Eigenvalues of quantum Gelfand invariants

Jing, Naihuan; Liu, Ming; Molev, Alexander

doi:10.1063/5.0211081

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 $g l_{n}$ ⁠. 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 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 $g l_{n}$ ⁠. Both the quantized enveloping algebra $U_{q} (g l_{n})$ and the quantum affine algebra $U_{q} ({\hat{g l}}_{n})$ 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 $U_{q} (g l_{n})$ and $U_{q} ({\hat{g l}}_{n})$ 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 $U_{q} (g l_{n})$ ⁠; see also Ref. 1. By taking the limit q → 1 one recovers the central elements of $U (g l_{n})$ going back to Ref. 8, which are known as the Gelfand invariants. Note that a different generalization of the Gelfand invariants for $g l_{n}$ as central elements in $U_{q} (g l_{n})$ ⁠, was given in Ref. 9, where their eigenvalues in irreducible highest weight representations were calculated. A new family of central elements in $U_{q} ({\hat{g l}}_{n})$ 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

U_{q} ({\hat{g l}}_{n})

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

U_{q} (g l_{n})

⁠. 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

g l_{n}

with the highest weight λ = (λ₁, …, 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

\begin{align} tr E^{m} \mapsto \sum_{k = 1}^{n} ℓ_{k}^{m} \frac{(ℓ_{1} - ℓ_{k} + 1) \dots (ℓ_{n} - ℓ_{k} + 1)}{(ℓ_{1} - ℓ_{k}) \dots \land \dots (ℓ_{n} - ℓ_{k})}, \end{align}

(1.1)

where ℓ_i = λ_i + n − i and the symbol ∧ indicates that the zero factor is skipped. Formula can be derived with the use of R-matrix calculations in the Yangian

Y (g l_{n})

or in the universal enveloping algebra

U (g l_{n})

⁠; 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 tr_q M^m of the powers of the generator matrix

M = L^{-} {(L^{+})}^{- 1}

in the representation L_q(λ) of

U_{q} (g l_{n})

(see Sec. III for the definitions). We use a standard notation for the q-numbers

{[k]}_{q} = \frac{q^{k} - q^{- k}}{q - q^{- 1}}, k \in Z .

Theorem 1.1.

The eigenvalue of the quantum Gelfand invariant tr_q M^m in L_q(λ) is found by

{tr}_{q} M^{m} \mapsto \sum_{k = 1}^{n} q^{2 ℓ_{k} m} \frac{{[ℓ_{1} - ℓ_{k} + 1]}_{q} \dots {[ℓ_{n} - ℓ_{k} + 1]}_{q}}{{[ℓ_{1} - ℓ_{k}]}_{q} \dots \land \dots {[ℓ_{n} - ℓ_{k}]}_{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 $U_{q} (g l_{n})$ which, however, turn out to coincide with the quantum Gelfand invariants, up to a possible replacement q ↦ q⁻¹.

In the limit q → 1 we have

\frac{M - 1}{q - q^{- 1}} \to E

(1.3)

by and below. Therefore, the Gelfand invariants in

U (g l_{n})

are recovered from the elements tr_q M^m in the limit, while the Perelomov–Popov formulas follow from ; 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.

II. LIOUVILLE FORMULAS

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

U_{q} ({\hat{g l}}_{n})

as introduced in Ref. 26. We follow⁴ and use the same settings as in our earlier work.¹⁷ Let

e_{i j} \in End C^{n}

denote the standard matrix units. Consider the R-matrix

R (x) = \frac{f (x)}{q - q^{- 1} x} (R - x \tilde{R}),

(2.1)

where

R = q \sum_{i} e_{i i} \otimes e_{i i} + \sum_{i \neq j} e_{i i} \otimes e_{j j} + (q - q^{- 1}) \sum_{i < j} e_{i j} \otimes e_{j i}

(2.2)

and

\tilde{R} = q^{- 1} \sum_{i} e_{i i} \otimes e_{i i} + \sum_{i \neq j} e_{i i} \otimes e_{j j} - (q - q^{- 1}) \sum_{i > j} e_{i j} \otimes e_{j i},

while the formal power series

f (x) = 1 + \sum_{k = 1}^{\infty} f_{k} x^{k}, f_{k} = f_{k} (q),

is uniquely determined by the relation

f (x q^{2 n}) = f (x) \frac{(1 - x q^{2}) (1 - x q^{2 n - 2})}{(1 - x) (1 - x q^{2 n})} .

