We consider the quantum Gelfand invariants which first appeared in a landmark paper by Reshetikhin et al. [Algebra Anal. 1(1), 178–206 (1989)]. We calculate the eigenvalues of the invariants acting in irreducible highest weight representations of the quantized enveloping algebra for gln. The calculation is based on Liouville-type formulas relating two families of central elements in the quantum affine algebras of type A.

The quantized enveloping algebras and quantum affine algebras associated with simple Lie algebras comprise remarkable families of quantum groups, as introduced by Drinfeld5 and Jimbo.13 These algebras and their representations have since found numerous connections with many areas in mathematics and physics.

In this paper we will be concerned with those families associated with the general linear Lie algebras gln. Both the quantized enveloping algebra Uq(gln) and the quantum affine algebra Uq(gl̂n) admit R-matrix (or RTT) presentations going back to the work of the Leningrad school headed by Faddeev; see e.g., Refs. 19 and 27 for reviews of the foundations of the R-matrix approach originated in the quantum inverse scattering method.

Central elements in both Uq(gln) and Uq(gl̂n) are constructed with the use of the R-matrix presentations and found as coefficients of the respective quantum determinants; see Refs. 3, 14, and 19 and also Ref. 6 for more general constructions of central elements in the quantized enveloping algebras and quantum affine algebras. As pointed out in Ref. 27, the quantum traces of powers of generator matrices are central in Uq(gln); see also Ref. 1. By taking the limit q → 1 one recovers the central elements of U(gln) going back to Ref. 8, which are known as the Gelfand invariants. Note that a different generalization of the Gelfand invariants for gln as central elements in Uq(gln), was given in Ref. 9, where their eigenvalues in irreducible highest weight representations were calculated. A new family of central elements in Uq(gl̂n) was given in Ref. 2 and they were related to the quantum determinants by Liouville-type formulas, although they were not accompanied by proofs. This result is quite analogous to the corresponding quantum Liouville formulas for the Yangians originated in Ref. 22 and we give a complete proof in this paper.

By taking the images of the new central elements of Uq(gl̂n) under the evaluation homomorphism, we recover the quantum Gelfand invariants of Ref. 27. The Liouville formulas will then allow us to calculate the eigenvalues of the quantum Gelfand invariants in irreducible highest weight representations of Uq(gln). We thus obtain q-analogues of the Perelomov–Popov formulas.24 To recall the eigenvalue formulas from Ref. 24, consider the irreducible highest weight representation L(λ) of gln with the highest weight λ = (λ1, …, λn) and combine the standard basis elements Eij into the matrix E = [Eij]. Then the eigenvalue of the Gelfand invariant tr Em in L(λ) is found by
trEmk=1nkm(1k+1)(nk+1)(1k)(nk),
(1.1)
where i = λi + ni and the symbol ∧ indicates that the zero factor is skipped. Formula (1.1) can be derived with the use of R-matrix calculations in the Yangian Y(gln) or in the universal enveloping algebra U(gln); see Ref. 20, Sec. 7.1 and Ref. 21, Sec. 4.8, respectively.
Our main result concerning the quantum Gelfand invariants is the following theorem, where we calculate the eigenvalues of the quantum traces trqMm of the powers of the generator matrix M=L(L+)1 in the representation Lq(λ) of Uq(gln) (see Sec. III for the definitions). We use a standard notation for the q-numbers
[k]q=qkqkqq1,kZ.

Theorem 1.1.
The eigenvalue of the quantum Gelfand invariant trqMm in Lq(λ) is found by
trqMmk=1nq2km[1k+1]q[nk+1]q[1k]q[nk]q.
(1.2)

We will prove Theorem 1.1 in Sec. III by deriving it from the Liouville formula given in Theorem 2.4 in a way similar to Ref. 20, Sec. 7.1. We also consider three more families of central elements of Uq(gln) which, however, turn out to coincide with the quantum Gelfand invariants, up to a possible replacement qq−1.

