Manin matrices, Casimir elements and

Sugawara operators

Alexander Molev

University of Sydney

1 I Origins and motivations.

I Basic properties of Manin matrices.

I Applications:

I Casimir elements for gln.

I Segal–Sugawara vectors for gln.

Plan

2 I Basic properties of Manin matrices.

I Applications:

I Casimir elements for gln.

I Segal–Sugawara vectors for gln.

Plan

I Origins and motivations.

2 I Applications:

I Casimir elements for gln.

I Segal–Sugawara vectors for gln.

Plan

I Origins and motivations.

I Basic properties of Manin matrices.

2 I Segal–Sugawara vectors for gln.

Plan

I Origins and motivations.

I Basic properties of Manin matrices.

I Applications:

I Casimir elements for gln.

2 Plan

I Origins and motivations.

I Basic properties of Manin matrices.

I Applications:

I Casimir elements for gln.

I Segal–Sugawara vectors for gln.

2 By the seminal works of V. Drinfeld (1985) and M. Jimbo (1985),

the universal enveloping algebra U(g) of a simple Lie algebra g

admits a deformation Uq(g) in the class of Hopf algebras.

The dual Hopf algebras are quantized algebras of functions

Funq(G) on the associated Lie group G

[N. Reshetikhin, L. Takhtajan and L. Faddeev 1990].

A detailed review of the theory and applications:

V. Chari and A. Pressley, A guide to quantum groups, 1994.

Quantum groups

3 The dual Hopf algebras are quantized algebras of functions

Funq(G) on the associated Lie group G

[N. Reshetikhin, L. Takhtajan and L. Faddeev 1990].

A detailed review of the theory and applications:

V. Chari and A. Pressley, A guide to quantum groups, 1994.

Quantum groups

By the seminal works of V. Drinfeld (1985) and M. Jimbo (1985),

the universal enveloping algebra U(g) of a simple Lie algebra g

admits a deformation Uq(g) in the class of Hopf algebras.

3 A detailed review of the theory and applications:

V. Chari and A. Pressley, A guide to quantum groups, 1994.

Quantum groups

By the seminal works of V. Drinfeld (1985) and M. Jimbo (1985),

the universal enveloping algebra U(g) of a simple Lie algebra g

admits a deformation Uq(g) in the class of Hopf algebras.

The dual Hopf algebras are quantized algebras of functions

Funq(G) on the associated Lie group G

[N. Reshetikhin, L. Takhtajan and L. Faddeev 1990].

3 Quantum groups

By the seminal works of V. Drinfeld (1985) and M. Jimbo (1985),

the universal enveloping algebra U(g) of a simple Lie algebra g

admits a deformation Uq(g) in the class of Hopf algebras.

The dual Hopf algebras are quantized algebras of functions

Funq(G) on the associated Lie group G

[N. Reshetikhin, L. Takhtajan and L. Faddeev 1990].

A detailed review of the theory and applications:

V. Chari and A. Pressley, A guide to quantum groups, 1994.

3 The algebra Funq(Mat2) is generated by four elements a, b, c, d, " ab # understood as the entries of the matrix , such that cd

ba = qab, dc = qcd, ca = qac, db = qbd,

and

bc = cb, ad − da + (q − q−1)bc = 0.

[L. Faddeev and L. Takhtajan 1986].

Basic example

4 ba = qab, dc = qcd, ca = qac, db = qbd,

and

bc = cb, ad − da + (q − q−1)bc = 0.

[L. Faddeev and L. Takhtajan 1986].

Basic example

The algebra Funq(Mat2) is generated by four elements a, b, c, d, " ab # understood as the entries of the matrix , such that cd

4 and

bc = cb, ad − da + (q − q−1)bc = 0.

[L. Faddeev and L. Takhtajan 1986].

Basic example

The algebra Funq(Mat2) is generated by four elements a, b, c, d, " ab # understood as the entries of the matrix , such that cd

ba = qab, dc = qcd, ca = qac, db = qbd,

4 [L. Faddeev and L. Takhtajan 1986].

Basic example

The algebra Funq(Mat2) is generated by four elements a, b, c, d, " ab # understood as the entries of the matrix , such that cd

ba = qab, dc = qcd, ca = qac, db = qbd,

and

bc = cb, ad − da + (q − q−1)bc = 0.

4 Basic example

The algebra Funq(Mat2) is generated by four elements a, b, c, d, " ab # understood as the entries of the matrix , such that cd

ba = qab, dc = qcd, ca = qac, db = qbd,

and

bc = cb, ad − da + (q − q−1)bc = 0.

[L. Faddeev and L. Takhtajan 1986].

4 A 2 × 2 matrix is q-Manin if the elements x0 and y0 defined by

" x0 # " ab #" x # = , y0 cd y

0 0 0 0 satisfy y x = qx y . The defining relations for Funq(Mat2) " ab # are equivalent to the conditions that the matrix cd and its transpose are q-Manin matrices.

As observed by Yu. Manin (1988), the relations are recovered via a “coaction” on the quantum plane: the algebra with generators x, y and the relation yx = qxy.

5 The defining relations for Funq(Mat2) " ab # are equivalent to the conditions that the matrix cd and its transpose are q-Manin matrices.

As observed by Yu. Manin (1988), the relations are recovered via a “coaction” on the quantum plane: the algebra with generators x, y and the relation yx = qxy.

A 2 × 2 matrix is q-Manin if the elements x0 and y0 defined by

" x0 # " ab #" x # = , y0 cd y satisfy y0x0 = qx0y0.

5 As observed by Yu. Manin (1988), the relations are recovered via a “coaction” on the quantum plane: the algebra with generators x, y and the relation yx = qxy.

A 2 × 2 matrix is q-Manin if the elements x0 and y0 defined by

" x0 # " ab #" x # = , y0 cd y

0 0 0 0 satisfy y x = qx y . The defining relations for Funq(Mat2) " ab # are equivalent to the conditions that the matrix cd and its transpose are q-Manin matrices.

5 Consider the characteristic polynomial of a matrix   M11 ... M1n    . . .  M =  . . .      Mn1 ... Mnn

with entries in C:

n det(I + tM) = 1 + ∆1 t + ··· + ∆n t .

