Higher Sugawara operators
Alexander Molev
University of Sydney
Plan of the talk
Plan of the talk
I
Casimir elements for the classical Lie algebras from
the Schur–Weyl duality.
Plan of the talk
I
Casimir elements for the classical Lie algebras from
the Schur–Weyl duality.
I
Affine Kac–Moody algebras: center at the critical level.
Symmetric group Sm
Let Sm denote the group of permutations of the set {1, . . . , m}.
Symmetric group Sm
Let Sm denote the group of permutations of the set {1, . . . , m}.
Let sab denote the transposition (a, b) for a < b.
Symmetric group Sm
Let Sm denote the group of permutations of the set {1, . . . , m}.
Let sab denote the transposition (a, b) for a < b.
The symmetrizer is the element
h(m) =
1 X
s ∈ C[Sm ].
m!
s∈Sm
Symmetric group Sm
Let Sm denote the group of permutations of the set {1, . . . , m}.
Let sab denote the transposition (a, b) for a < b.
The symmetrizer is the element
h(m) =
1 X
s ∈ C[Sm ].
m!
s∈Sm
The anti-symmetrizer is the element
a(m) =
1 X
sgn s · s ∈ C[Sm ].
m!
s∈Sm
Fusion formulas
Fusion formulas
Theorem [Jucys 1966].
The following factorization formulas hold:
h(m) =
1
m!
Y
16a<b6m
1+
sab ,
b−a
Fusion formulas
Theorem [Jucys 1966].
The following factorization formulas hold:
h(m) =
a(m) =
1
m!
1
m!
Y
1+
sab ,
b−a
1−
sab ,
b−a
16a<b6m
Y
16a<b6m
Fusion formulas
Theorem [Jucys 1966].
The following factorization formulas hold:
h(m) =
a(m) =
1
m!
1
m!
Y
1+
sab ,
b−a
1−
sab ,
b−a
16a<b6m
Y
16a<b6m
where both products are taken in the lexicographical order on
the set of pairs (a, b).
Action of Sm in tensors
Action of Sm in tensors
The symmetric group Sm acts in the tensor space
N
. . ⊗ C N}
|C ⊗ .{z
m
by the rule
sa b 7→ Pa b ,
1 6 a < b 6 m,
Action of Sm in tensors
The symmetric group Sm acts in the tensor space
N
. . ⊗ C N}
|C ⊗ .{z
m
by the rule
sa b 7→ Pa b ,
1 6 a < b 6 m,
where Pa b is the permutation operator
Pa b =
N
X
i,j=1
1⊗(a−1) ⊗ ei j ⊗ 1⊗(b−a−1) ⊗ ej i ⊗ 1⊗(m−b)
Action of Sm in tensors
The symmetric group Sm acts in the tensor space
N
. . ⊗ C N}
|C ⊗ .{z
m
by the rule
sa b 7→ Pa b ,
1 6 a < b 6 m,
where Pa b is the permutation operator
Pa b =
N
X
1⊗(a−1) ⊗ ei j ⊗ 1⊗(b−a−1) ⊗ ej i ⊗ 1⊗(m−b)
i,j=1
and ei j ∈ End C N are the matrix units.
The symmetrizer and anti-symmetrizer act as the operators
H (m) =
1
m!
Y
16a<b6m
1+
Pab b−a
The symmetrizer and anti-symmetrizer act as the operators
H (m) =
1
m!
Y
1+
Pab b−a
1−
Pab b−a
16a<b6m
and
A(m) =
1
m!
Y
16a<b6m
The symmetrizer and anti-symmetrizer act as the operators
H (m) =
1
m!
Y
1+
Pab b−a
1−
Pab b−a
16a<b6m
and
A(m) =
1
m!
Y
16a<b6m
which we regard as elements of the algebra
N
End (C N )⊗ m ∼
. . ⊗ End C N} .
= End
| C ⊗ .{z
m
Lie algebra glN
Lie algebra glN
We let Eij be the standard basis elements of glN .
Lie algebra glN
We let Eij be the standard basis elements of glN .
The universal enveloping algebra U(glN ) is the associative
algebra generated by the N 2 elements Eij subject to the defining
relations
Eij Ekl − Ekl Eij = δkj Eil − δil Ekl .
Lie algebra glN
We let Eij be the standard basis elements of glN .
The universal enveloping algebra U(glN ) is the associative
algebra generated by the N 2 elements Eij subject to the defining
relations
Eij Ekl − Ekl Eij = δkj Eil − δil Ekl .
We will combine the generators into the matrix E = [Eij ] which
will also be regarded as the element
E=
N
X
i,j=1
eij ⊗ Eij ∈ End C N ⊗ U(glN ).
Consider the algebra
N
. . ⊗ End C N} ⊗ U(glN )
|End C ⊗ .{z
m
Consider the algebra
N
. . ⊗ End C N} ⊗ U(glN )
|End C ⊗ .{z
m
and for a = 1, . . . , m introduce its elements by
Ea =
N
X
i,j=1
1⊗(a−1) ⊗ eij ⊗ 1⊗(m−a) ⊗ Eij .
Consider the algebra
N
. . ⊗ End C N} ⊗ U(glN )
|End C ⊗ .{z
m
and for a = 1, . . . , m introduce its elements by
Ea =
N
X
1⊗(a−1) ⊗ eij ⊗ 1⊗(m−a) ⊗ Eij .
i,j=1
Note the property Pa b Ea = Eb Pa b .
Consider the algebra
N
. . ⊗ End C N} ⊗ U(glN )
|End C ⊗ .{z
m
and for a = 1, . . . , m introduce its elements by
Ea =
N
X
1⊗(a−1) ⊗ eij ⊗ 1⊗(m−a) ⊗ Eij .
i,j=1
Note the property Pa b Ea = Eb Pa b .
Key Lemma. The defining relations of U(glN ) are equivalent
to the single relation
E1 E2 − E2 E1 = (E1 − E2 )P12 .
The trace is the linear map End C N → C
defined by tr : eij 7→ δij .
The trace is the linear map End C N → C
defined by tr : eij 7→ δij .
The partial trace tra acts on the a-th copy of End C N in
N
End
. . ⊗ End C N} ⊗ U(glN ).
| C ⊗ .{z
m
The trace is the linear map End C N → C
defined by tr : eij 7→ δij .
The partial trace tra acts on the a-th copy of End C N in
N
End
. . ⊗ End C N} ⊗ U(glN ).
| C ⊗ .{z
m
Theorem. For any s ∈ C[Sm ] and u1 , . . . , um ∈ C the element
tr1,...,m S (u1 + E1 ) . . . (um + Em )
belongs to the center Z(glN ) of U(glN ).
Proof. Consider the tensor product
End C N ⊗ End (C N )⊗m ⊗ U(glN )
with the copies of the algebra End C N labelled by 0, 1, . . . , m.
Proof. Consider the tensor product
End C N ⊗ End (C N )⊗m ⊗ U(glN )
with the copies of the algebra End C N labelled by 0, 1, . . . , m.
We will show that
E0 , tr1,...,m S (u1 + E1 ) . . . (um + Em ) = 0.
Proof. Consider the tensor product
End C N ⊗ End (C N )⊗m ⊗ U(glN )
with the copies of the algebra End C N labelled by 0, 1, . . . , m.
We will show that
E0 , tr1,...,m S (u1 + E1 ) . . . (um + Em ) = 0.
By the Key Lemma,
[E0 , ua + Ea ] = P0 a (ua + Ea ) − (ua + Ea )P0 a ,
where we used the relations Pa b Eb = Ea Pa b .
Hence
E0 , S (u1 + E1 ) . . . (um + Em )
=S
m
X
P0 a (u1 + E1 ) . . . (um + Em )
a=1
− S (u1 + E1 ) . . . (um + Em )
m
X
P0 a ,
a=1
because E0 S = SE0 and P0 a commutes with Eb for b 6= a.
Hence
E0 , S (u1 + E1 ) . . . (um + Em )
=S
m
X
P0 a (u1 + E1 ) . . . (um + Em )
a=1
− S (u1 + E1 ) . . . (um + Em )
m
X
P0 a ,
a=1
because E0 S = SE0 and P0 a commutes with Eb for b 6= a.
The sum of the permutation operators P0 a commutes with S
(the Schur–Weyl duality). Applying the trace tr1,...,m and using
its cyclic property we get 0.
Example: Capelli determinant.
Example: Capelli determinant.
Take m = N and introduce the Capelli determinant by
C(u) = tr1,...,N A(N) (u + E1 ) . . . (u + EN − N + 1).
Example: Capelli determinant.
Take m = N and introduce the Capelli determinant by
C(u) = tr1,...,N A(N) (u + E1 ) . . . (u + EN − N + 1).
Then C(u) coincides with the column-determinant


