The Elliptic 𝐺𝐿(𝑛) Dynamical Quantum Group as an 𝔥-Hopf Algebroid

Abstract

Using the language of 𝔥-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group, ℱ_ell(𝐺𝐿(𝑛)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra 𝔰𝔩_𝑛. We apply the generalized FRST construction and obtain an 𝔥-bialgebroid ℱ_ell(𝑀(𝑛)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing an antipode we obtain the 𝔥-Hopf algebroid ℱ_ell(𝐺𝐿(𝑛)).

1. Introduction

The quantum dynamical Yang-Baxter (QDYB) equation was introduced by Gervais and Neveu [1]. It was realized by Felder [2] that this equation is equivalent to the Star-Triangle relation in statistical mechanics. It is a generalization of the quantum Yang-Baxter equation, involving an extra, so-called dynamical, parameter. In [2] an interesting elliptic solution to the QDYB equation with spectral parameter was given, adapted from the 𝐴_𝑛⁽¹⁾ solution to the Star-Triangle relation constructed in [3]. Felder also defined a tensor category, which he suggested that it should be thought of as an elliptic analog of the category of representations of quantum groups. This category was further studied in [4] in the 𝔰𝔩₂ case.

In [5], the authors considered objects in Felder's category which were proposed as analogs of exterior and symmetric powers of the vector representation of 𝔤𝔩_𝑛. To each object in the tensor category they associate an algebra of vector-valued difference operators and prove that a certain operator, constructed from the analog of the top exterior power, commutes with all other difference operators. This is also proved in [6, Appendix B] in more detail and in [7] using a different approach.

An algebraic framework for studying dynamical R-matrices without spectral parameter was introduced in [8]. There the authors defined the notion of 𝔥-bialgebroids and 𝔥-Hopf algebroids, a special case of the Hopf algebroids defined by Lu [9]. See [10, Remark 2.1] for a comparison of Hopf algebroids to related structures. In [8] the authors also show, using a generalized version of the FRST construction, how to associate to every solution 𝑅 of the nonspectral quantum dynamical Yang-Baxter equation an 𝔥-bialgebroid. Under some extra condition they get an 𝔥-Hopf algebroid by adjoining formally the matrix elements of the inverse L-matrix. This correspondence gives a tensor equivalence between the category of representations of the R-matrix and the category of so-called dynamical representations of the 𝔥-bialgebroid.

In this paper we define an 𝔥-Hopf algebroid associated to the elliptic R-matrix from [2] with both dynamical and spectral parameters for 𝔤=𝔰𝔩_𝑛. This generalizes the spectral elliptic dynamical 𝐺𝐿(2) quantum group from [11] and the nonspectral trigonometric dynamical 𝐺𝐿(𝑛) quantum group from [12]. As in [11], this is done by first using the generalized FRST construction, modified to also include spectral parameters. In addition to the usual RLL relation, residual relations must be added “by hand” to be able to prove that different expressions for the determinant are equal.

Instead of adjoining formally all the matrix elements of the inverse L-matrix, we adjoin only the inverse of the determinant, as in [11]. Then we express the antipode using this inverse. The main problem is to find the correct formula for the determinant, to prove that it is central and to provide row and column expansion formulas for the determinant in the setting of 𝔥-bialgebroids.

It would be interesting to develop harmonic analysis for the elliptic 𝐺𝐿(𝑛) quantum group, similarly to [13]. In this context it is valuable to have an abstract algebra to work with and not only a tensor category analogous to a category of representations. For example, the analog of the Haar measure seems most naturally defined as a certain linear functional on the algebra.

The plan of this paper is as follows. After introducing some notation in Section 2.1, we recall the definition of the elliptic R-matrix in Section 2.2. In Section 3 we review the definition of 𝔥-bialgebroids and the generalized FRST construction with special emphasis on how to treat residual relations for a general R-matrix. We write down the relations explicitly in Section 4 for the algebra ℱ_ell(𝑀(𝑛)) obtained from the elliptic R-matrix. In particular we show that only one family of residual identities is needed.

Left and right analogs of the exterior algebra over ℂ^𝑛 are defined in Section 5 in a similar way as in [12]. They are certain comodule algebras over ℱ_ell(𝑀(𝑛)) and arise naturally from a single relation analogous to 𝑣⋀𝑣=0. The matrix elements of these corepresentations are generalized minors depending on a spectral parameter. Their properties are studied in Section 6. In particular we show that the left and right versions of the minors in fact coincide. In Section 6.3 we prove Laplace expansion formulas for these elliptic quantum minors.

In Section 7 we show that the 𝔥-bialgebroid ℱ_ell(𝑀(𝑛)) can be equipped with a cobraiding, in the sense of [14], extending the 𝑛=2 case from [10]. We use this and the ideas as in [5, 6] to prove that the determinant is central for all values of the spectral parameters. This implies that the determinant is central in the operator algebra as shown in [5].

Finally, in Section 7.4 we define ℱ_ell(𝐺𝐿(𝑛)) to be the localization of ℱ_ell(𝑀(𝑛)) at the determinant and show that it has an antipode giving it the structure of an 𝔥-Hopf algebroid.