The quantum affine algebra

U_{q} ({\hat{g l}}_{n})

is generated by elements

l_{i j}^{+} [- r], l_{i j}^{-} [r] with 1 ⩽ i, j ⩽ n, r = 0, 1, \dots,

and the invertible central element q^c, subject to the defining relations

\begin{array}{l} l_{j i}^{+} [0] & = l_{i j}^{-} [0] = 0 for 1 ⩽ i < j ⩽ n, \\ l_{i i}^{+} [0] l_{i i}^{-} [0] & = l_{i i}^{-} [0] l_{i i}^{+} [0] = 1 for i = 1, \dots, n, \end{array}

and

R (u / v) L_{1}^{\pm} (u) L_{2}^{\pm} (v) = L_{2}^{\pm} (v) L_{1}^{\pm} (u) R (u / v),

(2.3)

R (u q^{- c} / v) L_{1}^{+} (u) L_{2}^{-} (v) = L_{2}^{-} (v) L_{1}^{+} (u) R (u q^{c} / v) .

(2.4)

In the last two formulas we consider the matrices

L^{\pm} (u) = [l_{i j}^{\pm} (u)]

⁠, whose entries are formal power series in u and u⁻¹,

l_{i j}^{+} (u) = \sum_{r = 0}^{\infty} l_{i j}^{+} [- r] u^{r}, l_{i j}^{-} (u) = \sum_{r = 0}^{\infty} l_{i j}^{-} [r] u^{- r} .

Here and below we regard the matrices as elements

L^{\pm} (u) = \sum_{i, j = 1}^{n} e_{i j} \otimes l_{i j}^{\pm} (u) \in End C^{n} \otimes U_{q} ({\hat{g l}}_{n}) [[u^{\pm 1}]]

and use a subscript to indicate a copy of the matrix in the multiple tensor product algebra

\underset{k}{\underset{︸}{End C^{n} \otimes \dots \otimes End C^{n}}} \otimes U_{q} ({\hat{g l}}_{n}) [[u^{\pm 1}]]

(2.5)

so that

L_{a}^{\pm} (u) = \sum_{i, j = 1}^{n} 1^{\otimes (a - 1)} \otimes e_{i j} \otimes 1^{\otimes (k - a)} \otimes l_{i j}^{\pm} (u) .

We regard the usual matrix transposition also as the linear map

t : End C^{n} \to End C^{n}, e_{i j} \mapsto e_{j i} .

For any a ∈ {1, …, k} we will denote by t_a the corresponding partial transposition on the algebra which acts as t on the a-th copy of

End C^{n}

and as the identity map on all the other tensor factors.

The scalar factor in is necessary for the R-matrix to satisfy the crossing symmetry relations.⁷ We will use one of them given by

{(R_{12} {(x)}^{- 1})}^{t_{2}} D_{2} R_{12} {(x q^{2 n})}^{t_{2}} = D_{2},

(2.6)

where D denotes the diagonal n × n matrix

D = diag [q^{n - 1}, q^{n - 3}, \dots, q^{- n + 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

U_{q} ({\hat{g l}}_{n})

such that

\begin{align} L^{\pm} {(u q^{2 n})}^{t} D {(L^{\pm} {(u)}^{- 1})}^{t} & = z^{\pm} (u) D \end{align}

(2.7)

and

\begin{align} {(L^{\pm} {(u)}^{- 1})}^{t} D^{- 1} L^{\pm} {(u q^{2 n})}^{t} & = z^{\pm} (u) D^{- 1} . \end{align}

(2.8)

Moreover, the coefficients of the series z^±(u) belong to the center of the algebra

U_{q} ({\hat{g l}}_{n})

⁠.