In particular,

∆1 = tr M, ∆n = det M.

MacMahon Master Theorem

6 n det(I + tM) = 1 + ∆1 t + ··· + ∆n t .

In particular,

∆1 = tr M, ∆n = det M.

MacMahon Master Theorem

Consider the characteristic polynomial of a matrix   M11 ... M1n    . . .  M =  . . .      Mn1 ... Mnn

with entries in C:

6 In particular,

∆1 = tr M, ∆n = det M.

MacMahon Master Theorem

Consider the characteristic polynomial of a matrix   M11 ... M1n    . . .  M =  . . .      Mn1 ... Mnn

with entries in C:

n det(I + tM) = 1 + ∆1 t + ··· + ∆n t .

6 MacMahon Master Theorem

Consider the characteristic polynomial of a matrix   M11 ... M1n    . . .  M =  . . .      Mn1 ... Mnn

with entries in C:

n det(I + tM) = 1 + ∆1 t + ··· + ∆n t .

In particular,

∆1 = tr M, ∆n = det M.

6 Consider the algebra

n n End C ⊗ ... ⊗ End C ⊗ C | {z } k

and for a = 1,..., k set

n X Ma = I ⊗ ... ⊗ I ⊗ eij ⊗ I ⊗ ... ⊗ I ⊗ Mij. | {z } | {z } i,j=1 a−1 k−a

Regard the matrix M as the element

n X n M = eij ⊗ Mij ∈ End C ⊗ C. i,j=1

7 and for a = 1,..., k set

n X Ma = I ⊗ ... ⊗ I ⊗ eij ⊗ I ⊗ ... ⊗ I ⊗ Mij. | {z } | {z } i,j=1 a−1 k−a

Regard the matrix M as the element

n X n M = eij ⊗ Mij ∈ End C ⊗ C. i,j=1

Consider the algebra

n n End C ⊗ ... ⊗ End C ⊗ C | {z } k

7 Regard the matrix M as the element

n X n M = eij ⊗ Mij ∈ End C ⊗ C. i,j=1

Consider the algebra

n n End C ⊗ ... ⊗ End C ⊗ C | {z } k and for a = 1,..., k set

n X Ma = I ⊗ ... ⊗ I ⊗ eij ⊗ I ⊗ ... ⊗ I ⊗ Mij. | {z } | {z } i,j=1 a−1 k−a

7 Denote by H(k) and A(k) the respective images of the

symmetrizer and anti-symmetrizer

1 X 1 X σ and sgn σ · σ. k! k! σ∈Sk σ∈Sk

We regard H(k) and A(k) as elements of the algebra

n n End C ⊗ ... ⊗ End C . | {z } k

The symmetric group Sk acts on the tensor product space

n n C ⊗ ... ⊗ C | {z } k by permutations of tensor factors.

8 We regard H(k) and A(k) as elements of the algebra

n n End C ⊗ ... ⊗ End C . | {z } k

The symmetric group Sk acts on the tensor product space

n n C ⊗ ... ⊗ C | {z } k by permutations of tensor factors.

Denote by H(k) and A(k) the respective images of the symmetrizer and anti-symmetrizer

1 X 1 X σ and sgn σ · σ. k! k! σ∈Sk σ∈Sk

8 The symmetric group Sk acts on the tensor product space

n n C ⊗ ... ⊗ C | {z } k by permutations of tensor factors.

Denote by H(k) and A(k) the respective images of the symmetrizer and anti-symmetrizer

1 X 1 X σ and sgn σ · σ. k! k! σ∈Sk σ∈Sk

We regard H(k) and A(k) as elements of the algebra

n n End C ⊗ ... ⊗ End C . | {z } k 8 Introduce the sums n X k (k) Ferm = 1 + (−1) tr A M1 ... Mk = det(I − M), k=1

∞ X (k) Bos = 1 + tr H M1 ... Mk. k=1

Theorem [P. MacMahon 1916]. We have the identity

Bos × Ferm = 1.

We have

(k) ∆k = tr A M1 ... Mk, k = 1,..., n.

9 Theorem [P. MacMahon 1916]. We have the identity

Bos × Ferm = 1.

We have

(k) ∆k = tr A M1 ... Mk, k = 1,..., n.

Introduce the sums n X k (k) Ferm = 1 + (−1) tr A M1 ... Mk = det(I − M), k=1

∞ X (k) Bos = 1 + tr H M1 ... Mk. k=1

9 We have

(k) ∆k = tr A M1 ... Mk, k = 1,..., n.

Introduce the sums n X k (k) Ferm = 1 + (−1) tr A M1 ... Mk = det(I − M), k=1

∞ X (k) Bos = 1 + tr H M1 ... Mk. k=1

Theorem [P. MacMahon 1916]. We have the identity

Bos × Ferm = 1.

9 Consider the tensor product algebra A ⊗ C[x, y]. Look for 2 × 2 matrices over A such that x0 and y0 defined by

" x0 # " ab #" x # = , y0 cd y commute. We have

[ax + by, cx + dy] = [a, c] x2 + [a, d] + [b, c]
xy + [b, d] y2.

This leads to the definition of Manin matrices:

[a, c] = [b, d] = 0 and [a, d] = [c, b].

Manin matrices

10 Look for 2 × 2 matrices over A such that x0 and y0 defined by

" x0 # " ab #" x # = , y0 cd y commute. We have

[ax + by, cx + dy] = [a, c] x2 + [a, d] + [b, c]
xy + [b, d] y2.

This leads to the definition of Manin matrices:

[a, c] = [b, d] = 0 and [a, d] = [c, b].

Manin matrices

Consider the tensor product algebra A ⊗ C[x, y].

10 We have

[ax + by, cx + dy] = [a, c] x2 + [a, d] + [b, c]
xy + [b, d] y2.

This leads to the definition of Manin matrices:

[a, c] = [b, d] = 0 and [a, d] = [c, b].

Manin matrices

Consider the tensor product algebra A ⊗ C[x, y]. Look for 2 × 2 matrices over A such that x0 and y0 defined by

" x0 # " ab #" x # = , y0 cd y commute.

