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*.

## I. INTRODUCTION

The *quantized enveloping algebras* and *quantum affine algebras* associated with simple Lie algebras comprise remarkable families of quantum groups, as introduced by Drinfeld^{5} 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 $Uq(gl\u0302n)$ 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 $Uq(gl\u0302n)$ 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 $Uq(gl\u0302n)$ 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.

*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*= (

*λ*

_{1}, …,

*λ*

_{n}) and combine the standard basis elements

*E*

_{ij}into the matrix

*E*= [

*E*

_{ij}]. Then the eigenvalue of the Gelfand invariant tr

*E*

^{m}in

*L*(

*λ*) is found by

*ℓ*

_{i}=

*λ*

_{i}+

*n*−

*i*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.

_{q}

*M*

^{m}of the powers of the generator matrix $M=L\u2212(L+)\u22121$ in the representation

*L*

_{q}(

*λ*) of $Uq(gln)$ (see Sec. III for the definitions). We use a standard notation for the

*q*-numbers

*The eigenvalue of the quantum Gelfand invariant*tr

_{q}

*M*

^{m}

*in*

*L*

_{q}(

*λ*)

*is found by*

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 *q* ↦ *q*^{−1}.

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.

## II. LIOUVILLE FORMULAS

*q*as a nonzero complex number which is not a root of unity. Recall the

*R*-matrix presentation of the quantum affine algebra $Uq(gl\u0302n)$ as introduced in Ref. 26. We follow

^{4}and use the same settings as in our earlier work.

^{17}Let $eij\u2208EndCn$ denote the standard matrix units. Consider the

*R*-matrix

*quantum affine algebra*$Uq(gl\u0302n)$ is generated by elements

*q*

^{c}, subject to the defining relations

*u*and

*u*

^{−1},

*a*∈ {1, …,

*k*} we will denote by

*t*

_{a}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 following proposition was stated in Ref. 2, Eq. (4.28) without proof.

*There exist a series*

*z*

^{+}(

*u*)

*in*

*u*

*and a series*

*z*

^{−}(

*u*)

*in*

*u*

^{−1}

*with coefficients in the algebra*$Uq(gl\u0302n)$

*such that*

*and*

*Moreover, the coefficients of the series*

*z*

^{±}(

*u*)

*belong to the center of the algebra*$Uq(gl\u0302n)$.

*t*

_{2}to get

*R*-matrix by

*R*-matrix $R\u2212xR\u0303$ evaluated at

*x*= 1 equals (

*q*−

*q*

^{−1})

*P*, where

*P*is the permutation operator. Therefore,

*u*=

*vq*

^{2n}we get

*Q*is an operator in $EndCn\u2297EndCn$ 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

*z*

^{−}(

*v*) commutes with

*L*

^{+}(

*u*). We have

*z*

^{+}(

*v*) are verified in the same way.□

*We have the formulas*

*and*

*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 $Uq(gl\u0302n)$:

*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).

*We have the relations*

*quantum comatrices*$L\u0302\xb1(u)$ introduced in Refs. 23 and 28. They are defined by the relations

*q*-permutation operator $Pq\u2208EndCn\u2297EndCn$ is defined by

*i*= 1, …,

*k*− 1, where

*s*

_{i}denotes the transposition (

*i*,

*i*+ 1). If $\sigma =si1\cdots sil$ is a reduced decomposition of an element $\sigma \u2208Sk$ then we set $P\sigma q=Psi1q\cdots Psilq$. Denote by

*e*

_{1}, …,

*e*

_{n}the canonical basis vectors of $Cn$. Then for any indices

*a*

_{1}< ⋯ <

*a*

_{k}and any $\tau \u2208Sk$ we have

*q*-antisymmetrizer

*fusion procedure*

^{3}for the quantum affine algebra imply the relations

*a*

_{1}< ⋯ <

*a*

_{k}then

*b*

_{1}< ⋯ <

*b*

_{k}(and the

*a*

_{i}are arbitrary) then

*i*,

*j*) entry of the matrix $L\u0302\xb1(u)$ is given by

*c*= 0, and derive the desired relations in the quotient algebra $U\xb0q(gl\u0302n)$. The mapping