Proof.

Multiply both sides of by

L_{2}^{\pm} {(v)}^{- 1}

from the left and the right and apply the transposition t₂ to get

R {(u / v)}^{t_{2}} {(L_{2}^{\pm} {(v)}^{- 1})}^{t} L_{1}^{\pm} (u) = L_{1}^{\pm} (u) {(L_{2}^{\pm} {(v)}^{- 1})}^{t} R {(u / v)}^{t_{2}}

and hence

{(R {(u / v)}^{t_{2}})}^{- 1} L_{1}^{\pm} (u) {(L_{2}^{\pm} {(v)}^{- 1})}^{t} = {(L_{2}^{\pm} {(v)}^{- 1})}^{t} L_{1}^{\pm} (u) {(R {(u / v)}^{t_{2}})}^{- 1} .

(2.9)

Use the crossing symmetry relation to replace the R-matrix by

{(R {(u / v)}^{t_{2}})}^{- 1} = D_{2}^{- 1} {(R {(u / v q^{2 n})}^{- 1})}^{t_{2}} D_{2}

and get

{(R {(u / v q^{2 n})}^{- 1})}^{t_{2}} D_{2} L_{1}^{\pm} (u) {(L_{2}^{\pm} {(v)}^{- 1})}^{t} D_{2}^{- 1} = D_{2} {(L_{2}^{\pm} {(v)}^{- 1})}^{t} L_{1}^{\pm} (u) D_{2}^{- 1} {(R {(u / v q^{2 n})}^{- 1})}^{t_{2}} .

Now cancel the scalar factors appearing in on both sides of this relation and observe that the R-matrix

R - x \tilde{R}

evaluated at x = 1 equals (q − q⁻¹)P, where P is the permutation operator. Therefore,

{({(R - \tilde{R})}^{- 1})}^{t_{2}} = \frac{1}{q - q^{- 1}} Q with Q = \sum_{i, j = 1}^{n} e_{i j} \otimes e_{i j} .

Hence, by taking u = vq²ⁿ we get

Q D_{2} L_{1}^{\pm} (v q^{2 n}) {(L_{2}^{\pm} {(v)}^{- 1})}^{t} D_{2}^{- 1} = D_{2} {(L_{2}^{\pm} {(v)}^{- 1})}^{t} L_{1}^{\pm} (v q^{2 n}) D_{2}^{- 1} Q .

Since Q is an operator in

End C^{n} \otimes End C^{n}

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

Q X_{1} = Q X_{2}^{t}

and

X_{1} Q = X_{2}^{t} Q

which hold for an arbitrary matrix X, we can write the definition of z^±(v) as

\begin{align} Q L_{2}^{\pm} {(v q^{2 n})}^{t} D_{2} {(L_{2}^{\pm} {(v)}^{- 1})}^{t} & = Q D_{2} z^{\pm} (v) \end{align}

(2.10)

and

\begin{align} {(L_{2}^{\pm} {(v)}^{- 1})}^{t} D_{2}^{- 1} L_{2}^{\pm} {(v q^{2 n})}^{t} Q & = D_{2}^{- 1} Q z^{\pm} (v) . \end{align}

(2.11)

By taking trace over the first copy of

End C^{n}

on both sides of and we arrive at and , respectively.

We will now use to show that the series z⁻(v) commutes with L⁺(u). We have

L_{1}^{+} (u) z^{-} (v) = L_{1}^{+} (u) D_{2} {(L_{2}^{-} {(v)}^{- 1})}^{t} D_{2}^{- 1} L_{2}^{-} {(v q^{2 n})}^{t} .