E12
...
E1N

u + E11




u + E22 − 1 . . .
E2N

 E21
.
C(u) = cdet 

 .
..
..

 ..
.
.




EN1
EN2
. . . u + ENN − N + 1
Example: Capelli determinant.
Take m = N and introduce the Capelli determinant by
C(u) = tr1,...,N A(N) (u + E1 ) . . . (u + EN − N + 1).
Then C(u) coincides with the column-determinant


E12
...
E1N

u + E11




u + E22 − 1 . . .
E2N

 E21
.
C(u) = cdet 

 .
..
..

 ..
.
.




EN1
EN2
. . . u + ENN − N + 1
All coefficients of the polynomial C(u) are Casimir elements.
Indeed, observe that by the Key Lemma
1−
Pab (u + Ea − a + 1)(u + Eb − b + 1)
b−a
Pab .
= (u + Eb − b + 1)(u + Ea − a + 1) 1 −
b−a
Indeed, observe that by the Key Lemma
1−
Pab (u + Ea − a + 1)(u + Eb − b + 1)
b−a
Pab .
= (u + Eb − b + 1)(u + Ea − a + 1) 1 −
b−a
Hence, the fusion formula for A(N) gives
A(N) (u + E1 ) . . . (u + EN − N + 1) = (u + EN − N + 1) . . . (u + E1 ) A(N)
Indeed, observe that by the Key Lemma
1−
Pab (u + Ea − a + 1)(u + Eb − b + 1)
b−a
Pab .
= (u + Eb − b + 1)(u + Ea − a + 1) 1 −
b−a
Hence, the fusion formula for A(N) gives
A(N) (u + E1 ) . . . (u + EN − N + 1) = (u + EN − N + 1) . . . (u + E1 ) A(N)
and that this equals A(N) C(u).
Indeed, observe that by the Key Lemma
1−
Pab (u + Ea − a + 1)(u + Eb − b + 1)
b−a
Pab .
= (u + Eb − b + 1)(u + Ea − a + 1) 1 −
b−a
Hence, the fusion formula for A(N) gives
A(N) (u + E1 ) . . . (u + EN − N + 1) = (u + EN − N + 1) . . . (u + E1 ) A(N)
and that this equals A(N) C(u).
It remains to note that tr1,...,N A(N) = 1.
Example: Gelfand invariants.
Example: Gelfand invariants.
Take s = (m m − 1 . . . 1) in the Theorem. Then
S = Pm−1 m . . . P2 3 P1 2 .
Example: Gelfand invariants.
Take s = (m m − 1 . . . 1) in the Theorem. Then
S = Pm−1 m . . . P2 3 P1 2 .
We get the Casimir elements (Gelfand invariants):
tr1,...,m S E1 . . . Em = tr Em .
Example: Gelfand invariants.
Take s = (m m − 1 . . . 1) in the Theorem. Then
S = Pm−1 m . . . P2 3 P1 2 .
We get the Casimir elements (Gelfand invariants):
tr1,...,m S E1 . . . Em = tr Em .
For instance, for m = 2 we get
tr1,2 P12 E1 E2 = tr1,2 E2 P12 E2 = tr E2
because tr1 P12 = 1.
The Newton identity
Theorem [Perelomov–Popov, 1966].
We have the identity
1+
∞
X
m=0
(−1)m tr Em
C(u + 1)
.
=
(u − N + 1)m+1
C(u)
The Newton identity
Theorem [Perelomov–Popov, 1966].
We have the identity
1+
∞
X
m=0
(−1)m tr Em
C(u + 1)
.
=
(u − N + 1)m+1
C(u)
Proof. Verify
tr1,...,N A(N) (u + E1 ) . . . (u + EN−1 − N + 2)(u + EN + 1) = C(u + 1).
The Newton identity
Theorem [Perelomov–Popov, 1966].
We have the identity
1+
∞
X
m=0
(−1)m tr Em
C(u + 1)
.
=
(u − N + 1)m+1
C(u)
Proof. Verify
tr1,...,N A(N) (u + E1 ) . . . (u + EN−1 − N + 2)(u + EN + 1) = C(u + 1).
Hence,
C(u + 1) − C(u) = N tr1,...,N A(N) (u + E1 ) . . . (u + EN−1 − N + 2)
= N tr1,...,N A(N) C(u) (u + EN − N + 1)−1 .
Harish-Chandra isomorphism
Harish-Chandra isomorphism
Given an N-tuple of complex numbers λ = (λ1 , . . . , λN ), the
corresponding irreducible highest weight representation L(λ) of
the Lie algebra glN is generated by a nonzero vector ξ ∈ L(λ)
(the highest vector) such that
Harish-Chandra isomorphism
Given an N-tuple of complex numbers λ = (λ1 , . . . , λN ), the
corresponding irreducible highest weight representation L(λ) of
the Lie algebra glN is generated by a nonzero vector ξ ∈ L(λ)
(the highest vector) such that
Eij ξ = 0
for
1 6 i < j 6 N,
Eii ξ = λi ξ
for
1 6 i 6 N.
and
Any element z ∈ Z(glN ) acts in L(λ) by multiplying each vector
by a scalar χ(z).
Any element z ∈ Z(glN ) 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 l1 , . . . , lN , where
li = λi − i + 1.
Any element z ∈ Z(glN ) 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 l1 , . . . , lN , where
li = λi − i + 1.
The mapping z 7→ χ(z) defines an algebra isomorphism
χ : Z(glN ) → C[l1 , . . . , lN ]SN
known as the Harish-Chandra isomorphism.
Example. Under the Harish-Chandra isomorphism we have
χ : C(u) 7→ (u + l1 ) . . . (u + lN ),
li = λi − i + 1.
Example. Under the Harish-Chandra isomorphism we have
χ : C(u) 7→ (u + l1 ) . . . (u + lN ),
li = λi − i + 1.
This is immediate from the definition
C(u) =
X
σ∈SN
sgn σ · (u + E)σ(1)1 . . . (u + E − N + 1)σ(N)N .
Example. Under the Harish-Chandra isomorphism we have
χ : C(u) 7→ (u + l1 ) . . . (u + lN ),
li = λi − i + 1.
This is immediate from the definition
C(u) =
X
sgn σ · (u + E)σ(1)1 . . . (u + E − N + 1)σ(N)N .
σ∈SN
By the Newton formula, the Harish-Chandra images of the
Gelfand invariants are found by
∞
N
Y
X
u + li + 1
(−1)m χ(tr Em )
=
1+
.
(u − N + 1)m+1
u + li
m=0
i=1
Brauer algebra Bm (ω)
Brauer algebra Bm (ω)