2. Preliminaries

2.1. Notation

Let 𝑝,𝑞∈ℝ, 0<𝑝,𝑞<1. We assume 𝑝,𝑞 are generic in the sense that if 𝑝^𝑎𝑞^𝑏=1 for some 𝑎,𝑏∈ℤ, then 𝑎=𝑏=0.

Denote by 𝜃 the normalized Jacobi theta function: 𝜃(𝑧)=𝜃(𝑧;𝑝)=^∞_𝑗=0(1−𝑧𝑝^𝑗𝑝)(1−^𝑗+1𝑧). (2.1) It is holomorphic on ℂ^×∶=ℂ⧵{0} with zero set {𝑝^𝑘∶𝑘∈ℤ} and satisfies 𝜃(𝑧⁻¹)=𝜃(𝑝𝑧)=−𝑧⁻¹𝜃(𝑧) (2.2) and the addition formula 𝑥𝜃(𝑥𝑦,𝑦𝑧,𝑧𝑤,𝑤𝑥)=𝜃(𝑥𝑤,𝑤𝑧,𝑧𝑦,𝑦𝑧)+(𝑦𝑥)𝜃(𝑥𝑧,𝑧𝑦,𝑦𝑤,𝑤), (2.3) where we use the notation 𝜃(𝑧₁,…,𝑧_𝑛)=𝜃(𝑧₁)⋯𝜃(𝑧_𝑛). (2.4) Recall also the Jacobi triple product identity, which can be written _𝑘∈ℤ(−𝑧)^𝑘𝑝^{𝑘(𝑘−1)/2}=𝜃(𝑧)^∞_𝑗=1(1−𝑝^𝑗). (2.5) It will sometimes be convenient to use the auxiliary function 𝐸 given by 𝐸∶ℂ→ℂ,𝐸(𝑠)=𝑞^𝑠𝜃(𝑞^−2𝑠). (2.6) Relation (2.2) implies that 𝐸(−𝑠)=−𝐸(𝑠).

The set {1,2,…,𝑛} will be denoted by [1,𝑛].

2.2. The Elliptic R-Matrix

Let 𝔥 be a complex vector space, viewed as an abelian Lie algebra, 𝔥^∗ its dual space, and let 𝑉=⊕_𝜆∈𝔥^∗𝑉_𝜆 a diagonalizable 𝔥-module. A dynamical R-matrix is by definition a meromorphic function 𝑅∶𝔥^∗×ℂ^×→End_𝔥(𝑉⊗𝑉) (2.7) satisfying the quantum dynamical Yang-Baxter equation with spectral parameter (QDYBE): 𝑧𝑅(𝜆,²𝑧₃)⁽²³⁾𝑅(𝜆−ℎ²,𝑧₁𝑧₃)⁽¹³⁾𝑧𝑅(𝜆,¹𝑧₂)⁽¹²⁾=𝑅(𝜆−ℎ³,𝑧₁𝑧₂)⁽¹²⁾𝑧𝑅(𝜆,¹𝑧₃)⁽¹³⁾𝑅(𝜆−ℎ¹,𝑧₂𝑧₃)⁽²³⁾. (2.8) Equation (2.8) is an equality in the algebra of meromorphic functions 𝔥^∗×ℂ^×→End(𝑉^⊗3). Upper indices are leg-numbering notation, and ℎ indicates the action of 𝔥. For example, 𝑅(𝜆−ℎ³,𝑧₁𝑧₂)⁽¹²⁾𝑧(𝑢⊗𝑣⊗𝑤)=𝑅(𝜆−𝛼,¹𝑧₂)(𝑢⊗𝑣)⊗𝑤,if𝑤∈𝑉_𝛼. (2.9) An R-matrix 𝑅 is called unitary if 𝑅(𝜆,𝑧)𝑅(𝜆,𝑧⁻¹)⁽²¹⁾=Id_𝑉⊗𝑉 (2.10) as meromorphic functions on 𝔥^∗×ℂ^× with values in End_𝔥(𝑉⊗𝑉).

In the example we study, 𝔥 is the Cartan subalgebra of 𝔰𝔩(𝑛). Thus 𝔥 is the abelian Lie algebra of all traceless diagonal complex 𝑛×𝑛 matrices. Let 𝑉 be the 𝔥-module ℂ^𝑛 with standard basis 𝑒₁,…,𝑒_𝑛. Define 𝜔(𝑖)∈𝔥^∗ (𝑖=1,…,𝑛) by 𝜔(𝑖)(ℎ)=ℎ_𝑖,ifℎ=diag(ℎ₁,…,ℎ_𝑛)∈𝔥. (2.11) We have 𝑉=⊕^𝑛_𝑖=1𝑉_𝜔(𝑖) and 𝑉_𝜔(𝑖)=ℂ𝑒_𝑖. Define 𝑅∶𝔥^∗×ℂ^×→End(𝑉⊗𝑉) (2.12) by 𝑅(𝜆,𝑧)=^𝑛_𝑖=1𝐸_𝑖𝑖⊗𝐸_𝑖𝑖+_𝑖≠𝑗𝛼(𝜆_𝑖𝑗,𝑧)𝐸_𝑖𝑖⊗𝐸_𝑗𝑗+_𝑖≠𝑗𝛽(𝜆_𝑖𝑗,𝑧)𝐸_𝑖𝑗⊗𝐸_𝑗𝑖, (2.13) where 𝐸_𝑖𝑗∈End(𝑉) are the matrix units, 𝜆_𝑖𝑗 (𝜆∈𝔥^∗) is an abbrevation for 𝜆(𝐸_𝑖𝑖−𝐸_𝑗𝑗), and 𝛼,𝛽∶ℂ×ℂ^×→ℂ (2.14) are given by 𝛼(𝜆,𝑧)=𝛼(𝜆,𝑧;𝑝,𝑞)=𝜃(𝑧)𝜃(𝑞^2(𝜆+1))𝜃(𝑞²𝑧)𝜃(𝑞^2𝜆),𝛽(𝜆,𝑧)=𝛽(𝜆,𝑧;𝑝,𝑞)=𝜃(𝑞²)𝜃(𝑞^−2𝜆𝑧)𝜃(𝑞²𝑧)𝜃(𝑞^−2𝜆). (2.15)(2.16)