(2.12)

Transform relation in the same way as we did for in the beginning of the proof to get the following counterpart of :

{(R {(u q^{- c} / v)}^{t_{2}})}^{- 1} L_{1}^{+} (u) {(L_{2}^{-} {(v)}^{- 1})}^{t} = {(L_{2}^{-} {(v)}^{- 1})}^{t} L_{1}^{+} (u) {(R {(u q^{c} / v)}^{t_{2}})}^{- 1} .

Hence, the right hand side of equals

D_{2} R_{12} {(u q^{- c} / v)}^{t_{2}} {(L_{2}^{-} {(v)}^{- 1})}^{t} L_{1}^{+} (u) {(R_{12} {(u q^{c} / v)}^{t_{2}})}^{- 1} D_{2}^{- 1} L_{2}^{-} {(v q^{2 n})}^{t} .

(2.13)

Applying again , we can write

{(R_{12} {(u q^{c} / v)}^{t_{2}})}^{- 1} D_{2}^{- 1} = D_{2}^{- 1} {(R_{12} {(u q^{c} / v q^{2 n})}^{- 1})}^{t_{2}} .

Continue transforming by using the following consequence of :

L_{1}^{+} (u) {(R_{12} {(u q^{c} / v q^{2 n})}^{- 1})}^{t_{2}} L_{2}^{-} {(v q^{2 n})}^{t} = L_{2}^{-} {(v q^{2 n})}^{t} {(R_{12} {(u q^{- c} / v q^{2 n})}^{- 1})}^{t_{2}} L_{2}^{-} {(v q^{2 n})}^{t},

so that becomes

D_{2} R_{12} {(u q^{- c} / v)}^{t_{2}} {(L_{2}^{-} {(v)}^{- 1})}^{t} D_{2}^{- 1} L_{2}^{-} {(v q^{2 n})}^{t} {(R_{12} {(u q^{- c} / v q^{2 n})}^{- 1})}^{t_{2}} L_{1}^{+} (u) .

By this simplifies to

D_{2} R_{12} {(u q^{- c} / v)}^{t_{2}} D_{2}^{- 1} {(R_{12} {(u q^{- c} / v q^{2 n})}^{- 1})}^{t_{2}} z^{-} (v) L_{1}^{+} (u)

which equals

z^{-} (v) L_{1}^{+} (u)

by . This proves that

L_{1}^{+} (u) z^{-} (v) = z^{-} (v) L_{1}^{+} (u)

⁠. The relation

L_{1}^{-} (u) z^{-} (v) = z^{-} (v) L_{1}^{-} (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

\begin{align} z^{\pm} (u) & = \frac{1}{{[n]}_{q}} tr D L^{\pm} (u q^{2 n}) L^{\pm} {(u)}^{- 1} \end{align}

(2.14)

and

\begin{align} z^{\pm} (u) & = \frac{1}{{[n]}_{q}} tr D^{- 1} L^{\pm} {(u)}^{- 1} L^{\pm} (u q^{2 n}) . \end{align}

(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

U_{q} ({\hat{g l}}_{n})

⁠:

qdet L^{\pm} (u) = \sum_{σ \in S_{n}} {(- q)}^{- l (σ)} l_{σ (1) 1}^{\pm} (u q^{2 n - 2}) \dots l_{σ (n) n}^{\pm} (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^{\pm} (u) = \frac{qdet L^{\pm} (u q^{2})}{qdet L^{\pm} (u)} .

(2.16)

Proof.

We follow (Ref. 20, Sec. 1.9) and use the quantum comatrices

{\hat{L}}^{\pm} (u)

introduced in Refs. 23 and 28. They are defined by the relations

{\hat{L}}^{\pm} (u q^{2}) L^{\pm} (u) = qdet L^{\pm} (u) 1,

(2.17)

where 1 denotes the identity matrix. We will derive formulas for the entries of the matrices

{\hat{L}}^{\pm} (u)

by using the quantum minor relations reviewed e.g., in Ref. 11 which we outline below.