Multiplication of m-diagrams (m = 8):
1 2 3
r r
r
H HH
r r Hr
4
r
r
5 6 7 8
r r r
r
HH
H
r r r Hr
Brauer algebra Bm (ω)
Multiplication of m-diagrams (m = 8):
1 2 3
r
r
r
H HH
r
r Hr
r
r
r
r
4
r
r
r r
@ r @r
5 6 7 8
r r r
r
HH
H
r r r Hr
r
r
r
r
r
r
@
r @r
Brauer algebra Bm (ω)
Multiplication of m-diagrams (m = 8):
1 2 3
r
r
r
H HH
r
r Hr
r
r
r
r
4
r
r
r r
@ r @r
5 6 7 8
r r r
r
HH
H
r r r Hr
r
r
r
r
r
r
@
r @r
Brauer algebra Bm (ω)
Multiplication of m-diagrams (m = 8):
1 2 3
r
r
r
H HH
r
r Hr
r
r
r
r
4
r
r
r r
@ r @r
5 6 7 8
r r r
r
HH
H
r r r Hr
r
r
r
r
r
r
@
r @r
1 2 3 4
r
r
r
r
P
= ω PPP
Pr
r
r
rP
5
r
r
6 7 8
r r r
@ r @r r
The dimension of the Brauer algebra Bm (ω) is (2m − 1)!!.
The dimension of the Brauer algebra Bm (ω) is (2m − 1)!!.
For 1 6 a < b 6 m denote by sa b and ga b the diagrams
q ··· P
q q · · · qq
P
P
q ··· q q · · ·PqPq
1
a
b
···
···
q
q
m
and
q ···
q ···
q q ··· q q ···
q q ··· q q ···
1
a
b
q
q
m
The dimension of the Brauer algebra Bm (ω) is (2m − 1)!!.
For 1 6 a < b 6 m denote by sa b and ga b the diagrams
q ··· P
q q · · · qq
P
P
q ··· q q · · ·PqPq
1
a
b
···
···
q
q
and
m
q ···
q ···
q q ··· q q ···
q q ··· q q ···
1
a
b
The symmetrizer in Bm (ω) is the idempotent s(m) such that
sa b s(m) = s(m) sa b = s(m)
and
ga b s(m) = s(m) ga b = 0.
q
q
m
Explicitly,
s
(m)
−1 X
bm/2c
1 X
r ω/2 + m − 2
(−1)
d,
=
r
m!
(r)
r=0
d∈D
where D(r) ⊂ Bm (ω) denotes the set of diagrams which have
exactly r horizontal edges in the top.
Explicitly,
s
(m)
−1 X
bm/2c
1 X
r ω/2 + m − 2
(−1)
d,
=
r
m!
(r)
r=0
d∈D
where D(r) ⊂ Bm (ω) denotes the set of diagrams which have
exactly r horizontal edges in the top. Also,
s(m) =
Y
16a<b6m
and
1−
gab
h(m) ,
ω +a+b−3
Explicitly,
s
(m)
−1 X
bm/2c
1 X
r ω/2 + m − 2
(−1)
d,
=
r
m!
(r)
r=0
d∈D
where D(r) ⊂ Bm (ω) denotes the set of diagrams which have
exactly r horizontal edges in the top. Also,
s(m) =
Y
1−
16a<b6m
gab
h(m) ,
ω +a+b−3
and
s(m) =
1
m!
Y
16a<b6m
1+
sab
gab
−
,
b − a ω/2 + b − a − 1
where the products are in the lexicographic order.
Brauer–Schur–Weyl duality
Brauer–Schur–Weyl duality
There are commuting actions of the classical groups
in types B, C or D and the Brauer algebra with a specialized
parameter ω on the tensor product space
N
C
. . ⊗ C N} .
| ⊗ .{z
m
Brauer–Schur–Weyl duality
There are commuting actions of the classical groups
in types B, C or D and the Brauer algebra with a specialized
parameter ω on the tensor product space
N
C
. . ⊗ C N} .
| ⊗ .{z
m
The dual pairs are
Bm (N), ON
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
N
C
. . ⊗ C N} .
| ⊗ .{z
m
The dual pairs are
Bm (N), ON
and
Bm (−N), SpN
with
N = 2n.
Action in tensors
Action in tensors
In the case g = oN set ω = N. The generators of Bm (N) act
in the tensor space
N
. . ⊗ C N}
|C ⊗ .{z
m
by the rule
sa b 7→ Pa b ,
ga b 7→ Qa b ,
1 6 a < b 6 m,
Action in tensors
In the case g = oN set ω = N. The generators of Bm (N) act
in the tensor space
N
. . ⊗ C N}
|C ⊗ .{z
m
by the rule
sa b 7→ Pa b ,
ga b 7→ Qa b ,
1 6 a < b 6 m,
where i 0 = N − i + 1 and
Qa b =
N
X
i,j=1
1⊗(a−1) ⊗ ei j ⊗ 1⊗(b−a−1) ⊗ ei 0 j 0 ⊗ 1⊗(m−b) .
In the case g = spN with N = 2n set ω = −N. The
generators of Bm (−N) act in the tensor space (C N )⊗m by
sa b 7→ −Pa b ,
ga b 7→ −Qa b ,
1 6 a < b 6 m,
In the case g = spN with N = 2n set ω = −N. The
generators of Bm (−N) act in the tensor space (C N )⊗m by
sa b 7→ −Pa b ,
ga b 7→ −Qa b ,
1 6 a < b 6 m,
with εi = −εn+i = 1 for i = 1, . . . , n and
Qa b =
N
X
i,j=1
εi εj 1⊗(a−1) ⊗ ei j ⊗ 1⊗(b−a−1) ⊗ ei 0 j 0 ⊗ 1⊗(m−b) .
In the case g = spN with N = 2n set ω = −N. The
generators of Bm (−N) act in the tensor space (C N )⊗m by
sa b 7→ −Pa b ,
ga b 7→ −Qa b ,
1 6 a < b 6 m,
with εi = −εn+i = 1 for i = 1, . . . , n and
Qa b =
N
X
εi εj 1⊗(a−1) ⊗ ei j ⊗ 1⊗(b−a−1) ⊗ ei 0 j 0 ⊗ 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,
N
S(m) ∈ End
. . ⊗ End C N} .
| C ⊗ .{z
m
Explicitly, in the orthogonal case
S(m) =
1
m!
Y
16a<b6m
1+
Qab
Pab
−
,
b − a N/2 + b − a − 1
Explicitly, in the orthogonal case
S(m) =
1
m!
Y
1+
16a<b6m
Qab
Pab
−
,
b − a N/2 + b − a − 1
and in the symplectic case
S(m) =
1
m!
Y
16a<b6m
1−
Pab
Qab
−
.
b−a n−b+a+1
Explicitly, in the orthogonal case
S(m) =
1
m!
Y
1+
16a<b6m
Qab
Pab
−
,
b − a N/2 + b − a − 1
and in the symplectic case
S(m) =
1
m!
Y
16a<b6m
1−
Pab
Qab
−
.
b−a n−b+a+1
Remark. S(n+1) = 0 for g = sp2n .
Explicitly, in the orthogonal case
S(m) =
1
m!
Y
1+
16a<b6m
Qab
Pab
−
,
b − a N/2 + b − a − 1
and in the symplectic case
S(m) =
1
m!
Y
16a<b6m
1−
Pab
Qab
−
.
b−a n−b+a+1
Remark. S(n+1) = 0 for g = sp2n . Consider γm (−2n) S(m) ,