10 This leads to the definition of Manin matrices:

[a, c] = [b, d] = 0 and [a, d] = [c, b].

Manin matrices

Consider the tensor product algebra A ⊗ C[x, y]. Look for 2 × 2 matrices over A such that x0 and y0 defined by

" x0 # " ab #" x # = , y0 cd y commute. We have

[ax + by, cx + dy] = [a, c] x2 + [a, d] + [b, c]
xy + [b, d] y2.

10 Manin matrices

Consider the tensor product algebra A ⊗ C[x, y]. Look for 2 × 2 matrices over A such that x0 and y0 defined by

" x0 # " ab #" x # = , y0 cd y commute. We have

[ax + by, cx + dy] = [a, c] x2 + [a, d] + [b, c]
xy + [b, d] y2.

This leads to the definition of Manin matrices:

[a, c] = [b, d] = 0 and [a, d] = [c, b].

10     Mij, Mk l = Mk j, Mil , i, j, k, l ∈ {1,..., n}.

Equivalently, using the permutation operator

n n P ∈ End C ⊗ End C , P : ξ ⊗ η 7→ η ⊗ ξ

we find that M is a Manin matrix if and only if

(1 − P) M1 M2 (1 + P) = 0

n n in the algebra End C ⊗ End C ⊗ A.

Definition. An n × n matrix M over an associative algebra A is a Manin matrix if all its 2 × 2 submatrices are Manin matrices:

11 Equivalently, using the permutation operator

n n P ∈ End C ⊗ End C , P : ξ ⊗ η 7→ η ⊗ ξ

we find that M is a Manin matrix if and only if

(1 − P) M1 M2 (1 + P) = 0

n n in the algebra End C ⊗ End C ⊗ A.

Definition. An n × n matrix M over an associative algebra A is a Manin matrix if all its 2 × 2 submatrices are Manin matrices:

    Mij, Mk l = Mk j, Mil , i, j, k, l ∈ {1,..., n}.

11 we find that M is a Manin matrix if and only if

(1 − P) M1 M2 (1 + P) = 0

n n in the algebra End C ⊗ End C ⊗ A.

Definition. An n × n matrix M over an associative algebra A is a Manin matrix if all its 2 × 2 submatrices are Manin matrices:

    Mij, Mk l = Mk j, Mil , i, j, k, l ∈ {1,..., n}.

Equivalently, using the permutation operator

n n P ∈ End C ⊗ End C , P : ξ ⊗ η 7→ η ⊗ ξ

11 Definition. An n × n matrix M over an associative algebra A is a Manin matrix if all its 2 × 2 submatrices are Manin matrices:

    Mij, Mk l = Mk j, Mil , i, j, k, l ∈ {1,..., n}.

Equivalently, using the permutation operator

n n P ∈ End C ⊗ End C , P : ξ ⊗ η 7→ η ⊗ ξ we find that M is a Manin matrix if and only if

(1 − P) M1 M2 (1 + P) = 0

n n in the algebra End C ⊗ End C ⊗ A. 11 Theorem [Garoufalidis–Lê–Zeilberger 2006].

If M is a Manin matrix, then

Bos × Ferm = 1.

For any n × n matrix M over an associative algebra set

n X k (k) Ferm = 1 + (−1) tr A M1 ... Mk, k=1

∞ X (k) Bos = 1 + tr H M1 ... Mk. k=1

12 For any n × n matrix M over an associative algebra set

n X k (k) Ferm = 1 + (−1) tr A M1 ... Mk, k=1

∞ X (k) Bos = 1 + tr H M1 ... Mk. k=1

Theorem [Garoufalidis–Lê–Zeilberger 2006].

If M is a Manin matrix, then

Bos × Ferm = 1.

12 Proposition. If M is a Manin matrix, then n X k (k) cdet(I + tM) = t tr A M1 ... Mk, k=0 ∞  −1 X k (k) cdet(I − tM) = t tr H M1 ... Mk. k=0

Proof. For any σ ∈ Sk we have

(k) (k) A M1 ... Mk = sgn σ · A M1 ... Mk Pσ.

Introduce the column-determinant of a matrix M by

X cdet M = sgn σ · Mσ(1)1 ... Mσ(n)n. σ∈Sn

13 Proof. For any σ ∈ Sk we have

(k) (k) A M1 ... Mk = sgn σ · A M1 ... Mk Pσ.

Introduce the column-determinant of a matrix M by

X cdet M = sgn σ · Mσ(1)1 ... Mσ(n)n. σ∈Sn

Proposition. If M is a Manin matrix, then n X k (k) cdet(I + tM) = t tr A M1 ... Mk, k=0 ∞  −1 X k (k) cdet(I − tM) = t tr H M1 ... Mk. k=0

13 Introduce the column-determinant of a matrix M by

X cdet M = sgn σ · Mσ(1)1 ... Mσ(n)n. σ∈Sn

Proposition. If M is a Manin matrix, then n X k (k) cdet(I + tM) = t tr A M1 ... Mk, k=0 ∞  −1 X k (k) cdet(I − tM) = t tr H M1 ... Mk. k=0

Proof. For any σ ∈ Sk we have

(k) (k) A M1 ... Mk = sgn σ · A M1 ... Mk Pσ.

13 n n−1 n I Set C(u) = cdet(uI − M) = u − ∆1 u + ··· + (−1) ∆n. The Cayley–Hamilton identity holds: C(M) = 0.

I If M and cdet M are invertible, them M−1 is a Manin matrix. Moreover, cdet(M−1) = (cdet M)−1.

Further properties and generalizations: [Chervov, Falqui,

Foata, Han, M., Ragoucy, Rubtsov, Silantyev, . . . 2007–2020].

I The Newton identity holds:

∞ d X cdet(I + tM) = cdet(I + tM) (−t)k tr M k+1. dt k=0

14 The Cayley–Hamilton identity holds: C(M) = 0.

I If M and cdet M are invertible, them M−1 is a Manin matrix. Moreover, cdet(M−1) = (cdet M)−1.

Further properties and generalizations: [Chervov, Falqui,

Foata, Han, M., Ragoucy, Rubtsov, Silantyev, . . . 2007–2020].