Proposition 2.1 (see [2]). The map 𝑅 is a unitary R-matrix.

For the reader's convenience, we give the explicit relationship between the R-matrix (2.13) and Felders R-matrix as written in [5] which we denote by 𝑅₁. Thus 𝑅₁∶𝔥^∗₁×ℂ→End(𝑉⊗𝑉), where 𝔥₁ is the Cartan subalgebra of 𝔤𝔩(𝑛), is defined as in (2.13) with 𝛼,𝛽 replaced by 𝛼₁,𝛽₁∶ℂ²→ℂ, 𝛼₁(𝜆,𝑥)=𝛼₁𝜃(𝜆,𝑥;𝜏,𝛾)=¹(𝑥;𝜏)𝜃₁(𝜆+𝛾;𝜏)𝜃₁(𝑥−𝛾;𝜏)𝜃₁,𝛽(𝜆;𝜏)₁(𝜆,𝑥)=𝛽₁𝜃(𝜆,𝑥;𝜏,𝛾)=−¹(𝑥+𝜆;𝜏)𝜃₁(𝛾;𝜏)𝜃₁(𝑥−𝛾;𝜏)𝜃₁.(𝜆;𝜏) (2.17) Here 𝜏,𝛾∈ℂ with Im𝜏>0, and 𝜃₁ is the first Jacobi theta function: 𝜃₁(𝑥;𝜏)=−_{𝑗∈ℤ+1/2}𝑒^{𝜋𝑖𝑗2𝜏+2𝜋𝑖𝑗(𝑥+1/2)}. (2.18) As proved in [2], 𝑅₁ satisfies the following version of the QDYBE: 𝑅₁(𝜆−𝛾ℎ³,𝑥₁−𝑥₂)⁽¹²⁾𝑅₁(𝜆,𝑥₁−𝑥₃)⁽¹³⁾𝑅₁(𝜆−𝛾ℎ¹,𝑥₂−𝑥₃)⁽²³⁾=𝑅₁(𝜆,𝑥₂−𝑥₃)⁽²³⁾𝑅₁(𝜆−𝛾ℎ²,𝑥₁−𝑥₃)⁽¹³⁾𝑅₁(𝜆,𝑥₁−𝑥₂)⁽¹²⁾ (2.19) and the unitarity condition 𝑅₁(𝜆,𝑥)𝑅₁²¹(𝜆,−𝑥)=Id_𝑉⊗𝑉. (2.20) We can identify 𝔥^∗≃𝔥^∗₁/ℂtr where tr∈𝔥^∗₁ is the trace. Since 𝑅₁ has the form (2.13), it is constant, as a function of 𝜆∈𝔥^∗₁, on the cosets modulo ℂtr. So 𝑅₁ induces a map 𝔥^∗×ℂ→End(𝑉⊗𝑉), which we also denote by 𝑅₁, still satisfying (2.19), (2.20).

Let 𝜏,𝛾∈ℂ with Im𝜏>0 be such that 𝑝=𝑒^𝜋𝑖𝜏, 𝑞=𝑒^𝜋𝑖𝛾. Then, as meromorphic functions of (𝜆,𝑥)∈𝔥^∗×ℂ, 𝑅₁𝜏(𝛾𝜆,−𝑥;2,𝛾)=𝑅(𝜆,𝑧;𝑝,𝑞), (2.21) where 𝑧=𝑒^{2𝜋𝑖𝑥}. Indeed, using the Jacobi triple product identity (2.5) we have 𝜃₁𝜏(𝑥;2)=𝑖𝑒^{𝜋𝑖(𝜏/2−𝑥)}𝜃(𝑧)^∞_𝑗=1(1−𝑝^𝑗), (2.22) and substituting this into (2.17) gives 𝛼₁(𝛾𝜆,−𝑥;𝜏/2,𝛾)=𝛼(𝜆,𝑧;𝑝,𝑞) and 𝛽₁(𝛾𝜆,−𝑥;𝜏/2,𝛾)=𝛽(𝜆,𝑧;𝑝,𝑞) which proves (2.21).