 N
for g = oN
ω +m−2
γm (ω) =
,
ω=

ω + 2m − 2

−2n
for g = sp2n .
Lie algebras oN and spN
Lie algebras oN and spN
Let g = oN , spN with N = 2n or N = 2n + 1.
Lie algebras oN and spN
Let g = oN , spN with N = 2n or N = 2n + 1.
Set
Fi j = Ei j − Ej 0 i 0
respectively.
or
Fi j = Ei j − εi εj Ej 0 i 0 ,
Lie algebras oN and spN
Let g = oN , spN with N = 2n or N = 2n + 1.
Set
Fi j = Ei j − Ej 0 i 0
or
Fi j = Ei j − εi εj Ej 0 i 0 ,
respectively.
Introduce the N × N matrix F = [Fij ]
F=
N
X
i,j=1
eij ⊗ Fi j ∈ End C N ⊗ U(g).
Theorem. For any s ∈ Bm (ω) with ω = ±N
and u1 , . . . , um ∈ C the element
tr1,...,m S (u1 + F1 ) . . . (um + Fm )
belongs to the center Z(g) of U(g).
Theorem. For any s ∈ Bm (ω) with ω = ±N
and u1 , . . . , um ∈ C the element
tr1,...,m S (u1 + F1 ) . . . (um + Fm )
belongs to the center Z(g) of U(g).
In particular, there are analogues of the Capelli determinant
and Gelfand invariants.
Theorem. For any s ∈ Bm (ω) with ω = ±N
and u1 , . . . , um ∈ C the element
tr1,...,m S (u1 + F1 ) . . . (um + Fm )
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.
Proof of the theorem relies on the matrix form of the defining
relations for U(g):
Proof of the theorem relies on the matrix form of the defining
relations for U(g):
F1 F2 − F2 F1 = (P12 − Q12 ) F2 − F2 (P12 − Q12 )
Proof of the theorem relies on the matrix form of the defining
relations for U(g):
F1 F2 − F2 F1 = (P12 − Q12 ) F2 − F2 (P12 − Q12 )
where both sides are regarded as elements of the algebra
End C N ⊗ End C N ⊗ U(g) and
F1 =
N
X
i,j=1
eij ⊗ 1 ⊗ Fij ,
F2 =
N
X
i,j=1
1 ⊗ eij ⊗ Fij .
More constructions of Casimir elements for the Lie algebras
glN , oN and sp2n are known.
More constructions of Casimir elements for the Lie algebras
glN , oN and sp2n are known.
In particular, there is a linear basis of Z(glN ) formed by the
quantum immanants S λ with λ running over partitions with at
most N parts (Okounkov–Olshanski, 1996, 1998).
More constructions of Casimir elements for the Lie algebras
glN , oN and sp2n are known.
In particular, there is a linear basis of Z(glN ) 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.
Affine Kac–Moody algebras
Affine Kac–Moody algebras
Define an invariant bilinear form on a simple Lie algebra g,
hX, Yi =
1
tr(ad X ad Y),
2h∨
where h∨ is the dual Coxeter number.
Affine Kac–Moody algebras
Define an invariant bilinear form on a simple Lie algebra g,
hX, Yi =
1
tr(ad X ad Y),
2h∨
where h∨ is the dual Coxeter number.
For the classical types, hX, Yi = const · tr X Y,
h∨ =