*θ*we have

*θ*to both sides of relations (2.17) and then replace

*u*by

*u*

^{−1}

*q*

^{−2n+2}.□

*L*

^{±}(

*u*), we can write (2.7) as

*u*replaced by

*uq*

^{2}we find that this expression coincides with the right hand side of (2.16).

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 $Uq(gl\u0302n)$ also follows from (2.16) due to the respective properties of the quantum determinants qdet *L*^{+}(*u*) and qdet *L*^{−}(*u*).

## III. QUANTUM GELFAND INVARIANTS

^{4}to define the

*quantized enveloping algebra*$Uq(gln)$ in its

*R*-matrix presentation

^{27}as the algebra generated by elements $lij+$ and $lij\u2212$ with 1 ⩽

*i*,

*j*⩽

*n*subject to the relations

*R*-matrix

*R*is defined in (2.2), while

*L*

^{+}and

*L*

^{−}are the matrices

*q*→ 1 by the formulas

_{q}

*M*

^{m}= tr

*DM*

^{m}belong to the center of the algebra $Uq(gln)$. The elements tr

_{q}

*M*

^{m}act by multiplication by scalars in the irreducible highest weight representations

*L*

_{q}(

*λ*). The representation

*L*

_{q}(

*λ*) of $Uq(gln)$ is generated by a nonzero vector

*ξ*such that

*n*-tuple

*λ*= (

*λ*

_{1}, …,

*λ*

_{n}) of integers (or real numbers). This is a

*q*-deformation of the irreducible $gln$-module

*L*(

*λ*) with the highest weight

*λ*.

We will show that the eigenvalue of the quantum Gelfand invariant tr_{q} *M*^{m} in *L*_{q}(*λ*) is given by formula (1.2).

*L*

^{+}(

*u*) is found by

*ξ*of

*L*

_{q}(

*λ*), we find that its eigenvalue is given by

*ℓ*

_{i}=

*λ*

_{i}+

*n*−

*i*. Therefore, the eigenvalue of the image of the right hand side of (3.4) in

*L*

_{q}(

*λ*) is given by

*u*write it as

*a*

_{k}are given by

*z*

^{+}(

*u*) under the evaluation homomorphism (3.3) we get

*u*

^{m}on both sides of the power expansions. Note that the formula is also valid for

*m*= 0.□

*q*→ 1 of the expression

*q*-numbers specialize by the rule [

*r*]

_{q}→

*r*as

*q*→ 1. Hence, for each

*k*= 1, …,

*n*it suffices to find the limit value of the expression

_{q}

*M*

^{m}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

*L*

_{q}(

*λ*) are obtained by the replacement

*q*↦

*q*

^{−1}in (1.2):

## ACKNOWLEDGMENTS

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

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

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

## DATA AVAILABILITY

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

## REFERENCES

*q*-analogue of the centralizer construction and skew representations of the quantum affine algebra

*q*-difference analogue of $U(g)$ and the Yang–Baxter equation

*q*-analogue of $Uq(gl(N+1))$, Hecke algebra, and the Yang–Baxter equation

*R*-matrix and Drinfeld presentations of quantum affine algebra: Type

*C*

*R*-matrix and Drinfeld presentations of quantum affine algebra: Types

*B*and

*D*

*Integrable Quantum Field Theories*

*Lecture Notes in Physics Vol. 151*

*Yangians and Classical Lie Algebras*

*Mathematical Surveys and Monographs Vol. 143*

*Sugawara Operators for Classical Lie Algebras*

*Mathematical Surveys and Monographs Vol. 229*

*sl*(

*n*) trigonometric

*R*-matrices