By replacing 𝜆, 𝑥_𝑖 in (2.19) by 𝛾𝜆, −𝑥_𝑖 and using (2.21) we obtain (2.8) with 𝑧_𝑖=𝑒^{2𝜋𝑖𝑥𝑖}. Similarly the unitarity (2.10) of 𝑅 is obtained from (2.20).

2.3. Useful Identities

We end this section by recording some useful identities. Recall the definitions of 𝛼,𝛽 in (2.15). It is immediate that 𝛼(𝜆,𝑞²)=𝛽(−𝜆,𝑞²). (2.23) By induction, one generalizes (2.2) to 𝜃(𝑝^𝑠𝑧)=(−1)^𝑠(𝑝^{𝑠(𝑠−1)/2}𝑧^𝑠)⁻¹𝜃(𝑧),for𝑠∈ℤ. (2.24) Applying (2.24) to the definitions of 𝛼, 𝛽 we get 𝛼(𝜆,𝑝^𝑘𝑧)=𝑞^2𝑘𝛼(𝜆,𝑧),𝛽(𝜆,𝑝^𝑘𝑧)=𝑞^{2𝑘(𝜆+1)}𝛽(𝜆,𝑧), (2.25) and, using also 𝜃(𝑧⁻¹)=−𝑧⁻¹𝜃(𝑧), lim_{𝑧→𝑝−𝑘𝑞−2}𝑞⁻¹𝜃(𝑞²𝑧)𝑞𝜃(𝑞⁻²𝑧)𝛼(𝜆,𝑧)=𝛼(𝜆,𝑝^𝑘𝑞²),lim_{𝑧→𝑝−𝑘𝑞−2}𝑞⁻¹𝜃(𝑞²𝑧)𝑞𝜃(𝑞⁻²𝑧)𝛽(𝜆,𝑧)=−𝛽(−𝜆,𝑝^𝑘𝑞²), (2.26) for 𝜆∈ℂ, 𝑧∈ℂ^×, and 𝑘∈ℤ. By the addition formula (2.3) with (𝑥,𝑦,𝑧,𝑤)=(𝑧^1/2𝑞^−𝜆+1,𝑧^1/2𝑞^𝜆−1,𝑧^1/2𝑞^𝜆+1,𝑧^1/2𝑞^−𝜆−1), (2.27) we have 𝛼(𝜆,𝑧)𝛼(−𝜆,𝑧)−𝛽(𝜆,𝑧)𝛽(−𝜆,𝑧)=𝑞²𝜃(𝑞⁻²𝑧)𝜃(𝑞².𝑧) (2.28)

3. 𝔥-Bialgebroids

3.1. Definitions

We recall some definitions from [8]. Let 𝔥^∗ be a finite-dimensional complex vector space (e.g., the dual space of an abelian Lie algebra), and let 𝑀_𝔥^∗ be the field of meromorphic functions on 𝔥^∗.

Definition 3.1. An 𝔥-algebra is a complex associative algebra 𝐴 with 1 which is bigraded over 𝔥^∗, 𝐴=⊕_{𝛼,𝛽∈𝔥}^∗𝐴_𝛼𝛽, and equipped with two algebra embeddings 𝜇_𝑙,𝜇_𝑟∶𝑀_𝔥^∗→𝐴, called the left and right moment maps, such that 𝜇_𝑙(𝑓)𝑎=𝑎𝜇_𝑙(𝑇_𝛼𝑓),𝜇_𝑟(𝑓)𝑎=𝑎𝜇_𝑟(𝑇_𝛽𝑓),for𝑎∈𝐴_𝛼𝛽,𝑓∈𝑀_𝔥^∗, (3.1) where 𝑇_𝛼 denotes the automorphism (𝑇_𝛼𝑓)(𝜁)=𝑓(𝜁+𝛼) of 𝑀_𝔥^∗. A morphism of 𝔥-algebras is an algebra homomorphism preserving the bigrading and the moment maps.

The matrix tensor product 𝐴⊗𝐵 of two 𝔥-algebras 𝐴, 𝐵 is the 𝔥^∗-bigraded vector space with (𝐴⊗𝐵)_𝛼𝛽=⊕_𝛾∈𝔥^∗(𝐴_𝛼𝛾⊗_𝑀𝔥∗𝐵_𝛾𝛽), where ⊗_𝑀𝔥∗ denotes tensor product over ℂ modulo the relations: 𝜇^𝐴_𝑟(𝑓)𝑎⊗𝑏=𝑎⊗𝜇^𝐵_𝑙(𝑓)𝑏,∀𝑎∈𝐴,𝑏∈𝐵,𝑓∈𝑀_𝔥^∗. (3.2) The multiplication (𝑎⊗𝑏)(𝑐⊗𝑑)=𝑎𝑐⊗𝑏𝑑 for 𝑎,𝑐∈𝐴 and 𝑏,𝑑∈𝐵 and the moment maps 𝜇_𝑙(𝑓)=𝜇^𝐴_𝑙(𝑓)⊗1 and 𝜇_𝑟(𝑓)=1⊗𝜇^𝐵_𝑟(𝑓) make 𝐴⊗𝐵 into an 𝔥-algebra.