N




N−2






n + 1
for g = slN ,
const = 1
for g = oN ,
const =
for g = sp2n ,
1
2
const = 1.
The affine Kac–Moody algebra b
g is the central extension
b
g = g[t, t−1 ] ⊕ CK
The affine Kac–Moody algebra b
g is the central extension
b
g = g[t, t−1 ] ⊕ CK
with the commutation relations
X[r], Y[s] = [X, Y][r + s] + r δr,−s hX, Yi K,
where X[r] = X t r for any X ∈ g and r ∈ Z.
The affine Kac–Moody algebra b
g is the central extension
b
g = g[t, t−1 ] ⊕ CK
with the commutation relations
X[r], Y[s] = [X, Y][r + s] + r δr,−s hX, Yi K,
where X[r] = X t r for any X ∈ g and r ∈ Z.
Problem: What are Casimir elements for b
g?
The universal enveloping algebra at the critical level U−h∨ (b
g) is
the quotient of U(b
g) by the ideal generated by K + h∨ .
The universal enveloping algebra at the critical level U−h∨ (b
g) is
the quotient of U(b
g) by the ideal generated by K + h∨ .
By [Kac 1974], the canonical quadratic Casimir element
e −h∨ (b
belongs to a completion U
g) of U−h∨ (b
g) with respect to the
left ideals Im , m > 0, generated by tm g[t].
e −h∨ (b
Let Z(b
g) be the center of the completed algebra U
g).
e −h∨ (b
Let Z(b
g) be the center of the completed algebra U
g).
Known results:
I
Algebraic structure of Z(b
g).
e −h∨ (b
Let Z(b
g) be the center of the completed algebra U
g).
Known results:
I
Algebraic structure of Z(b
g).
I
Explicit generators for classical types A, B, C, D.
e −h∨ (b
Let Z(b
g) be the center of the completed algebra U
g).
Known results:
I
Algebraic structure of Z(b
g).
I
Explicit generators for classical types A, B, C, D.
Questions:
I
Extension to Lie superalgebras.
e −h∨ (b
Let Z(b
g) be the center of the completed algebra U
g).
Known results:
I
Algebraic structure of Z(b
g).
I
Explicit generators for classical types A, B, C, D.
Questions:
I
Extension to Lie superalgebras.
I
Extension to quantum affine algebras.
b N ):
Example: g = glN . Defining relations for U(gl
Eij [r] Ekl [s] − Ekl [s] Eij [r]
δij δkl K.
= δkj Ei l [r + s] − δi l Ekj [r + s] + r δr,−s δkj δi l −
N
b N ):
Example: g = glN . Defining relations for U(gl
Eij [r] Ekl [s] − Ekl [s] Eij [r]
δij δkl K.
= δkj Ei l [r + s] − δi l Ekj [r + s] + r δr,−s δkj δi l −
N
The critical level is K = −N.
b N ):
Example: g = glN . Defining relations for U(gl
Eij [r] Ekl [s] − Ekl [s] Eij [r]
δij δkl K.
= δkj Ei l [r + s] − δi l Ekj [r + s] + r δr,−s δkj δi l −
N
The critical level is K = −N.
For all r ∈ Z the sums
N
X
i=1
are Casimir elements.
Eii [r]
For r ∈ Z set
Cr =
N X
X
i,j=1
s<0
Eij [s] Eji [r − s] +
X
s>0
Eji [r − s] Eij [s] .
For r ∈ Z set
Cr =
N X
X
i,j=1
s<0
Eij [s] Eji [r − s] +
X
Eji [r − s] Eij [s] .
s>0
All Cr are Casimir elements at the critical level, they belong to
b N ).
e −N (gl
the completed universal enveloping algebra U
Introduce the (formal) Laurent series
Eij (z) =
X
r∈Z
Eij [r] z−r−1
Introduce the (formal) Laurent series
Eij (z) =
X
Eij [r] z−r−1
r∈Z
and use the notation
Eij (z)+ =
X
r<0
Eij [r] z−r−1 ,
Eij (z)− =
X
r>0
Eij [r] z−r−1 .
Introduce the (formal) Laurent series
Eij (z) =
X
Eij [r] z−r−1
r∈Z
and use the notation
Eij (z)+ =
X
Eij [r] z−r−1 ,
Eij (z)− =
r<0
X
Eij [r] z−r−1 .
r>0
Given two Laurent series a(z) and b(z),
their normally ordered product is defined by
: a(z)b(z) : = a(z)+ b(z) + b(z) a(z)− .
Note
X
r∈Z
Cr z−r−2 =
N X
i,j=1
Eij (z)+ Eji (z) + Eji (z)Eij (z)− .
Note
X
Cr z−r−2 =
N X
Eij (z)+ Eji (z) + Eji (z)Eij (z)− .
i,j=1
r∈Z
Hence, all coefficients of the series
2
tr : E(z) : =
N
X
i,j=1
are Casimir elements.
: Eij (z)Eji (z) :
Similarly, all coefficients of the series
tr : E(z)3 : =
N
X
: Eij (z) Ejk (z) Eki (z) :
i,j,k=1
are Casimir elements, where the normal ordering is applied
from right to left.
Similarly, all coefficients of the series
tr : E(z)3 : =
N
X
: Eij (z) Ejk (z) Eki (z) :
i,j,k=1
are Casimir elements, where the normal ordering is applied
from right to left.
However, the claim does not extend to tr : E(z)4 : !
Similarly, all coefficients of the series
tr : E(z)3 : =
N
X
: Eij (z) Ejk (z) Eki (z) :
i,j,k=1
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
2
tr : E(z)4 : − tr : ∂z E(z) :
are Casimir elements.
Invariants of the vacuum module
Invariants of the vacuum module
The vacuum module at the critical level is the b
g-module
V(g) = U−h∨ (b
g)/U−h∨ (b
g)g[t].
Invariants of the vacuum module
The vacuum module at the critical level is the b
g-module
V(g) = U−h∨ (b
g)/U−h∨ (b
g)g[t].
The Feigin–Frenkel center z(b
g) is the algebra of g[t]-invariants
z(b
g) = {v ∈ V(g) | g[t]v = 0}.
Invariants of the vacuum module
The vacuum module at the critical level is the b
g-module
V(g) = U−h∨ (b
g)/U−h∨ (b
g)g[t].
The Feigin–Frenkel center z(b
g) is the algebra of g[t]-invariants
z(b
g) = {v ∈ V(g) | g[t]v = 0}.
Note V(g) ∼
= U t−1 g[t−1 ]
as a vector space.
Invariants of the vacuum module
The vacuum module at the critical level is the b
g-module
V(g) = U−h∨ (b
g)/U−h∨ (b
g)g[t].
The Feigin–Frenkel center z(b
g) is the algebra of g[t]-invariants
z(b
g) = {v ∈ V(g) | g[t]v = 0}.
Note V(g) ∼
= U t−1 g[t−1 ]
as a vector space.
Hence, z(b
g) is a subalgebra of U t−1 g[t−1 ] .
Properties:
I
The subalgebra z(b
g) of U t−1 g[t−1 ]
is commutative.
Properties:
I
The subalgebra z(b
g) of U t−1 g[t−1 ]
I
It is invariant with respect to the translation operator T
defined as the derivation T = −d/dt.
is commutative.
Properties:
I
The subalgebra z(b
g) of U t−1 g[t−1 ]
I
It is invariant with respect to the translation operator T
is commutative.
defined as the derivation T = −d/dt.
Any element of z(b
g) is called a Segal–Sugawara vector.
Theorem (Feigin–Frenkel, 1992,
Frenkel, 2007).
There exist Segal–Sugawara vectors S1 , . . . , Sn ∈ U t−1 g[t−1 ] ,
n = rank g, such that
Theorem (Feigin–Frenkel, 1992,
Frenkel, 2007).
There exist Segal–Sugawara vectors S1 , . . . , Sn ∈ U t−1 g[t−1 ] ,
n = rank g, such that
z(b
g) = C[T k Sl | l = 1, . . . , n, k > 0].
Theorem (Feigin–Frenkel, 1992,
Frenkel, 2007).
There exist Segal–Sugawara vectors S1 , . . . , Sn ∈ U t−1 g[t−1 ] ,
n = rank g, such that
z(b
g) = C[T k Sl | l = 1, . . . , n, k > 0].
We call S1 , . . . , Sn a complete set of Segal–Sugawara vectors.
Theorem (Feigin–Frenkel, 1992,
Frenkel, 2007).
There exist Segal–Sugawara vectors S1 , . . . , Sn ∈ U t−1 g[t−1 ] ,
n = rank g, such that
z(b
g) = C[T k Sl | l = 1, . . . , n, k > 0].
We call S1 , . . . , Sn a complete set of Segal–Sugawara vectors.
Explicit constructions of such sets and a new proof of
the theorem for the classical types A, B, C, D:
[Chervov–Talalaev, 2006,
Chervov–M., 2009,
M. 2013].
Example: g = glN .
Example: g = glN .
Set τ = −d/dt and consider the N × N matrix