Recall that the q-permutation operator

P^{q} \in End C^{n} \otimes End C^{n}

is defined by

P^{q} = \sum_{i = 1}^{n} e_{i i} \otimes e_{i i} + q \sum_{i > j} e_{i j} \otimes e_{j i} + q^{- 1} \sum_{i < j} e_{i j} \otimes e_{j i} .

The action of the symmetric group

S_{k}

on the space

{(C^{n})}^{\otimes k}

can be defined by setting

s_{i} \mapsto P_{s_{i}}^{q} ≔ P_{i, i + 1}^{q}

for i = 1, …, k − 1, where s_i denotes the transposition (i, i + 1). If

σ = s_{i_{1}} \dots s_{i_{l}}

is a reduced decomposition of an element

σ \in S_{k}

then we set

P_{σ}^{q} = P_{s_{i_{1}}}^{q} \dots P_{s_{i_{l}}}^{q}

⁠. Denote by e₁, …, e_n the canonical basis vectors of

C^{n}

⁠. Then for any indices a₁ < ⋯ < a_k and any

τ \in S_{k}

we have

P_{σ}^{q} (e_{a_{τ (1)}} \otimes \dots \otimes e_{a_{τ (k)}}) = q^{l (σ τ^{- 1}) - l (τ)} e_{a_{τ σ^{- 1} (1)}} \otimes \dots \otimes e_{a_{τ σ^{- 1} (k)}} .

(2.18)

We denote by

A_{q}^{(k)}

the q-antisymmetrizer

A_{q}^{(k)} = \sum_{σ \in S_{k}} sgn σ \cdot P_{σ}^{q} .

(2.19)

The defining relations and the fusion procedure³ for the quantum affine algebra imply the relations

A_{q}^{(k)} L_{1}^{\pm} (u q^{2 k - 2}) \dots L_{k}^{\pm} (u) = L_{k}^{\pm} (u) \dots L_{1}^{\pm} (u q^{2 k - 2}) A_{q}^{(k)} .

(2.20)

The quantum minors are the series

{l^{\pm}}_{b_{1} \dots b_{k}}^{a_{1} \dots a_{k}} (u)

with coefficients in

U_{q} ({\hat{g l}}_{n})

defined by the expansion of the elements as

\sum_{a_{i}, b_{i}} e_{a_{1} b_{1}} \otimes \dots \otimes e_{a_{k} b_{k}} \otimes {l^{\pm}}_{b_{1} \dots b_{k}}^{a_{1} \dots a_{k}} (u) .

Explicit formulas for the quantum minors have the form: if a₁ < ⋯ < a_k then

{l^{\pm}}_{b_{1} \dots b_{k}}^{a_{1} \dots a_{k}} (u) = \sum_{σ \in S_{k}} {(- q)}^{- l (σ)} \cdot l_{a_{σ (1)} b_{1}}^{\pm} (u q^{2 k - 2}) \dots l_{a_{σ (k)} b_{k}}^{\pm} (u) .

(2.21)

If b₁ < ⋯ < b_k (and the a_i are arbitrary) then

{l^{\pm}}_{b_{1} \dots b_{k}}^{a_{1} \dots a_{k}} (u) = \sum_{σ \in S_{k}} {(- q)}^{l (σ)} \cdot l_{a_{k} b_{σ (k)}}^{\pm} (u) \dots l_{a_{1} b_{σ (1)}}^{\pm} (u q^{2 k - 2})

and for any

τ \in S_{k}

we have

{l^{\pm}}_{b_{τ (1)} \dots b_{τ (k)}}^{a_{1} \dots a_{k}} (u) = {(- q)}^{- l (τ)} {l^{\pm}}_{b_{1} \dots b_{k}}^{a_{1} \dots a_{k}} (u) .