Example 3.2. Let 𝐷_𝔥 be the algebra of operators on 𝑀_𝔥^∗ of the form ∑_𝑖𝑓_𝑖𝑇_𝛼𝑖 with 𝑓_𝑖∈𝑀_𝔥^∗ and 𝛼_𝑖∈𝔥^∗. It is an 𝔥-algebra with bigrading 𝑓𝑇_−𝛼∈(𝐷_𝔥)_𝛼𝛼, and both moment maps equal to the natural embedding.

For any 𝔥-algebra 𝐴, there are canonical isomorphisms 𝐴≃𝐴⊗𝐷_𝔥≃𝐷_𝔥⊗𝐴 defined by 𝑥≃𝑥⊗𝑇_−𝛽≃𝑇_−𝛼⊗𝑥,for𝑥∈𝐴_𝛼𝛽. (3.3)

Definition 3.3. An 𝔥-bialgebroid is an 𝔥-algebra 𝐴 equipped with two 𝔥-algebra morphisms, the comultiplication Δ∶𝐴→𝐴⊗𝐴 and the counit 𝜀∶𝐴→𝐷_𝔥 such that (Δ⊗Id)∘Δ=(Id⊗Δ)∘Δ and (𝜀⊗Id)∘Δ=Id=(Id⊗𝜀)∘Δ, under the identifications (3.3).

3.2. The Generalized FRST Construction

In [8] the authors gave a generalized FRST construction which attaches an 𝔥-bialgebroid to each solution of the quantum dynamical Yang-Baxter equation without spectral parameter. One way of extending to the case including a spectral parameter is described in [11]. However, when specifying the R-matrix to (2.13) with 𝑛=2, they had to impose in addition certain so-called residual relations in order to prove, for example, that the determinant is central. Such relations were also required in [4] in a different algebraic setting. In the setting of operator algebras, where the algebras consist of linear operators on a vector space depending meromorphically on the spectral variables, as in [5], such relations are consequences of the ordinary RLL relations by taking residues.

Another motivation for our procedure is that 𝔥-bialgebroids associated to gauge equivalent R-matrices should be isomorphic. In particular one should be allowed to multiply the R-matrix by any nonzero meromorphic function of the spectral variable without changing the isomorphism class of the associated algebra (for the full definition of gauge equivalent R-matrices see [8]).

These considerations suggest the following procedure for constructing an 𝔥-bialgebroid from a quantum dynamical R-matrix with spectral parameter.

Let 𝔥 be a finite-dimensional abelian Lie algebra, 𝑉=⊕_𝛼∈𝔥^∗𝑉_𝛼 a finite-dimensional diagonalizable 𝔥-module, and 𝑅∶𝔥^∗×ℂ^×→End_𝔥(𝑉⊗𝑉) a meromorphic function. We attach to this data an 𝔥-bialgebroid 𝐴_𝑅 as follows. Let {𝑒_𝑥}_𝑥∈𝑋 be a homogeneous basis of 𝑉, where 𝑋 is an index set. The matrix elements 𝑅^𝑎𝑏_𝑥𝑦∶𝔥^∗×ℂ^×→ℂ of 𝑅 are given by 𝑅(𝜁,𝑧)(𝑒_𝑎⊗𝑒_𝑏)=_{𝑥,𝑦∈𝑋}𝑅^𝑎𝑏_𝑥𝑦(𝜁,𝑧)𝑒_𝑥⊗𝑒_𝑦. (3.4) They are meromorphic on 𝔥^∗×ℂ^×. Define 𝜔∶𝑋→𝔥^∗ by 𝑒_𝑥∈𝑉_𝜔(𝑥). Let 𝐴_𝑅 be the complex associative algebra with 1 generated by {𝐿_𝑥𝑦(𝑧)∶𝑥,𝑦∈𝑋,𝑧∈ℂ^×} and two copies of 𝑀_𝔥^∗, whose elements are denoted by 𝑓(𝜆) and 𝑓(𝜌), respectively, with defining relations 𝑓(𝜆)𝑔(𝜌)=𝑔(𝜌)𝑓(𝜆) for 𝑓,𝑔∈𝑀_𝔥^∗ and 𝑓(𝜆)𝐿_𝑥𝑦(𝑧)=𝐿_𝑥𝑦(𝑧)𝑓(𝜆+𝜔(𝑥)),𝑓(𝜌)𝐿_𝑥𝑦(𝑧)=𝐿_𝑥𝑦(𝑧)𝑓(𝜌+𝜔(𝑦)), (3.5) for all 𝑥,𝑦∈𝑋, 𝑧∈ℂ^× and 𝑓∈𝑀_𝔥^∗. The bigrading on 𝐴_𝑅 is given by 𝐿_𝑥𝑦(𝑧)∈(𝐴_𝑅)_{𝜔(𝑥),𝜔(𝑦)} for 𝑥,𝑦∈𝑋, 𝑧∈ℂ^× and 𝑓(𝜆),𝑓(𝜌)∈(𝐴_𝑅)₀₀ for 𝑓∈𝑀_𝔥^∗. The moment maps are defined by 𝜇_𝑙(𝑓)=𝑓(𝜆), 𝜇_𝑟(𝑓)=𝑓(𝜌). The counit and comultiplication are defined by 𝜀(𝐿_𝑎𝑏(𝑧))=𝛿_𝑎𝑏𝑇_{−𝜔(𝑎)},𝜀(𝑓(𝜆))=𝜀(𝑓(𝜌))=𝑓𝑇₀,Δ(𝐿_𝑎𝑏(𝑧))=_𝑥∈𝑋𝐿_𝑎𝑥(𝑧)⊗𝐿_𝑥𝑏(𝑧),Δ(𝑓(𝜆))=𝑓(𝜆)⊗1,Δ(𝑓(𝜌))=1⊗𝑓(𝜌). (3.6) This makes 𝐴_𝑅 into an 𝔥-bialgebroid.