E12 [−1]
...
τ + E11 [−1]


τ + E22 [−1] . . .
 E21 [−1]
τ + E[−1] = 

..
..
..

.
.
.


EN1 [−1]
EN2 [−1]
...

E1N [−1] 


E2N [−1] 
.

..

.


τ + ENN [−1]
The coefficients φ1 , . . . , φN of the polynomial
cdet τ + E[−1] = τ N + φ1 τ N−1 + · · · + φN−1 τ + φN
form a complete set of Segal–Sugawara vectors.
The coefficients φ1 , . . . , φN of the polynomial
cdet τ + E[−1] = τ N + φ1 τ N−1 + · · · + φN−1 τ + φN
form a complete set of Segal–Sugawara vectors.
For N = 2
cdet τ + E[−1] = τ + E11 [−1] τ + E22 [−1] − E21 [−1]E12 [−1]
= τ 2 + φ1 τ + φ2
The coefficients φ1 , . . . , φN of the polynomial
cdet τ + E[−1] = τ N + φ1 τ N−1 + · · · + φN−1 τ + φN
form a complete set of Segal–Sugawara vectors.
For N = 2
cdet τ + E[−1] = τ + E11 [−1] τ + E22 [−1] − E21 [−1]E12 [−1]
= τ 2 + φ1 τ + φ2
with
φ1 = E11 [−1] + E22 [−1],
φ2 = E11 [−1] E22 [−1] − E21 [−1] E12 [−1] + E22 [−2].
To get another family of Segal–Sugawara vectors, expand
m
tr τ + E[−1] = θm 0 τ m + θm 1 τ m−1 + · · · + θm m
To get another family of Segal–Sugawara vectors, expand
m
tr τ + E[−1] = θm 0 τ m + θm 1 τ m−1 + · · · + θm m
b ).
All coefficients θm i belong to the Feigin–Frenkel center z(gl
N
To get another family of Segal–Sugawara vectors, expand
m
tr τ + E[−1] = θm 0 τ m + θm 1 τ m−1 + · · · + θm m
b ).
All coefficients θm i belong to the Feigin–Frenkel center z(gl
N
The elements θ1 1 , . . . , θN N form
a complete set of Segal–Sugawara vectors.
To get another family of Segal–Sugawara vectors, expand
m
tr τ + E[−1] = θm 0 τ m + θm 1 τ m−1 + · · · + θm m
b ).
All coefficients θm i belong to the Feigin–Frenkel center z(gl
N
The elements θ1 1 , . . . , θN N form
a complete set of Segal–Sugawara vectors.
The following are Segal–Sugawara vectors for glN :
tr E[−1],
tr E[−1]2 ,
tr E[−1]3 ,
tr E[−1]4 − tr E[−2]2 .
b N ) are recovered
e −N (gl
The corresponding central elements in U
by the state-field correspondence map Y which takes
elements of the vacuum module V(glN ) to Laurent series in z;
b N ) are recovered
e −N (gl
The corresponding central elements in U
by the state-field correspondence map Y which takes
elements of the vacuum module V(glN ) to Laurent series in z;
its application to Segal–Sugawara vectors yields Laurent series
whose coefficients are Casimir elements.
b N ) are recovered
e −N (gl
The corresponding central elements in U
by the state-field correspondence map Y which takes
elements of the vacuum module V(glN ) to Laurent series in z;
its application to Segal–Sugawara vectors yields Laurent series
whose coefficients are Casimir elements.
By definition,
Y : Eij [−1] 7→ Eij (z) =
X
r∈Z
Eij [r] z−r−1 .
Also,
Y : Eij [−r − 1] 7→
1 r
∂ Eij (z),
r! z
r > 0,
Also,
Y : Eij [−r − 1] 7→
1 r
∂ Eij (z),
r! z
r > 0,
and
Y : Eij [−1] Ekl [−1] 7→ : Eij (z) Ekl (z) :
Also,
Y : Eij [−r − 1] 7→
1 r
∂ Eij (z),
r! z
r > 0,
and
Y : Eij [−1] Ekl [−1] 7→ : Eij (z) Ekl (z) :
We have
Y : tr E[−1] 7→ tr E(z)
Y : tr E[−1]2 7→ tr : E(z)2 :
Y : tr E[−1]3 7→ tr : E(z)3 :
Y : tr E[−1]4 − tr E[−2]2 7→ tr : E(z)4 : − tr : ∂z E(z)
2
:
Write
tr : ∂z + E(z)
m
: = θm 0 (z) ∂zm + · · · + θm m (z).
Write
tr : ∂z + E(z)
m
: = θm 0 (z) ∂zm + · · · + θm m (z).
Theorem. The coefficients of the Laurent series
θ1 1 (z), . . . , θN N (z)
b N ).
e −N (gl
are topological generators of the center of U
Write
tr : ∂z + E(z)
m
: = θm 0 (z) ∂zm + · · · + θm m (z).
Theorem. The coefficients of the Laurent series
θ1 1 (z), . . . , θN N (z)
b N ).
e −N (gl
are topological generators of the center of U
Remark. The theorem holds in the same form for any complete
set of Segal–Sugawara vectors.
Proving the Feigin–Frenkel theorem for the classical types:
Proving the Feigin–Frenkel theorem for the classical types:
I
Produce Segal–Sugawara vectors S1 , . . . , Sn explicitly.
Proving the Feigin–Frenkel theorem for the classical types:
I
Produce Segal–Sugawara vectors S1 , . . . , Sn explicitly.
I
Show that all elements T k Sl with l = 1, . . . , n and k > 0 are
algebraically independent and generate z(b
g).
Proving the Feigin–Frenkel theorem for the classical types:
I
Produce Segal–Sugawara vectors S1 , . . . , Sn explicitly.
I
Show that all elements T k Sl with l = 1, . . . , n and k > 0 are
algebraically independent and generate z(b
g).
Use the classical limit:
gr U t−1 g[t−1 ] ∼
= S t−1 g[t−1 ]
Proving the Feigin–Frenkel theorem for the classical types:
I
Produce Segal–Sugawara vectors S1 , . . . , Sn explicitly.
I