In the limit q → 1 we have
M1qq1E
(1.3)
by (3.1) and (3.2) below. Therefore, the Gelfand invariants in U(gln) are recovered from the elements trqMm in the limit, while the Perelomov–Popov formulas (1.1) follow from (1.2); see Remark 3.1 below.

We point out a related recent work,18 where explicit formulas for certain central elements in the reflection equation algebras were given and their relation with the quantum Gelfand invariants were reviewed. This includes the connection with an earlier construction of central elements in Ref. 25 and with the Cayley–Hamilton theorem and Newton identities of Refs. 10, 12, and 23.

This work was completed during the first and third named authors’ visits to the South China University of Technology and to the Shanghai University. They are grateful to the Departments of Mathematics in both universities for the warm hospitality.

We will regard q as a nonzero complex number which is not a root of unity. Recall the R-matrix presentation of the quantum affine algebra Uq(gl̂n) as introduced in Ref. 26. We follow4 and use the same settings as in our earlier work.17 Let eijEndCn denote the standard matrix units. Consider the R-matrix
R(x)=f(x)qq1x(RxR̃),
(2.1)
where
R=qieiieii+ijeiiejj+(qq1)i<jeijeji
(2.2)
and
R̃=q1ieiieii+ijeiiejj(qq1)i>jeijeji,
while the formal power series
f(x)=1+k=1fkxk,fk=fk(q),
is uniquely determined by the relation
f(xq2n)=f(x)(1xq2)(1xq2n2)(1x)(1xq2n).
The quantum affine algebra Uq(gl̂n) is generated by elements
lij+[r],lij[r]with1i,jn,r=0,1,,
and the invertible central element qc, subject to the defining relations
lji+[0]=lij[0]=0for1i<jn,lii+[0]lii[0]=lii[0]lii+[0]=1fori=1,,n,
and
R(u/v)L1±(u)L2±(v)=L2±(v)L1±(u)R(u/v),
(2.3)
R(uqc/v)L1+(u)L2(v)=L2(v)L1+(u)R(uqc/v).
(2.4)
In the last two formulas we consider the matrices L±(u)=lij±(u), whose entries are formal power series in u and u−1,
lij+(u)=r=0lij+[r]ur,lij(u)=r=0lij[r]ur.
Here and below we regard the matrices as elements
L±(u)=i,j=1neijlij±(u)EndCnUq(gl̂n)[[u±1]]
and use a subscript to indicate a copy of the matrix in the multiple tensor product algebra
EndCnEndCnkUq(gl̂n)[[u±1]]
(2.5)
so that
La±(u)=i,j=1n1(a1)eij1(ka)lij±(u).
We regard the usual matrix transposition also as the linear map
t:EndCnEndCn,eijeji.
For any a ∈ {1, …, k} we will denote by ta the corresponding partial transposition on the algebra (2.5) which acts as t on the a-th copy of EndCn and as the identity map on all the other tensor factors.
The scalar factor in (2.1) is necessary for the R-matrix to satisfy the crossing symmetry relations.7 We will use one of them given by
R12(x)1t2D2R12(xq2n)t2=D2,
(2.6)
where D denotes the diagonal n × n matrix
D=diagqn1,qn3,,qn+1.

The following proposition was stated in Ref. 2, Eq. (4.28) without proof.

Proposition 2.1.
There exist a series z+(u) in u and a series z(u) in u−1 with coefficients in the algebra Uq(gl̂n) such that
L±(uq2n)tD(L±(u)1)t=z±(u)D
(2.7)
and
(L±(u)1)tD1L±(uq2n)t=z±(u)D1.
(2.8)
Moreover, the coefficients of the series z±(u) belong to the center of the algebra Uq(gl̂n).