I The Newton identity holds:

∞ d X cdet(I + tM) = cdet(I + tM) (−t)k tr M k+1. dt k=0

n n−1 n I Set C(u) = cdet(uI − M) = u − ∆1 u + ··· + (−1) ∆n.

14 I If M and cdet M are invertible, them M−1 is a Manin matrix. Moreover, cdet(M−1) = (cdet M)−1.

Further properties and generalizations: [Chervov, Falqui,

Foata, Han, M., Ragoucy, Rubtsov, Silantyev, . . . 2007–2020].

I The Newton identity holds:

∞ d X cdet(I + tM) = cdet(I + tM) (−t)k tr M k+1. dt k=0

n n−1 n I Set C(u) = cdet(uI − M) = u − ∆1 u + ··· + (−1) ∆n. The Cayley–Hamilton identity holds: C(M) = 0.

14 Further properties and generalizations: [Chervov, Falqui,

Foata, Han, M., Ragoucy, Rubtsov, Silantyev, . . . 2007–2020].

I The Newton identity holds:

∞ d X cdet(I + tM) = cdet(I + tM) (−t)k tr M k+1. dt k=0

n n−1 n I Set C(u) = cdet(uI − M) = u − ∆1 u + ··· + (−1) ∆n. The Cayley–Hamilton identity holds: C(M) = 0.

I If M and cdet M are invertible, them M−1 is a Manin matrix. Moreover, cdet(M−1) = (cdet M)−1.

14 I The Newton identity holds:

∞ d X cdet(I + tM) = cdet(I + tM) (−t)k tr M k+1. dt k=0

n n−1 n I Set C(u) = cdet(uI − M) = u − ∆1 u + ··· + (−1) ∆n. The Cayley–Hamilton identity holds: C(M) = 0.

I If M and cdet M are invertible, them M−1 is a Manin matrix. Moreover, cdet(M−1) = (cdet M)−1.

Further properties and generalizations: [Chervov, Falqui,

Foata, Han, M., Ragoucy, Rubtsov, Silantyev, . . . 2007–2020].

14 n The Lie algebra gln is the vector space End C with the bracket

A, B = AB − BA.

The matrix units eij form its basis with the commutation relations

  eij, ekl = δkj eil − δil ekj.

−1 The group GLn acts on gln by conjugation: X 7→ gX g ,

and the action extends to the symmetric algebra S(gln) which

2 can be viewed as the algebra of polynomials in n variables Eij.

Applications: Casimir elements

15 The matrix units eij form its basis with the commutation relations

  eij, ekl = δkj eil − δil ekj.

−1 The group GLn acts on gln by conjugation: X 7→ gX g ,

and the action extends to the symmetric algebra S(gln) which

2 can be viewed as the algebra of polynomials in n variables Eij.

Applications: Casimir elements

n The Lie algebra gln is the vector space End C with the bracket

A, B = AB − BA.

15 −1 The group GLn acts on gln by conjugation: X 7→ gX g ,

and the action extends to the symmetric algebra S(gln) which

2 can be viewed as the algebra of polynomials in n variables Eij.

Applications: Casimir elements

n The Lie algebra gln is the vector space End C with the bracket

A, B = AB − BA.

The matrix units eij form its basis with the commutation relations

  eij, ekl = δkj eil − δil ekj.

15 and the action extends to the symmetric algebra S(gln) which

2 can be viewed as the algebra of polynomials in n variables Eij.

Applications: Casimir elements

n The Lie algebra gln is the vector space End C with the bracket

A, B = AB − BA.

The matrix units eij form its basis with the commutation relations

  eij, ekl = δkj eil − δil ekj.

−1 The group GLn acts on gln by conjugation: X 7→ gX g ,

15 Applications: Casimir elements

n The Lie algebra gln is the vector space End C with the bracket

A, B = AB − BA.

The matrix units eij form its basis with the commutation relations

  eij, ekl = δkj eil − δil ekj.

−1 The group GLn acts on gln by conjugation: X 7→ gX g ,

and the action extends to the symmetric algebra S(gln) which

2 can be viewed as the algebra of polynomials in n variables Eij.

15 Write

n n−1 det(u + E) = u + ∆1 u + ··· + ∆n.

We have

GLn S(gln) = C[∆1,..., ∆n].

Consider the matrix   E11 ... E1n    . . .  E =  . . .      En1 ... Enn with entries in the symmetric algebra S(gln).

16 We have

GLn S(gln) = C[∆1,..., ∆n].

Consider the matrix   E11 ... E1n    . . .  E =  . . .      En1 ... Enn with entries in the symmetric algebra S(gln). Write

n n−1 det(u + E) = u + ∆1 u + ··· + ∆n.

16 Consider the matrix   E11 ... E1n    . . .  E =  . . .      En1 ... Enn with entries in the symmetric algebra S(gln). Write

n n−1 det(u + E) = u + ∆1 u + ··· + ∆n.

We have

GLn S(gln) = C[∆1,..., ∆n].

16 The symmetrization map

∼ $ : S(gln) → U(gln),

is a GLn-module isomorphism, defined by

1 X $ : X ... X 7→ X ... X , X ∈ gl , 1 k k! σ(1) σ(k) i n σ∈Sk

[Poincaré–Birkhoff–Witt Theorem].

The universal enveloping algebra U(gln) is the associative

2 algebra with n generators Eij and the defining relations

Eij Ekl − Ekl Eij = δkj Eil − δil Ekj.

17 is a GLn-module isomorphism, defined by

1 X $ : X ... X 7→ X ... X , X ∈ gl , 1 k k! σ(1) σ(k) i n σ∈Sk

[Poincaré–Birkhoff–Witt Theorem].

The universal enveloping algebra U(gln) is the associative

2 algebra with n generators Eij and the defining relations

Eij Ekl − Ekl Eij = δkj Eil − δil Ekj.

The symmetrization map

∼ $ : S(gln) → U(gln),

17 The universal enveloping algebra U(gln) is the associative

2 algebra with n generators Eij and the defining relations

Eij Ekl − Ekl Eij = δkj Eil − δil Ekj.