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

Lemma 2.5.

The (i, j) entry of the matrix

{\hat{L}}^{\pm} (u)

is given by

{\hat{l}}_{i j}^{\pm} (u) = {(- q)}^{j - i} {l^{\pm}}_{1 \dots \hat{i} \dots n}^{1 \dots \hat{j} \dots n} (u),

(2.22)

where the hats on the right-hand side indicate the indices to be omitted.

Proof.

According to , the quantum determinants are defined by the relations

A_{q}^{(n)} L_{1}^{\pm} (u q^{2 n - 2}) \dots L_{n}^{\pm} (u) = A_{q}^{(n)} qdet L^{\pm} (u) .

Hence, the definition of the quantum comatrices implies

A_{q}^{(n)} L_{1}^{\pm} (u q^{2 n - 4}) \dots L_{n - 1}^{\pm} (u) = A_{q}^{(n)} {\hat{L}}_{n}^{\pm} (u) .

Apply both sides to the vector

e_{1} \otimes \dots \otimes {\hat{e}}_{i} \otimes \dots \otimes e_{n} \otimes e_{j}

and use and to equate the coefficients of the vector

A_{q}^{(n)} (e_{1} \otimes \dots \otimes e_{n})

to obtain .□

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

Lemma 2.6.

We have the relations

D {\hat{L}}^{\pm} {(u)}^{t} D^{- 1} L^{\pm} {(u q^{2 n - 2})}^{t} = qdet L^{\pm} (u) 1 .

Proof.

Since each relation only involves generators belonging to the subalgebra of

U_{q} ({\hat{g l}}_{n})

generated by the coefficients of the series

l_{i j}^{+} (u)

or

l_{i j}^{-} (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} ({\hat{g l}}_{n})

⁠. The mapping

θ : L^{\pm} (u) \to L^{\mp} {(u^{- 1})}^{t}

defines an automorphism of the algebra

{U °}_{q} ({\hat{g l}}_{n})

⁠. By Lemma 2.5 and the quantum minor formulas, for the images under θ we have

θ : {\hat{L}}^{\pm} (u) \mapsto D {\hat{L}}^{\mp} {(u^{- 1} q^{- 2 n + 4})}^{t} D^{- 1}

and

θ : qdet L^{\pm} (u) \mapsto qdet L^{\mp} (u^{- 1} q^{- 2 n + 2}) .

It remains to apply θ to both sides of relations and then replace u by u⁻¹q⁻²ⁿ⁺².□

Now, using and the centrality of qdet L^±(u), we can write as

z^{\pm} (u) = L^{\pm} {(u q^{2 n})}^{t} D {(L^{\pm} {(u)}^{- 1})}^{t} D^{- 1} = {(qdet L^{\pm} (u))}^{- 1} L^{\pm} {(u q^{2 n})}^{t} D {\hat{L}}^{\pm} {(u q^{2})}^{t} D^{- 1} .

By applying Lemma 2.6 with u replaced by uq² we find that this expression coincides with the right hand side of .

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 $U_{q} ({\hat{g l}}_{n})$ 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

We will follow⁴ to define the quantized enveloping algebra

U_{q} (g l_{n})

in its R-matrix presentation²⁷ as the algebra generated by elements

l_{i j}^{+}

and

l_{i j}^{-}

with 1 ⩽ i, j ⩽ n subject to the relations

\begin{aligned} l_{i j}^{-} & = l_{j i}^{+} = 0, 1 ⩽ i < j ⩽ n, \\ l_{i i}^{-} l_{i i}^{+} & = l_{i i}^{+} l_{i i}^{-} = 1, 1 ⩽ i ⩽ n, \\ R L_{1}^{\pm} L_{2}^{\pm} & = L_{2}^{\pm} L_{1}^{\pm} R, R L_{1}^{+} L_{2}^{-} = L_{2}^{-} L_{1}^{+} R, \end{aligned}

where the R-matrix R is defined in , while L⁺ and L⁻ are the matrices

L^{\pm} = \sum_{i, j} e_{i j} \otimes l_{i j}^{\pm} \in End C^{n} \otimes U_{q} (g l_{n}) .