Show that all elements T k Sl with l = 1, . . . , n and k > 0 are
algebraically independent and generate z(b
g).
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] .
Let X1 , . . . , Xd be a basis of g and let P = P(X1 , . . . , Xd ) be a
g-invariant in the symmetric algebra S(g).
Let X1 , . . . , Xd be a basis of g and let P = P(X1 , . . . , Xd ) be a
g-invariant in the symmetric algebra S(g). Then each element
P(r) = T r P X1 [−1], . . . , Xd [−1] ,
is a
r > 0,
g[t]-invariant in the symmetric algebra S t−1 g[t−1 ] .
Let X1 , . . . , Xd be a basis of g and let P = P(X1 , . . . , Xd ) be a
g-invariant in the symmetric algebra S(g). Then each element
P(r) = T r P X1 [−1], . . . , Xd [−1] ,
is a
r > 0,
g[t]-invariant in the symmetric algebra S t−1 g[t−1 ] .
Theorem (Raïs–Tauvel 1992,
Beilinson–Drinfeld 1997).
If P1 , . . . , Pn are algebraically independent generators of S(g)g ,
then the elements P1,(r) , . . . , Pn,(r) with r > 0 are algebraically
g[t]
independent generators of S t−1 g[t−1 ]
.
Explicit generators of z(b
g). Type A
Explicit generators of z(b
g). Type A
Set
Eij [r] = Eij tr ∈ glN [t, t−1 ]
Explicit generators of z(b
g). Type A
Set
Eij [r] = Eij tr ∈ glN [t, t−1 ]
and
E[r] =
N
X
i,j=1
eij ⊗ Eij [r] ∈ End C N ⊗ U glN [t, t−1 ] .
Explicit generators of z(b
g). Type A
Set
Eij [r] = Eij tr ∈ glN [t, t−1 ]
and
E[r] =
N
X
eij ⊗ Eij [r] ∈ End C N ⊗ U glN [t, t−1 ] .
i,j=1
Consider the algebra
−1
N
N
⊗
U
gl
[t,
t
]
End
C
⊗
.
.
.
⊗
End
C
N
|
{z
}
m
Explicit generators of z(b
g). Type A
Set
Eij [r] = Eij tr ∈ glN [t, t−1 ]
and
E[r] =
N
X
eij ⊗ Eij [r] ∈ End C N ⊗ U glN [t, t−1 ] .
i,j=1
Consider the algebra
−1
N
N
⊗
U
gl
[t,
t
]
End
C
⊗
.
.
.
⊗
End
C
N
|
{z
}
m
and recall its elements H (m) and A(m) .
Theorem.
All coefficients of the polynomials in τ = −d/d t
Theorem.
All coefficients of the polynomials in τ = −d/d t
tr1,...,m A(m) τ + E[−1]1 . . . τ + E[−1]m
= φm 0 τ m + φm 1 τ m−1 + · · · + φm m ,
tr1,...,m H (m) τ + E[−1]1 . . . τ + E[−1]m
= ψm 0 τ m + ψm 1 τ m−1 + · · · + ψm m ,
and
tr (τ + E[−1])m = θm 0 τ m + θm 1 τ m−1 + · · · + θm m
Theorem.
All coefficients of the polynomials in τ = −d/d t
tr1,...,m A(m) τ + E[−1]1 . . . τ + E[−1]m
= φm 0 τ m + φm 1 τ m−1 + · · · + φm m ,
tr1,...,m H (m) τ + E[−1]1 . . . τ + E[−1]m
= ψm 0 τ m + ψm 1 τ m−1 + · · · + ψm m ,
and
tr (τ + E[−1])m = θm 0 τ m + θm 1 τ m−1 + · · · + θm m
b ).
belong to the Feigin–Frenkel center z(gl
N
b N ):
Proof. Use the matrix form of the defining relations of U−N (gl
b N ):
Proof. Use the matrix form of the defining relations of U−N (gl
For any r ∈ Z set
E[r] =
N
X
i,j=1
b N ).
eij ⊗ Eij [r] ∈ End C N ⊗ U−N (gl
b N ):
Proof. Use the matrix form of the defining relations of U−N (gl
For any r ∈ Z set
E[r] =
N
X
b N ).
eij ⊗ Eij [r] ∈ End C N ⊗ U−N (gl
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 + r δr,−s 1 − N P12 .
The required relations in the vacuum module are
E[0]0 tr1,...,m A(m) τ + E[−1]1 . . . τ + E[−1]m = 0
The required relations in the vacuum module are
E[0]0 tr1,...,m A(m) τ + E[−1]1 . . . τ + E[−1]m = 0
and
E[1]0 tr1,...,m A(m) τ + E[−1]1 . . . τ + E[−1]m = 0.
The required relations in the vacuum module are
E[0]0 tr1,...,m A(m) τ + E[−1]1 . . . τ + E[−1]m = 0
and
E[1]0 tr1,...,m A(m) τ + E[−1]1 . . . τ + E[−1]m = 0.
The elements ψm a and θm a are expressed in terms of the
φm a through the MacMahon Master Theorem and the Newton
identities, respectively.
The coefficients of the column-determinant are related to the
φm a through the relation
cdet τ + E[−1] = tr1,...,N A(N) τ + E[−1]1 . . . τ + E[−1]N .
The coefficients of the column-determinant are related to the
φm a through the relation
cdet τ + E[−1] = tr1,...,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) ,
The coefficients of the column-determinant are related to the
φm a through the relation
cdet τ + E[−1] = tr1,...,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.
Types B, C and D
Types B, C and D
Recall the symmetrizers associated with oN and sp2n :
S(m) =
1
m!
Y
16a<b6m
1+
Qab
Pab
−
,
b − a N/2 + b − a − 1
Types B, C and D
Recall the symmetrizers associated with oN and sp2n :
S(m) =
1
m!
Y
1+
16a<b6m
Qab
Pab
−
,
b − a N/2 + b − a − 1
and
S(m) =
1
m!
Y
16a<b6m
1−
Pab
Qab
−
.
b−a n−b+a+1
Types B, C and D
Recall the symmetrizers associated with oN and sp2n :
S(m) =
1
m!
Y
1+
16a<b6m
Qab
Pab
−
,
b − a N/2 + b − a − 1
and
S(m) =
1
m!
Y
16a<b6m
1−
Pab
Qab
−
.
b−a n−b+a+1
Also,
γm (ω) =
ω +m−2
,
ω + 2m − 2
ω=