The symmetrization map

∼ $ : S(gln) → U(gln),

is a GLn-module isomorphism, defined by

1 X $ : X ... X 7→ X ... X , X ∈ gl , 1 k k! σ(1) σ(k) i n σ∈Sk

[Poincaré–Birkhoff–Witt Theorem]. 17 Hence

  Z(gln) = C $(∆1), . . . , $(∆n) .

By Schur’s Lemma, any element z ∈ Z(gln) acts as scalar

multiplication in any finite-dimensional simple gln-module.

Question: What are the scalars corresponding to $(∆i)?

This implies the isomorphism

GLn ∼ S(gln) = Z(gln),

where Z(gln) is the center of U(gln).

18 By Schur’s Lemma, any element z ∈ Z(gln) acts as scalar

multiplication in any finite-dimensional simple gln-module.

Question: What are the scalars corresponding to $(∆i)?

This implies the isomorphism

GLn ∼ S(gln) = Z(gln),

where Z(gln) is the center of U(gln). Hence

  Z(gln) = C $(∆1), . . . , $(∆n) .

18 Question: What are the scalars corresponding to $(∆i)?

This implies the isomorphism

GLn ∼ S(gln) = Z(gln),

where Z(gln) is the center of U(gln). Hence

  Z(gln) = C $(∆1), . . . , $(∆n) .

By Schur’s Lemma, any element z ∈ Z(gln) acts as scalar multiplication in any finite-dimensional simple gln-module.

18 This implies the isomorphism

GLn ∼ S(gln) = Z(gln),

where Z(gln) is the center of U(gln). Hence

  Z(gln) = C $(∆1), . . . , $(∆n) .

By Schur’s Lemma, any element z ∈ Z(gln) acts as scalar multiplication in any finite-dimensional simple gln-module.

Question: What are the scalars corresponding to $(∆i)?

18 such that

Eij ξ = 0 for 1 6 i < j 6 n, and

Eii ξ = λi ξ for 1 6 i 6 n,

for certain λi ∈ C satisfying the conditions λi − λi+1 ∈ Z+.

Any element z ∈ Z(gln) acts in L by multiplying each vector by a

scalar χ(z). As a function of the parameters λi, the scalar χ(z)

is a shifted symmetric polynomial in the variables λ1, . . . , λn.

Any finite-dimensional simple gln-module L is generated by a nonzero vector ξ ∈ L

19 for certain λi ∈ C satisfying the conditions λi − λi+1 ∈ Z+.

Any element z ∈ Z(gln) acts in L by multiplying each vector by a

scalar χ(z). As a function of the parameters λi, the scalar χ(z)

is a shifted symmetric polynomial in the variables λ1, . . . , λn.

Any finite-dimensional simple gln-module L is generated by a nonzero vector ξ ∈ L such that

Eij ξ = 0 for 1 6 i < j 6 n, and

Eii ξ = λi ξ for 1 6 i 6 n,

19 Any element z ∈ Z(gln) acts in L by multiplying each vector by a

scalar χ(z). As a function of the parameters λi, the scalar χ(z)

is a shifted symmetric polynomial in the variables λ1, . . . , λn.

Any finite-dimensional simple gln-module L is generated by a nonzero vector ξ ∈ L such that

Eij ξ = 0 for 1 6 i < j 6 n, and

Eii ξ = λi ξ for 1 6 i 6 n, for certain λi ∈ C satisfying the conditions λi − λi+1 ∈ Z+.

19 As a function of the parameters λi, the scalar χ(z)

is a shifted symmetric polynomial in the variables λ1, . . . , λn.

Any finite-dimensional simple gln-module L is generated by a nonzero vector ξ ∈ L such that

Eij ξ = 0 for 1 6 i < j 6 n, and

Eii ξ = λi ξ for 1 6 i 6 n, for certain λi ∈ C satisfying the conditions λi − λi+1 ∈ Z+.

Any element z ∈ Z(gln) acts in L by multiplying each vector by a scalar χ(z).

19 Any finite-dimensional simple gln-module L is generated by a nonzero vector ξ ∈ L such that

Eij ξ = 0 for 1 6 i < j 6 n, and

Eii ξ = λi ξ for 1 6 i 6 n, for certain λi ∈ C satisfying the conditions λi − λi+1 ∈ Z+.

Any element z ∈ Z(gln) acts in L by multiplying each vector by a scalar χ(z). As a function of the parameters λi, the scalar χ(z) is a shifted symmetric polynomial in the variables λ1, . . . , λn.

19 The map χ is the Harish-Chandra isomorphism between

Z(gln) and the algebra of shifted symmetric polynomials.

Algebraically independent generators:

elementary shifted symmetric polynomials

∗ X em(λ1, . . . , λn) = λi1 (λi2 − 1) ... (λim − m + 1) i1<···

with m = 1,..., n.

The polynomial χ(z) is symmetric in the shifted variables

λ1, λ2 − 1, . . . , λn − n + 1.

20 Algebraically independent generators:

elementary shifted symmetric polynomials

∗ X em(λ1, . . . , λn) = λi1 (λi2 − 1) ... (λim − m + 1) i1<···

with m = 1,..., n.

The polynomial χ(z) is symmetric in the shifted variables

λ1, λ2 − 1, . . . , λn − n + 1.

The map χ is the Harish-Chandra isomorphism between

Z(gln) and the algebra of shifted symmetric polynomials.

20 The polynomial χ(z) is symmetric in the shifted variables

λ1, λ2 − 1, . . . , λn − n + 1.

The map χ is the Harish-Chandra isomorphism between

Z(gln) and the algebra of shifted symmetric polynomials.

Algebraically independent generators: elementary shifted symmetric polynomials

∗ X em(λ1, . . . , λn) = λi1 (λi2 − 1) ... (λim − m + 1) i1<···

20 m The Stirling number of the second kind counts the number k of partitions of the set {1,..., m} into k nonempty subsets.

m Equivalently, k! is the number of surjective functions k {1,..., m} → {1,..., k}.

Recurrence relation:

m m − 1 m − 1 = + k . k k − 1 k

Stirling numbers

21 m Equivalently, k! is the number of surjective functions k {1,..., m} → {1,..., k}.