The universal enveloping algebra

U (g l_{n})

is recovered from

U_{q} (g l_{n})

in the limit q → 1 by the formulas

\begin{align} \frac{l_{i j}^{-}}{q - q^{- 1}} & \to E_{i j}, \frac{l_{j i}^{+}}{q - q^{- 1}} \to - E_{j i} for i > j, \end{align}

(3.1)

and

\begin{align} \frac{l_{i i}^{-} - 1}{q - 1} & \to E_{i i}, \frac{l_{i i}^{+} - 1}{q - 1} \to - E_{i i} for i = 1, \dots, n . \end{align}

(3.2)

Set

M = L^{-} {(L^{+})}^{- 1}

⁠. By Ref. 27, the quantum traces defined by tr_q M^m = tr DM^m belong to the center of the algebra

U_{q} (g l_{n})

⁠. 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

U_{q} (g l_{n})

is generated by a nonzero vector ξ such that

\begin{array}{l} l_{i j}^{+} ξ & = 0 for 1 ⩽ i < j ⩽ n, \\ l_{i i}^{-} ξ & = q^{λ_{i}} ξ for 1 ⩽ i ⩽ n, \end{array}

for an n-tuple λ = (λ₁, …, λ_n) of integers (or real numbers). This is a q-deformation of the irreducible

g l_{n}

-module L(λ) with the highest weight λ.

Proof of Theorem 1.1.

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).

Recall the evaluation homomorphism

U_{q} ({\hat{g l}}_{n}) \to U_{q} (g l_{n})

defined by

L^{+} (u) \mapsto L^{+} - L^{-} u, L^{-} (u) \mapsto L^{-} - L^{+} u^{- 1}, q^{c} \mapsto 1,

(3.3)

and apply it to both sides of the Liouville formula

z^{+} (u) = \frac{qdet L^{+} (u q^{2})}{qdet L^{+} (u)}

(3.4)

proved in Theorem 2.4. The image of the quantum determinant qdet L⁺(u) is found by

\sum_{σ \in S_{n}} {(- q)}^{- l (σ)} (l_{σ (1) 1}^{+} - l_{σ (1) 1}^{-} u q^{2 n - 2}) \dots (l_{σ (n) n}^{+} - l_{σ (n) n}^{-} u) .

By applying this element to the highest vector ξ of L_q(λ), we find that its eigenvalue is given by

(q^{- λ_{1}} - q^{λ_{1} + 2 n - 2} u) \dots (q^{- λ_{n}} - q^{λ_{n}} u) = q^{n (n - 1) / 2} (q^{- ℓ_{1}} - q^{ℓ_{1}} u) \dots (q^{- ℓ_{n}} - q^{ℓ_{n}} u)

with ℓ_i = λ_i + n − i. Therefore, the eigenvalue of the image of the right hand side of in L_q(λ) is given by

\frac{(q^{- ℓ_{1}} - q^{ℓ_{1} + 2} u) \dots (q^{- ℓ_{n}} - q^{ℓ_{n} + 2} u)}{(q^{- ℓ_{1}} - q^{ℓ_{1}} u) \dots (q^{- ℓ_{n}} - q^{ℓ_{n}} u)} .

To expand this rational function into a series in u write it as

C + \frac{a_{1}}{1 - q^{2 ℓ_{1}} u} + \dots + \frac{a_{n}}{1 - q^{2 ℓ_{n}} u}

to find that the constants a_k are given by

a_{k} = (q^{n - 1} - q^{n + 1}) \frac{{[ℓ_{1} - ℓ_{k} + 1]}_{q} \dots {[ℓ_{n} - ℓ_{k} + 1]}_{q}}{{[ℓ_{1} - ℓ_{k}]}_{q} \dots \land \dots {[ℓ_{n} - ℓ_{k}]}_{q}} .