Consider the ideal in 𝐴_𝑅 generated by the RLL relations: _{𝑥,𝑦∈𝑋}𝑅^𝑥𝑦_𝑎𝑐𝑧(𝜆,¹𝑧₂)𝐿_𝑥𝑏(𝑧₁)𝐿_𝑦𝑑(𝑧₂)=_{𝑥,𝑦∈𝑋}𝑅^𝑏𝑑_𝑥𝑦𝑧(𝜌,¹𝑧₂)𝐿_𝑐𝑦(𝑧₂)𝐿_𝑎𝑥(𝑧₁), (3.7) where 𝑎,𝑏,𝑐,𝑑∈𝑋, and 𝑧₁,𝑧₂∈ℂ^×. More precisely, to account for possible singularities of 𝑅, we let 𝐼_𝑅 be the ideal in 𝐴_𝑅 generated by all relations of the form _{𝑥,𝑦∈𝑋}lim_{𝑤→𝑧1/𝑧2}(𝜑(𝑤)𝑅^𝑥𝑦_𝑎𝑐(𝜆,𝑤))𝐿_𝑥𝑏(𝑧₁)𝐿_𝑦𝑑(𝑧₂)=_{𝑥,𝑦∈𝑋}lim_{𝑤→𝑧1/𝑧2}(𝜑(𝑤)𝑅^𝑏𝑑_𝑥𝑦(𝜌,𝑤))𝐿_𝑐𝑦(𝑧₂)𝐿_𝑎𝑥(𝑧₁), (3.8) where 𝑎,𝑏,𝑐,𝑑∈𝑋, 𝑧₁,𝑧₂∈ℂ^×, and 𝜑∶ℂ^×→ℂ is a meromorphic function such that the limits exist.

We define 𝐴_𝑅 to be 𝐴_𝑅/𝐼_𝑅. The bigrading descends to 𝐴_𝑅 because (3.8) is homogeneous, of bidegree 𝜔(𝑎)+𝜔(𝑐),𝜔(𝑏)+𝜔(𝑑), by the 𝔥-invariance of 𝑅. One checks that Δ(𝐼_𝑅)⊆𝐴_𝑅⊗𝐼_𝑅+𝐼_𝑅⊗𝐴_𝑅 and 𝜀(𝐼_𝑅)=0. Thus 𝐴_𝑅 is an 𝔥-bialgebroid with the induced maps.

Remark 3.4. Objects in Felder's tensor category associated to an R-matrix 𝑅 are certain meromorphic functions 𝐿∶𝔥^∗×ℂ^×→End_𝔥(ℂ^𝑛⊗𝑊) where 𝑊 is a finite-dimensional 𝔥-module [2]. After regularizing 𝐿 with respect to the spectral parameter it will give rise to a dynamical representation of the 𝔥-bialgebroid 𝐴_𝑅 in the same way as in the nonspectral case treated in [8]. The residual relations incorporated in (3.8) are crucial for this fact to be true in the present, spectral, case.

3.3. Operator form of the RLL Relations

It is well known that the RLL relation (3.7) can be written as a matrix relation. We show how this is done in the present setting. It will be used later in Section 6.2.