Recurrence relation:

m m − 1 m − 1 = + k . k k − 1 k

Stirling numbers

m The Stirling number of the second kind counts the number k of partitions of the set {1,..., m} into k nonempty subsets.

21 Recurrence relation:

m m − 1 m − 1 = + k . k k − 1 k

Stirling numbers

m The Stirling number of the second kind counts the number k of partitions of the set {1,..., m} into k nonempty subsets.

m Equivalently, k! is the number of surjective functions k {1,..., m} → {1,..., k}.

21 Stirling numbers

m The Stirling number of the second kind counts the number k of partitions of the set {1,..., m} into k nonempty subsets.

m Equivalently, k! is the number of surjective functions k {1,..., m} → {1,..., k}.

Recurrence relation:

m m − 1 m − 1 = + k . k k − 1 k

21 m Stirling triangle: is in row m and column k k

1

11

131

1761

1 15 25 101

1 31 90 65 151

1 63 301 350 140 211 ......

22   Proof. Regard the matrix E = Eij as the element n X n
E = eij ⊗ Eij ∈ End C ⊗ U gln . i,j=1 Consider the algebra

n n
End C ⊗ ... ⊗ End C ⊗ U gln | {z } m Observe that

(m) $(∆m) = tr A E1 ... Em.

Theorem. For the Harish-Chandra images we have

m −1 X mnn χ : $(∆ ) 7→ e∗(λ , . . . , λ ). m k m k k 1 n k=1

23 Consider the algebra

n n
End C ⊗ ... ⊗ End C ⊗ U gln | {z } m Observe that

(m) $(∆m) = tr A E1 ... Em.

Theorem. For the Harish-Chandra images we have

m −1 X mnn χ : $(∆ ) 7→ e∗(λ , . . . , λ ). m k m k k 1 n k=1

  Proof. Regard the matrix E = Eij as the element n X n
E = eij ⊗ Eij ∈ End C ⊗ U gln . i,j=1

23 Observe that

(m) $(∆m) = tr A E1 ... Em.

Theorem. For the Harish-Chandra images we have

m −1 X mnn χ : $(∆ ) 7→ e∗(λ , . . . , λ ). m k m k k 1 n k=1

  Proof. Regard the matrix E = Eij as the element n X n
E = eij ⊗ Eij ∈ End C ⊗ U gln . i,j=1 Consider the algebra

n n
End C ⊗ ... ⊗ End C ⊗ U gln | {z } m

23 Theorem. For the Harish-Chandra images we have

m −1 X mnn χ : $(∆ ) 7→ e∗(λ , . . . , λ ). m k m k k 1 n k=1

  Proof. Regard the matrix E = Eij as the element n X n
E = eij ⊗ Eij ∈ End C ⊗ U gln . i,j=1 Consider the algebra

n n
End C ⊗ ... ⊗ End C ⊗ U gln | {z } m Observe that

(m) $(∆m) = tr A E1 ... Em. 23 in the tensor product algebra

n n End C ⊗ End C ⊗ U(gln).

±∂u Introduce the extended algebra U(gln) ⊗ C[u, e ], where

the element e∂u satisfies e∂u f (u) = f (u + 1)e∂u .

Key observation:

M = (uI + E)e−∂u

is a Manin matrix.

The defining relations of the algebra U(gln) can be written as

E1 E2 − E2 E1 = (E1 − E2)P

24 ±∂u Introduce the extended algebra U(gln) ⊗ C[u, e ], where

the element e∂u satisfies e∂u f (u) = f (u + 1)e∂u .

Key observation:

M = (uI + E)e−∂u

is a Manin matrix.

The defining relations of the algebra U(gln) can be written as

E1 E2 − E2 E1 = (E1 − E2)P in the tensor product algebra

n n End C ⊗ End C ⊗ U(gln).

24 Key observation:

M = (uI + E)e−∂u

is a Manin matrix.

The defining relations of the algebra U(gln) can be written as

E1 E2 − E2 E1 = (E1 − E2)P in the tensor product algebra

n n End C ⊗ End C ⊗ U(gln).

±∂u Introduce the extended algebra U(gln) ⊗ C[u, e ], where the element e∂u satisfies e∂u f (u) = f (u + 1)e∂u .

24 The defining relations of the algebra U(gln) can be written as

E1 E2 − E2 E1 = (E1 − E2)P in the tensor product algebra

n n End C ⊗ End C ⊗ U(gln).

±∂u Introduce the extended algebra U(gln) ⊗ C[u, e ], where the element e∂u satisfies e∂u f (u) = f (u + 1)e∂u .

Key observation:

M = (uI + E)e−∂u is a Manin matrix. 24 This implies the relation for the Capelli determinant (1890),

  u + E11 E12 ... E1n        E21 u + E22 − 1 ... E2n  cdet    . . . .   . . .. .      En1 ...... u + Enn − n + 1

(n) = tr A (u + E1)(u + E2 − 1) ... (u + En − n + 1).

The Harish-Chandra image is (u + λ1) ... (u + λn − n + 1).

Hence

(n) cdet M = tr A M1 ... Mn.

25 The Harish-Chandra image is (u + λ1) ... (u + λn − n + 1).

Hence

(n) cdet M = tr A M1 ... Mn.

This implies the relation for the Capelli determinant (1890),

  u + E11 E12 ... E1n        E21 u + E22 − 1 ... E2n  cdet    . . . .   . . .. .      En1 ...... u + Enn − n + 1

(n) = tr A (u + E1)(u + E2 − 1) ... (u + En − n + 1).

25 Hence

(n) cdet M = tr A M1 ... Mn.

This implies the relation for the Capelli determinant (1890),

  u + E11 E12 ... E1n        E21 u + E22 − 1 ... E2n  cdet    . . . .   . . .. .      En1 ...... u + Enn − n + 1

(n) = tr A (u + E1)(u + E2 − 1) ... (u + En − n + 1).

The Harish-Chandra image is (u + λ1) ... (u + λn − n + 1).