 N
for g = oN


−2n
for g = sp2n .
Let g = oN , spN with N = 2n or N = 2n + 1.
Let g = oN , spN with N = 2n or N = 2n + 1.
As before,
Fi j = Ei j − Ej 0 i 0
or
Fi j = Ei j − εi εj Ej 0 i 0
Let g = oN , spN with N = 2n or N = 2n + 1.
As before,
Fi j = Ei j − Ej 0 i 0
or
Fi j = Ei j − εi εj Ej 0 i 0
and
Fi j [r] = Fi j tr ∈ g[t, t−1 ].
Let g = oN , spN with N = 2n or N = 2n + 1.
As before,
Fi j = Ei j − Ej 0 i 0
or
Fi j = Ei j − εi εj Ej 0 i 0
and
Fi j [r] = Fi j tr ∈ g[t, t−1 ].
Combine into a matrix
F[r] =
N
X
i,j=1
eij ⊗ Fi j [r] ∈ End C N ⊗ U g[t, t−1 ] .
Theorem.
All coefficients of the polynomial in τ = −d/d t
γm (ω) tr1,...,m S(m) τ + F[−1]1 . . . τ + F[−1]m
= φm 0 τ m + φm 1 τ m−1 + · · · + φm m
Theorem.
All coefficients of the polynomial in τ = −d/d t
γm (ω) tr1,...,m S(m) τ + F[−1]1 . . . τ + F[−1]m
= φm 0 τ m + φm 1 τ m−1 + · · · + φm m
belong to the Feigin–Frenkel center z(b
g).
Theorem.
All coefficients of the polynomial in τ = −d/d t
γm (ω) tr1,...,m S(m) τ + F[−1]1 . . . τ + F[−1]m
= φm 0 τ m + φm 1 τ m−1 + · · · + φm m
belong to the Feigin–Frenkel center z(b
g).
In addition, in the case g = o2n , the Pfaffian
Pf F[−1] =
1 X
sgn σ · Fσ(1) σ(2)0 [−1] . . . Fσ(2n−1) σ(2n)0 [−1]
2n n!
σ∈S2n
belongs to z(b
o2n ).
Moreover, φ2 2 , φ4 4 , . . . , φ2n 2n is a complete set of
Segal–Sugawara vectors for o2n+1 and sp2n , whereas
Moreover, φ2 2 , φ4 4 , . . . , φ2n 2n is a complete set of
Segal–Sugawara vectors for o2n+1 and sp2n , whereas
φ2 2 , φ4 4 , . . . , φ2n−2 2n−2 , φn0 is a complete set of
Segal–Sugawara vectors for o2n , where φn0 = Pf F[−1].