Assume 𝑅^𝑎𝑏_𝑥𝑦(𝜁,𝑧) are defined, as meromorphic functions of 𝜁∈𝔥^∗ for any 𝑧∈ℂ^×. Define 𝖱(𝜆,𝑧),𝖱(𝜌,𝑧)∈End(𝑉⊗𝑉⊗𝐴_𝑅) by 𝖱(𝜆,𝑧)(𝑒_𝑎⊗𝑒_𝑏⊗𝑢)=_{𝑥,𝑦∈𝑋}𝑒_𝑥⊗𝑒_𝑦⊗𝑅^𝑎𝑏_𝑥𝑦(𝜆,𝑧)𝑢,𝖱(𝜌,𝑧)(𝑒_𝑎⊗𝑒_𝑏⊗𝑢)=_{𝑥,𝑦∈𝑋}𝑒_𝑥⊗𝑒_𝑦⊗𝑅^𝑎𝑏_𝑥𝑦(𝜌,𝑧)𝑢, (3.9) for 𝑎,𝑏∈𝑋, 𝑢∈𝐴_𝑅. Note that the 𝜆 and 𝜌 in the left-hand side are not variables but merely indicate which moment map is to be used. For 𝑧∈ℂ^× we also define 𝖫(𝑧)∈End(𝑉⊗𝐴_𝑅) by 𝖫(𝑧)=_{𝑥,𝑦∈𝑋}𝐸_𝑥𝑦⊗𝐿_𝑥𝑦(𝑧). (3.10) Here 𝐸_𝑥𝑦 are the matrix units in End(𝑉), and 𝐴_𝑅 acts on itself by left multiplication. The RLL relation (3.7) is equivalent to 𝑧𝖱(𝜆,¹𝑧₂)𝖫¹(𝑧₁)𝖫²(𝑧₂)=𝖫²(𝑧₂)𝖫¹(𝑧₁)𝖱(𝜌+ℎ¹+ℎ²,𝑧₁𝑧₂) (3.11) in End(𝑉⊗𝑉⊗𝐴_𝑅), where 𝖫^𝑖(𝑧)=𝖫(𝑧)^(𝑖,3)∈End(𝑉⊗𝑉⊗𝐴_𝑅) for 𝑖=1,2 and the operator 𝖱(𝜌+ℎ¹+ℎ²,𝑧₁/𝑧₂)∈End(𝑉⊗𝑉⊗𝐴_𝑅) is given by 𝑒_𝑎⊗𝑒_𝑏⊗𝑢↦_{𝑥,𝑦∈𝑋}𝑒_𝑥⊗𝑒_𝑦⊗𝑅^𝑎𝑏_𝑥𝑦𝑧(𝜌+𝜔(𝑎)+𝜔(𝑏),¹𝑧₂)𝑢, (3.12) where 𝑅^𝑎𝑏_𝑥𝑦(𝜌+𝜔(𝑎)+𝜔(𝑏),𝑧₁/𝑧₂) means the image in 𝐴_𝑅 of the meromorphic function 𝔥^∗∋𝜁↦𝑅^𝑎𝑏_𝑥𝑦(𝜁+𝜔(𝑎)+𝜔(𝑏),𝑧₁/𝑧₂) under the right moment map 𝜇_𝑟. This equivalence can be seen by acting on 𝑒_𝑏⊗𝑒_𝑑⊗1 in both sides of (3.11) and collecting and equating terms of the form 𝑒_𝑎⊗𝑒_𝑐⊗𝑢. The matrix elements of the R-matrix in the right-hand side can then be moved to the left using that 𝑅 is 𝔥-invariant and using relation (3.5).

4. The Algebra ℱ_ell(𝑀(𝑛))

We now specialize to the case where 𝔥 is the Cartan subalgebra of 𝔰𝔩(𝑛), 𝑉=ℂ^𝑛, and 𝑅 is given by (2.13)–(2.16). The case 𝑛=2 was considered in [11]. We will show that (3.8) contains precisely one additional family of relations, as compared to (3.7), and we write down all relations explicitly.

When we apply the generalized FRST construction to this data we obtain an 𝔥-bialgebroid which we denote by ℱ_ell(𝑀(𝑛)). The generators 𝐿_𝑖𝑗(𝑧) will be denoted by 𝑒_𝑖𝑗(𝑧). Thus ℱ_ell(𝑀(𝑛)) is the unital associative ℂ-algebra generated by 𝑒_𝑖𝑗(𝑧), 𝑖,𝑗∈[1,𝑛], 𝑧∈ℂ^×, and two copies of 𝑀_𝔥^∗, whose elements are denoted by 𝑓(𝜆) and 𝑓(𝜌) for 𝑓∈𝑀_𝔥^∗, subject to the following relations: 𝑓(𝜆)𝑒_𝑖𝑗(𝑧)=𝑒_𝑖𝑗(𝑧)𝑓(𝜆+𝜔(𝑖)),𝑓(𝜌)𝑒_𝑖𝑗(𝑧)=𝑒_𝑖𝑗(𝑧)𝑓(𝜌+𝜔(𝑗)), (4.1) for all 𝑓∈𝑀_𝔥^∗, 𝑖,𝑗∈[1,𝑛], and 𝑧∈ℂ^×, and ^𝑛_𝑥,𝑦=1𝑅^𝑥𝑦_𝑎𝑐𝑧(𝜆,¹𝑧₂)𝑒_𝑥𝑏(𝑧₁)𝑒_𝑦𝑑(𝑧₂)=^𝑛_𝑥,𝑦=1𝑅^𝑏𝑑_𝑥𝑦𝑧(𝜌,¹𝑧₂)𝑒_𝑐𝑦(𝑧₂)𝑒_𝑎𝑥(𝑧₁), (4.2) for all 𝑎,𝑏,𝑐,𝑑∈[1,𝑛]. More explicitly, from (2.13) we have 𝑅^𝑎𝑏_𝑥𝑦(𝜁,𝑧)=1,𝑎=𝑏=𝑥=𝑦,𝛼(𝜁_𝑥𝑦,𝑧),𝑎≠𝑏,𝑥=𝑎,𝑦=𝑏,𝛽(𝜁_𝑥𝑦,𝑧),𝑎≠𝑏,𝑥=𝑏,𝑦=𝑎,0,otherwise, (4.3) which substituted into (4.2) yields four families of relations: 𝑒_𝑎𝑏(𝑧₁)𝑒_𝑎𝑏(𝑧₂)=𝑒_𝑎𝑏(𝑧₂)𝑒_𝑎𝑏(𝑧₁),𝑒_𝑎b(𝑧₁)𝑒_𝑎𝑑(𝑧₂)=𝛼(𝜌_𝑏𝑑,𝑧₁𝑧₂)𝑒_𝑎𝑑(𝑧₂)𝑒_𝑎𝑏(𝑧₁)+𝛽(𝜌_𝑑𝑏,𝑧₁𝑧₂)𝑒_𝑎𝑏(𝑧₂)𝑒_𝑎𝑑(𝑧₁),𝛼(𝜆_𝑎𝑐,𝑧₁𝑧₂)𝑒_𝑎𝑏(𝑧₁)𝑒_𝑐𝑏(𝑧₂)+𝛽(𝜆_𝑎𝑐,𝑧₁𝑧₂)𝑒_𝑐𝑏(𝑧₁)𝑒_𝑎𝑏(𝑧₂)=𝑒_𝑐𝑏(𝑧₂)𝑒_𝑎𝑏(𝑧₁),𝛼(𝜆_𝑎𝑐,𝑧₁𝑧₂)𝑒_𝑎𝑏(𝑧₁)𝑒_𝑐𝑑(𝑧₂)+𝛽(𝜆_a𝑐,𝑧₁𝑧₂)𝑒_𝑐𝑏(𝑧₁)𝑒_𝑎𝑑(𝑧₂)=𝛼(𝜌_𝑏𝑑,𝑧₁𝑧₂)𝑒_𝑐𝑑(𝑧₂)𝑒_𝑎𝑏(𝑧₁)+𝛽(𝜌_𝑑𝑏,𝑧₁𝑧₂)𝑒_𝑐𝑏(𝑧₂)𝑒_𝑎𝑑(𝑧₁), (4.4a)(4.4b)(4.4c)(4.4d) where 𝑎,𝑏,𝑐,𝑑∈[1,𝑛], 𝑎≠𝑐, and 𝑏≠𝑑. Since 𝜃 has zeros precisely at 𝑝^𝑘,𝑘∈ℤ, 𝛼 and 𝛽 have poles at 𝑧=𝑞⁻²𝑝^𝑘,𝑘∈ℤ. Thus (4.4b)–(4.4d) are to hold for 𝑧₁,𝑧₂∈ℂ^× with 𝑧₁/𝑧₂∉{𝑝^𝑘𝑞⁻²∶𝑘∈ℤ}.