25 Using the identities for the Stirling numbers

m X m xm = x(x − 1) ... (x − k + 1), k k=1 we derive

m X m tr A(m)E ... E = tr A(m) E (E − 1) ... (E − k + 1). 1 m k 1 2 k k=1

It remains to calculate the partial traces of A(m).

Similarly,

(m) ∗ χ : tr A E1(E2 − 1) ... (Em − m + 1) 7→ em(λ1, . . . , λn).

26 we derive

m X m tr A(m)E ... E = tr A(m) E (E − 1) ... (E − k + 1). 1 m k 1 2 k k=1

It remains to calculate the partial traces of A(m).

Similarly,

(m) ∗ χ : tr A E1(E2 − 1) ... (Em − m + 1) 7→ em(λ1, . . . , λn).

Using the identities for the Stirling numbers

m X m xm = x(x − 1) ... (x − k + 1), k k=1

26 It remains to calculate the partial traces of A(m).

Similarly,

(m) ∗ χ : tr A E1(E2 − 1) ... (Em − m + 1) 7→ em(λ1, . . . , λn).

Using the identities for the Stirling numbers

m X m xm = x(x − 1) ... (x − k + 1), k k=1 we derive

m X m tr A(m)E ... E = tr A(m) E (E − 1) ... (E − k + 1). 1 m k 1 2 k k=1

26 Similarly,

(m) ∗ χ : tr A E1(E2 − 1) ... (Em − m + 1) 7→ em(λ1, . . . , λn).

Using the identities for the Stirling numbers

m X m xm = x(x − 1) ... (x − k + 1), k k=1 we derive

m X m tr A(m)E ... E = tr A(m) E (E − 1) ... (E − k + 1). 1 m k 1 2 k k=1

It remains to calculate the partial traces of A(m).

26 The affine Kac–Moody algebra bg is the central extension

−1 bg = g[t, t ] ⊕ CK

with the commutation relations for X[r] = X tr:

  X[r], Y[s] = [X, Y][r + s] + r δr,−shX, Yi K.

d Note that τ = − is a derivation of g. dt b

Feigin–Frenkel center

27 with the commutation relations for X[r] = X tr:

  X[r], Y[s] = [X, Y][r + s] + r δr,−shX, Yi K.

d Note that τ = − is a derivation of g. dt b

Feigin–Frenkel center

The affine Kac–Moody algebra bg is the central extension

−1 bg = g[t, t ] ⊕ CK

27 d Note that τ = − is a derivation of g. dt b

Feigin–Frenkel center

The affine Kac–Moody algebra bg is the central extension

−1 bg = g[t, t ] ⊕ CK

with the commutation relations for X[r] = X tr:

  X[r], Y[s] = [X, Y][r + s] + r δr,−shX, Yi K.

27 Feigin–Frenkel center

The affine Kac–Moody algebra bg is the central extension

−1 bg = g[t, t ] ⊕ CK

with the commutation relations for X[r] = X tr:

  X[r], Y[s] = [X, Y][r + s] + r δr,−shX, Yi K.

d Note that τ = − is a derivation of g. dt b

27 The Feigin–Frenkel center z(bg) is defined by

z(bg) = {v ∈ V(g) | g[t]v = 0}.

We have V(g) =∼ Ut−1g[t−1]
as vector spaces, and

−1 −1
z(bg) is a τ-invariant commutative subalgebra of U t g[t ] .

Consider the vacuum module at the critical level over bg,

V(g) = U(bg)/I, where the left ideal I is generated by g[t] and K + h∨.

28 We have V(g) =∼ Ut−1g[t−1]
as vector spaces, and

−1 −1
z(bg) is a τ-invariant commutative subalgebra of U t g[t ] .

Consider the vacuum module at the critical level over bg,

V(g) = U(bg)/I, where the left ideal I is generated by g[t] and K + h∨.

The Feigin–Frenkel center z(bg) is defined by

z(bg) = {v ∈ V(g) | g[t]v = 0}.

28 and

−1 −1
z(bg) is a τ-invariant commutative subalgebra of U t g[t ] .

Consider the vacuum module at the critical level over bg,

V(g) = U(bg)/I, where the left ideal I is generated by g[t] and K + h∨.

The Feigin–Frenkel center z(bg) is defined by

z(bg) = {v ∈ V(g) | g[t]v = 0}.

We have V(g) =∼ Ut−1g[t−1]
as vector spaces,

28 Consider the vacuum module at the critical level over bg,

V(g) = U(bg)/I, where the left ideal I is generated by g[t] and K + h∨.

The Feigin–Frenkel center z(bg) is defined by

z(bg) = {v ∈ V(g) | g[t]v = 0}.

We have V(g) =∼ Ut−1g[t−1]
as vector spaces, and

−1 −1
z(bg) is a τ-invariant commutative subalgebra of U t g[t ] .

28 As a differential algebra, it possesses free generators

S1,..., Sn (a complete set of Segal–Sugawara vectors),

where n = rank g and S1 = S.

[B. Feigin and E. Frenkel 1992, L. Rybnikov 2008].

Equivalently, z(bg) can be defined as the centralizer of the canonical Segal–Sugawara vector

d X 2 S = Xi[−1] , i=1 where X1,..., Xd is an orthonormal basis of g.

29 [B. Feigin and E. Frenkel 1992, L. Rybnikov 2008].

Equivalently, z(bg) can be defined as the centralizer of the canonical Segal–Sugawara vector

d X 2 S = Xi[−1] , i=1 where X1,..., Xd is an orthonormal basis of g.

As a differential algebra, it possesses free generators

S1,..., Sn (a complete set of Segal–Sugawara vectors), where n = rank g and S1 = S.

29 Equivalently, z(bg) can be defined as the centralizer of the canonical Segal–Sugawara vector

d X 2 S = Xi[−1] , i=1 where X1,..., Xd is an orthonormal basis of g.

As a differential algebra, it possesses free generators

S1,..., Sn (a complete set of Segal–Sugawara vectors), where n = rank g and S1 = S.

[B. Feigin and E. Frenkel 1992, L. Rybnikov 2008].

29 I The eigenvalues of the Hamiltonians on the Bethe vectors

are found from the Harish-Chandra images of S1,..., Sn.