Proof.
Multiply both sides of (2.3) by L2±(v)1 from the left and the right and apply the transposition t2 to get
R(u/v)t2L2±(v)1tL1±(u)=L1±(u)L2±(v)1tR(u/v)t2
and hence
R(u/v)t21L1±(u)L2±(v)1t=L2±(v)1tL1±(u)R(u/v)t21.
(2.9)
Use the crossing symmetry relation (2.6) to replace the R-matrix by
R(u/v)t21=D21R(u/vq2n)1t2D2
and get
R(u/vq2n)1t2D2L1±(u)L2±(v)1tD21=D2L2±(v)1tL1±(u)D21R(u/vq2n)1t2.
Now cancel the scalar factors appearing in (2.1) on both sides of this relation and observe that the R-matrix RxR̃ evaluated at x = 1 equals (qq−1)P, where P is the permutation operator. Therefore,
(RR̃)1t2=1qq1QwithQ=i,j=1neijeij.
Hence, by taking u = vq2n we get
QD2L1±(vq2n)L2±(v)1tD21=D2L2±(v)1tL1±(vq2n)D21Q.
Since Q is an operator in EndCnEndCn with a one-dimensional image, both sides must be equal to Qz±(v) for series z±(v) with coefficients in the quantum affine algebra. Using the relations QX1=QX2t and X1Q=X2tQ which hold for an arbitrary matrix X, we can write the definition of z±(v) as
QL2±(vq2n)tD2L2±(v)1t=QD2z±(v)
(2.10)
and
L2±(v)1tD21L2±(vq2n)tQ=D21Qz±(v).
(2.11)
By taking trace over the first copy of EndCn on both sides of (2.10) and (2.11) we arrive at (2.7) and (2.8), respectively.
We will now use (2.8) to show that the series z(v) commutes with L+(u). We have
L1+(u)z(v)=L1+(u)D2L2(v)1tD21L2(vq2n)t.
(2.12)
Transform relation (2.4) in the same way as we did for (2.3) in the beginning of the proof to get the following counterpart of (2.9):
R(uqc/v)t21L1+(u)L2(v)1t=L2(v)1tL1+(u)R(uqc/v)t21.
Hence, the right hand side of (2.12) equals
D2R12(uqc/v)t2L2(v)1tL1+(u)R12(uqc/v)t21D21L2(vq2n)t.
(2.13)
Applying again (2.6), we can write
R12(uqc/v)t21D21=D21R12(uqc/vq2n)1t2.
Continue transforming (2.13) by using the following consequence of (2.4):
L1+(u)R12(uqc/vq2n)1t2L2(vq2n)t=L2(vq2n)tR12(uqc/vq2n)1t2L2(vq2n)t,
so that (2.13) becomes
D2R12(uqc/v)t2L2(v)1tD21L2(vq2n)tR12(uqc/vq2n)1t2L1+(u).
By (2.8) this simplifies to
D2R12(uqc/v)t2D21R12(uqc/vq2n)1t2z(v)L1+(u)
which equals z(v)L1+(u) by (2.6). This proves that L1+(u)z(v)=z(v)L1+(u). The relation L1(u)z(v)=z(v)L1(u) and the centrality of z+(v) are verified in the same way.□

Remark 2.2.

The counterparts of the series z±(u) for the quantum affine algebras of types B, C and D appear in Ref. 15, Proposition 3.3 and Ref. 16, Proposition 3.3, where they were introduced by relations analogous to (2.7) and (2.8).

Corollary 2.3.
We have the formulas
z±(u)=1[n]qtrDL±(uq2n)L±(u)1
(2.14)
and
z±(u)=1[n]qtrD1L±(u)1L±(uq2n).
(2.15)

Proof.

The formulas follow by taking trace on both sides of the respective matrix relations (2.7) and (2.8).□

Recall that the quantum determinants qdet L+(u) and qdet L(u) are series in u and u−1, respectively, whose coefficients belong to the center of the quantum affine algebra Uq(gl̂n):
qdetL±(u)=σSn(q)l(σ)lσ(1)1±(uq2n2)lσ(n)n±(u),
where l(σ) denotes the length of the permutation σ.

