (−1) ar − as s6=r i,j Example.

The quadratic Gaudin Hamiltonian H(z) = S22(z) is given by

k k X H(r) X ∆(r) H( ) = + , z 2 2 z − ar (z − ar ) r=1 r=1 where the ai are assumed to be distinct,

(r) X 1 X (r) (s) ¯ H = eij eji (−1) ar − as s6=r i,j and ∆(r) denotes the eigenvalue of the Casimir element

P ¯ P (r) eij eji (−1) + eii of glm|n in the representation M .