− −
I Applying homomorphisms U t 1g[t 1] → U(g) one gets commutative subalgebras of U(g) thus solving

Vinberg’s quantization problem.

I The Segal–Sugawara vectors S1,..., Sn give rise to higher order Hamiltonians in the Gaudin model.

30 − −
I Applying homomorphisms U t 1g[t 1] → U(g) one gets commutative subalgebras of U(g) thus solving

Vinberg’s quantization problem.

I The Segal–Sugawara vectors S1,..., Sn give rise to higher order Hamiltonians in the Gaudin model.

I The eigenvalues of the Hamiltonians on the Bethe vectors

are found from the Harish-Chandra images of S1,..., Sn.

30 I The Segal–Sugawara vectors S1,..., Sn give rise to higher order Hamiltonians in the Gaudin model.

I The eigenvalues of the Hamiltonians on the Bethe vectors

are found from the Harish-Chandra images of S1,..., Sn.

− −
I Applying homomorphisms U t 1g[t 1] → U(g) one gets commutative subalgebras of U(g) thus solving

Vinberg’s quantization problem.

30 Working with the algebra

n n −1 −1
End C ⊗ ... ⊗ End C ⊗ U t gln[t ] , | {z } m write

(m)

Sm = tr A τ + E1[−1] ... τ + Em[−1] 1,

where

n X n −1 −1
E[r] = eij ⊗ Eij[r] ∈ End C ⊗ U t gln[t ] . i,j=1

Type A

31 write

(m)

Sm = tr A τ + E1[−1] ... τ + Em[−1] 1,

where

n X n −1 −1
E[r] = eij ⊗ Eij[r] ∈ End C ⊗ U t gln[t ] . i,j=1

Type A

Working with the algebra

n n −1 −1
End C ⊗ ... ⊗ End C ⊗ U t gln[t ] , | {z } m

31 where

n X n −1 −1
E[r] = eij ⊗ Eij[r] ∈ End C ⊗ U t gln[t ] . i,j=1

Type A

Working with the algebra

n n −1 −1
End C ⊗ ... ⊗ End C ⊗ U t gln[t ] , | {z } m write

(m)

Sm = tr A τ + E1[−1] ... τ + Em[−1] 1,

31 Type A

Working with the algebra

n n −1 −1
End C ⊗ ... ⊗ End C ⊗ U t gln[t ] , | {z } m write

(m)

Sm = tr A τ + E1[−1] ... τ + Em[−1] 1,

where

n X n −1 −1
E[r] = eij ⊗ Eij[r] ∈ End C ⊗ U t gln[t ] . i,j=1

31 Hence the elements Sm are also found by   τ + E11[−1] ... E1n[−1]    . . .  n n−1 cdet  . .. .  = τ + S1 τ + ··· + Sn.     En1[−1] ...τ + Enn[−1]

Theorem S1,..., Sn are free generators of z(glb n).

[A. Chervov and D. Talalaev 2006, A. Chervov and A. M. 2009].

Key lemma: M = τ I + E[−1] is a Manin matrix.

32 Theorem S1,..., Sn are free generators of z(glb n).

[A. Chervov and D. Talalaev 2006, A. Chervov and A. M. 2009].

Key lemma: M = τ I + E[−1] is a Manin matrix.

Hence the elements Sm are also found by   τ + E11[−1] ... E1n[−1]    . . .  n n−1 cdet  . .. .  = τ + S1 τ + ··· + Sn.     En1[−1] ...τ + Enn[−1]

32 Key lemma: M = τ I + E[−1] is a Manin matrix.

Hence the elements Sm are also found by   τ + E11[−1] ... E1n[−1]    . . .  n n−1 cdet  . .. .  = τ + S1 τ + ··· + Sn.     En1[−1] ...τ + Enn[−1]

Theorem S1,..., Sn are free generators of z(glb n).

[A. Chervov and D. Talalaev 2006, A. Chervov and A. M. 2009].

32 where the parts of partitions λ are λ1 > ··· > λ` > 0 and

cλ is the number of permutations of {1,..., m} of cycle type λ.

d Eliminate τ = − to get dt

(m)

Sm = tr A τ + E1[−1] ... τ + Em[−1] 1

X (m) = cλ tr A E1[−λ1] ... E`[−λ`], λ`m

33 cλ is the number of permutations of {1,..., m} of cycle type λ.

d Eliminate τ = − to get dt

(m)

Sm = tr A τ + E1[−1] ... τ + Em[−1] 1

X (m) = cλ tr A E1[−λ1] ... E`[−λ`], λ`m

where the parts of partitions λ are λ1 > ··· > λ` > 0 and

33 d Eliminate τ = − to get dt

(m)

Sm = tr A τ + E1[−1] ... τ + Em[−1] 1

X (m) = cλ tr A E1[−λ1] ... E`[−λ`], λ`m

where the parts of partitions λ are λ1 > ··· > λ` > 0 and

cλ is the number of permutations of {1,..., m} of cycle type λ.

33 × ∗ Applications [L. Rybnikov 2006]: for any z ∈ C and µ ∈ gln

the image of z(glb n) under the homomorphism

−1 −1
r % µ,z : U t gln[t ] → U(gln), X[r] 7→ X z + δr,−1 µ(X),

is a commutative subalgebra Aµ of U(gln) independent of z.

Theorem [O. Yakimova 2019, A. M. 2020].

The Segal–Sugawara vectors are given by

−1 X NN S = c tr A(`)E [−λ ] ... E [−λ ]. m m ` λ 1 1 ` ` λ`m

34 Theorem [O. Yakimova 2019, A. M. 2020].

The Segal–Sugawara vectors are given by

−1 X NN S = c tr A(`)E [−λ ] ... E [−λ ]. m m ` λ 1 1 ` ` λ`m

× ∗ Applications [L. Rybnikov 2006]: for any z ∈ C and µ ∈ gln the image of z(glb n) under the homomorphism

−1 −1
r % µ,z : U t gln[t ] → U(gln), X[r] 7→ X z + δr,−1 µ(X),

is a commutative subalgebra Aµ of U(gln) independent of z.

34