The following is a q-analogue of the quantum Liouville formula of Ref. 22. It was stated in Ref. 2, Eq. (4.32).

Theorem 2.4.
We have the relations
z±(u)=qdetL±(uq2)qdetL±(u).
(2.16)

Proof.
We follow (Ref. 20, Sec. 1.9) and use the quantum comatrices L̂±(u) introduced in Refs. 23 and 28. They are defined by the relations
L̂±(uq2)L±(u)=qdetL±(u)1,
(2.17)
where 1 denotes the identity matrix. We will derive formulas for the entries of the matrices L̂±(u) by using the quantum minor relations reviewed e.g., in Ref. 11 which we outline below.
Recall that the q-permutation operator PqEndCnEndCn is defined by
Pq=i=1neiieii+qi>jeijeji+q1i<jeijeji.
The action of the symmetric group Sk on the space (Cn)k can be defined by setting siPsiqPi,i+1q for i = 1, …, k − 1, where si denotes the transposition (i, i + 1). If σ=si1sil is a reduced decomposition of an element σSk then we set Pσq=Psi1qPsilq. Denote by e1, …, en the canonical basis vectors of Cn. Then for any indices a1 < ⋯ < ak and any τSk we have
Pσq(eaτ(1)eaτ(k))=ql(στ1)l(τ)eaτσ1(1)eaτσ1(k).
(2.18)
We denote by Aq(k) the q-antisymmetrizer
Aq(k)=σSksgnσPσq.
(2.19)
The defining relations (2.3) and the fusion procedure3 for the quantum affine algebra imply the relations
Aq(k)L1±(uq2k2)Lk±(u)=Lk±(u)L1±(uq2k2)Aq(k).
(2.20)
The quantum minors are the series l±b1bka1ak(u) with coefficients in Uq(gl̂n) defined by the expansion of the elements (2.20) as
ai,biea1b1eakbkl±b1bka1ak(u).
Explicit formulas for the quantum minors have the form: if a1 < ⋯ < ak then
l±b1bka1ak(u)=σSk(q)l(σ)laσ(1)b1±(uq2k2)laσ(k)bk±(u).
(2.21)
If b1 < ⋯ < bk (and the ai are arbitrary) then
l±b1bka1ak(u)=σSk(q)l(σ)lakbσ(k)±(u)la1bσ(1)±(uq2k2)
and for any τSk we have
l±bτ(1)bτ(k)a1ak(u)=(q)l(τ)l±b1bka1ak(u).

The following lemma was pointed out in Refs. 23 and 28.

Lemma 2.5.
The (i, j) entry of the matrix L̂±(u) is given by
l̂ij±(u)=(q)jil±1în1ĵn(u),
(2.22)
where the hats on the right-hand side indicate the indices to be omitted.

Proof.
According to (2.21), the quantum determinants are defined by the relations
Aq(n)L1±(uq2n2)Ln±(u)=Aq(n)qdetL±(u).
Hence, the definition (2.17) of the quantum comatrices implies
Aq(n)L1±(uq2n4)Ln1±(u)=Aq(n)L̂n±(u).
Apply both sides to the vector e1êienej and use (2.18) and (2.19) to equate the coefficients of the vector Aq(n)(e1en) to obtain (2.22).□

We will also use the relations for the transposed comatrices; see Refs. 23 and 28.

Lemma 2.6.
We have the relations
DL̂±(u)tD1L±(uq2n2)t=qdetL±(u)1.