Then write

\frac{1}{1 - q^{2 ℓ_{k}} u} = \sum_{m = 0}^{\infty} q^{2 ℓ_{k} m} u^{m} .

On the other hand, using , for the image of z⁺(u) under the evaluation homomorphism we get

\frac{1}{{[n]}_{q}} tr D (L^{+} - L^{-} u q^{2 n}) {(L^{+} - L^{-} u)}^{- 1} = \frac{1}{{[n]}_{q}} tr D (1 - M u q^{2 n}) {(1 - M u)}^{- 1}

which equals

1 + (q^{n - 1} - q^{n + 1}) \sum_{m = 1}^{\infty} {tr}_{q} M^{m} u^{m} .

Formula now follows by equating the coefficients of the powers u^m on both sides of the power expansions. Note that the formula is also valid for m = 0.□

Remark 3.1.

Due to , the Gelfand invariant

tr E^{m} \in U (g l_{n})

is obtained as the limit value as q → 1 of the expression

\frac{1}{{(q - q^{- 1})}^{m}} tr D {(M - 1)}^{m} = \frac{1}{{(q - q^{- 1})}^{m}} \sum_{r = 0}^{m} (\binom{m}{r}) {(- 1)}^{m - r} {tr}_{q} M^{r} .

We will look at the limit value of the corresponding linear combination of the eigenvalues given by and note that the 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

\frac{1}{{(q - q^{- 1})}^{m}} \sum_{r = 0}^{m} (\binom{m}{r}) {(- 1)}^{m - r} q^{2 ℓ_{k} r} = \frac{{(q^{2 ℓ_{k}} - 1)}^{m}}{{(q - q^{- 1})}^{m}},

which equals

ℓ_{k}^{m}

⁠, thus yielding formula .□

Since the evaluation homomorphism is surjective, the centrality property of the elements tr_q M^m also follows from the above calculations due to Proposition 2.1. Similarly, by using the three remaining formulas in and , we get three more families of central elements in

U_{q} (g l_{n})

together with the relations between them given by

tr D^{- 1} {({(L^{+})}^{- 1} L^{-})}^{m} = {tr}_{q} M^{m} and tr D^{- 1} {({(L^{-})}^{- 1} L^{+})}^{m} = tr D {(L^{+} {(L^{-})}^{- 1})}^{m} .

Moreover, additional relations between the families are provided by the isomorphism

U_{q} (g l_{n}) \to U_{q^{- 1}} (g l_{n}), L^{\pm} \mapsto {(L^{\pm})}^{- 1} .

This implies that the formulas for the eigenvalues of the other two central elements of

U_{q} (g l_{n})

in L_q(λ) are obtained by the replacement q ↦ q⁻¹ in :

tr D {(L^{+} {(L^{-})}^{- 1})}^{m} \mapsto \sum_{k = 1}^{n} q^{- 2 ℓ_{k} m} \frac{{[ℓ_{1} - ℓ_{k} + 1]}_{q} \dots {[ℓ_{n} - ℓ_{k} + 1]}_{q}}{{[ℓ_{1} - ℓ_{k}]}_{q} \dots \land \dots {[ℓ_{n} - ℓ_{k}]}_{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 instead of .

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.

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Published open access through an agreement with The University of Sydney - Camperdown and Darlington Campus

Eigenvalues of quantum Gelfand invariants

I. INTRODUCTION

II. LIOUVILLE FORMULAS

III. QUANTUM GELFAND INVARIANTS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Eigenvalues of quantum Gelfand invariants Open Access

I. INTRODUCTION

II. LIOUVILLE FORMULAS

III. QUANTUM GELFAND INVARIANTS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Eigenvalues of quantum Gelfand invariants