Proof.
Since each relation only involves generators belonging to the subalgebra of Uq(gl̂n) generated by the coefficients of the series lij+(u) or lij(u), we may assume that the central charge is specialized to zero, c = 0, and derive the desired relations in the quotient algebra U°q(gl̂n). The mapping
θ:L±(u)L(u1)t
defines an automorphism of the algebra U°q(gl̂n). By Lemma 2.5 and the quantum minor formulas, for the images under θ we have
θ:L̂±(u)DL̂(u1q2n+4)tD1
and
θ:qdetL±(u)qdetL(u1q2n+2).
It remains to apply θ to both sides of relations (2.17) and then replace u by u−1q−2n+2.□

Now, using (2.17) and the centrality of qdet L±(u), we can write (2.7) as
z±(u)=L±(uq2n)tD(L±(u)1)tD1=(qdetL±(u))1L±(uq2n)tDL̂±(uq2)tD1.
By applying Lemma 2.6 with u replaced by uq2 we find that this expression coincides with the right hand side of (2.16).

Remark 2.7.

Since the second part of Proposition 2.1 was not used in the Proof of Theorem 2.4, the fact that the coefficients of both series z+(u) and z(u) belong to the center of the quantum affine algebra Uq(gl̂n) also follows from (2.16) due to the respective properties of the quantum determinants qdet L+(u) and qdet L(u).

We will follow4 to define the quantized enveloping algebra Uq(gln) in its R-matrix presentation27 as the algebra generated by elements lij+ and lij with 1 ⩽ i, jn subject to the relations
lij=lji+=0,1i<jn,liilii+=lii+lii=1,1in,RL1±L2±=L2±L1±R,RL1+L2=L2L1+R,
where the R-matrix R is defined in (2.2), while L+ and L are the matrices
L±=i,jeijlij±EndCnUq(gln).
The universal enveloping algebra U(gln) is recovered from Uq(gln) in the limit q → 1 by the formulas
lijqq1Eij,lji+qq1Ejifori>j,
(3.1)
and
lii1q1Eii,lii+1q1Eiifori=1,,n.
(3.2)
Set M=L(L+)1. By Ref. 27, the quantum traces defined by trqMm = tr DMm belong to the center of the algebra Uq(gln). The elements trqMm act by multiplication by scalars in the irreducible highest weight representations Lq(λ). The representation Lq(λ) of Uq(gln) is generated by a nonzero vector ξ such that
lij+ξ=0for1i<jn,liiξ=qλiξfor1in,
for an n-tuple λ = (λ1, …, λn) of integers (or real numbers). This is a q-deformation of the irreducible gln-module L(λ) with the highest weight λ.

Proof of Theorem 1.1.

We will show that the eigenvalue of the quantum Gelfand invariant trqMm in Lq(λ) is given by formula (1.2).

Recall the evaluation homomorphism Uq(gl̂n)Uq(gln) defined by
L+(u)L+Lu,L(u)LL+u1,qc1,
(3.3)
and apply it to both sides of the Liouville formula
z+(u)=qdetL+(uq2)qdetL+(u)
(3.4)
proved in Theorem 2.4. The image of the quantum determinant qdet L+(u) is found by
σSn(q)l(σ)lσ(1)1+lσ(1)1uq2n2lσ(n)n+lσ(n)nu.
By applying this element to the highest vector ξ of Lq(λ), we find that its eigenvalue is given by
qλ1qλ1+2n2uqλnqλnu=qn(n1)/2(q1q1u)(qnqnu)
with i = λi + ni. Therefore, the eigenvalue of the image of the right hand side of (3.4) in Lq(λ) is given by
(q1q1+2u)(qnqn+2u)(q1q1u)(qnqnu).
To expand this rational function into a series in u write it as
C+a11q21u++an1q2nu
to find that the constants ak are given by
ak=(qn1qn+1)[1k+1]q[nk+1]q[1k]q[nk]q.
Then write
11q2ku=m=0q2kmum.
On the other hand, using (2.14), for the image of z+(u) under the evaluation homomorphism (3.3) we get
1[n]qtrD(L+Luq2n)(L+Lu)1=1[n]qtrD(1Muq2n)(1Mu)1
which equals
1+(qn1qn+1)m=1trqMmum.
Formula (1.2) now follows by equating the coefficients of the powers um on both sides of the power expansions. Note that the formula is also valid for m = 0.□

Remark 3.1.
Due to (1.3), the Gelfand invariant trEmU(gln) is obtained as the limit value as q → 1 of the expression
1(qq1)mtrD(M1)m=1(qq1)mr=0mmr(1)mrtrqMr.
We will look at the limit value of the corresponding linear combination of the eigenvalues given by (1.2) and note that the q-numbers specialize by the rule [r]qr as q → 1. Hence, for each k = 1, …, n it suffices to find the limit value of the expression
1(qq1)mr=0mmr(1)mrq2kr=(q2k1)m(qq1)m,
which equals km, thus yielding formula (1.1).□

Since the evaluation homomorphism (3.3) is surjective, the centrality property of the elements trqMm also follows from the above calculations due to Proposition 2.1. Similarly, by using the three remaining formulas in (2.14) and (2.15), we get three more families of central elements in Uq(gln) together with the relations between them given by
trD1((L+)1L)m=trqMmandtrD1((L)1L+)m=trD(L+(L)1)m.
Moreover, additional relations between the families are provided by the isomorphism
Uq(gln)Uq1(gln),L±(L±)1.
This implies that the formulas for the eigenvalues of the other two central elements of Uq(gln) in Lq(λ) are obtained by the replacement qq−1 in (1.2):
trD(L+(L)1)mk=1nq2km[1k+1]q[nk+1]q[1k]q[nk]q.
Alternatively, these eigenvalue formulas can be derived in the same way as in the Proof of Theorem 1.1 by working with the other Liouville formula in (2.16) instead of (3.4).

This work was supported by the Australian Research Council, Grant No. DP240101572.

The authors have no conflicts to disclose.

Naihuan Jing: Methodology (equal). Ming Liu: Methodology (equal). Alexander Molev: Methodology (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

1.
Baumann
,
P.
, “
On the center of quantized enveloping algebras
,”
J. Algebra
203
,
244
260
(
1998
).
2.
Belliard
,
S.
and
Ragoucy
,
E.
, “
The nested Bethe ansatz for ‘all’ open spin chains with diagonal boundary conditions
,”
J. Phys. A: Math. Theor.
42
(
20
),
205203
(
2009
).
3.
Cherednik
,
I. V.
, “
A new interpretation of Gelfand–Tzetlin bases
,”
Duke Math. J.
54
,
563
577
(
1987
).
4.
Ding
,
J.
and
Frenkel
,
I. B.
, “
Isomorphism of two realizations of quantum affine algebra Uq(gl̂(n))
,”
Commun. Math. Phys.
156
,
277
300
(
1993
).
5.
Drinfeld
,
V. G.
, “
Hopf algebras and the quantum Yang–Baxter equation
,”
Sov. Math. Dokl.
32
,
254
258
(
1985
).
6.
Etingof
,
P. I.
, “
Central elements for quantum affine algebras and affine MacDonald’s operators
,”
Math. Res. Lett.
2
,
611
628
(
1995
).
7.
Frenkel
,
I. B.
and
Reshetikhin
,
N. Y.
, “
Quantum affine algebras and holonomic difference equations
,”
Commun. Math. Phys.
146
,
1
60
(
1992
).
8.
Gelfand
,
I. M.
, “
Center of the infinitesimal group ring
,”
Mat. Sb.
26
,
103
112
(
1950
).
9.
Gould
,
M. D.
,
Zhang
,
R. B.
, and
Bracken
,
A. J.
, “
Generalized Gel’fand invariants and characteristic identities for quantum groups
,”
J. Math. Phys.
32
,
2298
2303
(
1991
).
10.
Gurevich
,
D. I.
,
Pyatov
,
P. N.
, and
Saponov
,
P. A.
, “
Hecke symmetries and characteristic relations on reflection equation algebras
,”
Lett. Math. Phys.
41
,
255
264
(
1997
).
11.
Hopkins
,
M. J.
and
Molev
,
A. I.
, “
A q-analogue of the centralizer construction and skew representations of the quantum affine algebra
,”
SIGMA
2
,
092
(
2006
).
12.
Isaev
,
A.
,
Ogievetsky
,
O.
, and
Pyatov
,
P.
, “
On quantum matrix algebras satisfying the Cayley–Hamilton–Newton identities
,”
J. Phys. A: Math. Gen.
32
,
L115
L121
(
1999
).
13.
Jimbo
,
M.
, “
A q-difference analogue of U(g) and the Yang–Baxter equation
,”
Lett. Math. Phys.
10
,
63
69
(
1985
).
14.
Jimbo
,
M.
, “
A q-analogue of Uq(gl(N+1)), Hecke algebra, and the Yang–Baxter equation
,”
Lett. Math. Phys.
11
,
247
252
(
1986
).
15.
Jing
,
N.
,
Liu
,
M.
, and
Molev
,
A.
, “
Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: Type C
,”
J. Math. Phys.
61
(
3
),
031701
(
2020
).
16.
Jing
,
N.
,
Liu
,
M.
, and
Molev
,
A.
, “
Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: Types B and D
,”
SIGMA
16
,
043
(
2020
).
17.
Jing
,
N.
,
Liu
,
M.
, and
Molev
,
A.
, “
Quantum Sugawara operators in type A
,” arXiv:2212.11435.
18.
Jordan
,
D.
and
White
,
N.
, “
The center of the reflection equation algebra via quantum minors
,”
J. Algebra
542
,
308
342
(
2020
).
19.
Kulish
,
P. P.
and
Sklyanin
,
E. K.
, “
Quantum spectral transform method: Recent developments
,” in
Integrable Quantum Field Theories
,
Lecture Notes in Physics Vol. 151
(
Springer
,
Berlin
,
1982
), pp.
61
119
.
20.
Molev
,
A.
,
Yangians and Classical Lie Algebras
,
Mathematical Surveys and Monographs Vol. 143
(
American Mathematical Society
,
Providence, RI
,
2007
).
21.
Molev
,
A.
,
Sugawara Operators for Classical Lie Algebras
,
Mathematical Surveys and Monographs Vol. 229
(
American Mathematical Society
,
Providence, RI
,
2018
).
22.
Nazarov
,
M. L.
, “
Quantum Berezinian and the classical Capelli identity
,”
Lett. Math. Phys.
21
,
123
131
(
1991
).
23.
Nazarov
,
M.
and
Tarasov
,
V.
, “
Yangians and Gelfand–Zetlin bases
,”
Publ. Res. Inst. Math. Sci.
30
,
459
478
(
1994
).
24.
Perelomov
,
A. M.
and
Popov
,
V. S.
, “
Casimir operators for semisimple Lie algebras
,”
Izv. Akad. Nauk SSSR Ser. Mat.
32
,
1368
1390
(
1968
).
25.
Reshetikhin
,
N. Y.
, “
Quasitriangular Hopf algebras and invariants of links
,”
Algebra Anal.
1
(
2
),
169
188
(
1989
).
26.
Reshetikhin
,
N. Y.
and
Semenov-Tian-Shansky
,
M. A.
, “
Central extensions of quantum current groups
,”
Lett. Math. Phys.
19
,
133
142
(
1990
).
27.
Reshetikhin
,
N. Y.
,
Takhtadzhyan
,
L. A.
, and
Faddeev
,
L. D.
, “
Quantization of Lie groups and Lie algebras
,”
Algebra Anal.
1
(
1
),
178
206
(
1989
).
28.
Tarasov
,
V. O.
, “
Cyclic monodromy matrices for sl(n) trigonometric R-matrices
,”
Commun. Math. Phys.
158
,
459
484
(
1993
).
Published open access through an agreement with The University of Sydney - Camperdown and Darlington Campus