In (3.8), assuming 𝑎≠𝑐, 𝑏≠𝑑, taking 𝑧₁=𝑧, 𝑧₂=𝑝^𝑘𝑞²𝑧, 𝜑(𝑤)=𝑞⁻¹𝜃(𝑞²𝑤)/𝑞𝜃(𝑞⁻²𝑤), and using the limit formulas (2.26), we obtain the relation 𝛼(𝜆_𝑎𝑐,𝑞²)(𝑒_𝑎𝑏(𝑧)𝑒_𝑐𝑑(𝑝^𝑘𝑞²𝑧)−𝑞^{2𝑘𝜆𝑐𝑎}𝑒_𝑐𝑏(𝑧)𝑒_𝑎𝑑(𝑝^𝑘𝑞²𝑧))=𝛼(𝜌_𝑏𝑑,𝑞²)𝑒_𝑐𝑑(𝑝^𝑘𝑞²𝑧)𝑒_𝑎𝑏(𝑧)−𝑞^{2𝑘𝜌𝑏𝑑}𝛽(𝜌_𝑏𝑑,𝑞²)𝑒_𝑐𝑏(𝑝^𝑘𝑞²𝑧)𝑒_𝑎𝑑(𝑧). (4.5) This identity does not follow from (4.4a)–(4.4d) in an obvious way. It will be called the residual RLL relation.

Proposition 4.1. Relations (4.4a)–(4.4d), and (4.5) generate the ideal 𝐼_𝑅. Hence (4.1), (4.4a)–(4.4d), and (4.5) consitute the defining relations of the algebra ℱ_ell(𝑀(𝑛)).

Proof. Assume that we have a relation of the form (3.8) and that a limit in one of the terms, lim_𝑤→𝑧𝜑(𝑤)𝑅^𝑎𝑏_𝑥𝑦(𝜆,𝑤), say, exists and is nonzero. Then one of the following cases occurs.
(1)At 𝑤=𝑧, 𝜑(𝑤) and 𝑅^𝑎𝑏_𝑥𝑦(𝜆,𝑤) are both regular. If this holds for all terms, then the relation is just a multiple of one of (4.4a)–(4.4d).(2)At 𝑤=𝑧, 𝜑(𝑤) has a pole while 𝑅^𝑎𝑏_𝑥𝑦(𝜆,𝑤) is regular. Then 𝑅^𝑎𝑏_𝑥𝑦(𝜆,𝑤) must vanish identically at 𝑤=𝑧. The only case where this is possible is when 𝑥≠𝑦 and 𝑅^𝑎𝑏_𝑥𝑦(𝜆,𝑤)=𝛼(𝜆_𝑥𝑦,𝑤), and 𝑧=𝑝^𝑘. But then there is another term containing 𝛽 which is never identically zero for any 𝑧, and hence the limit in that term does not exist.(3)At 𝑤=𝑧, 𝜑(𝑤) is regular while 𝑅^𝑎𝑏_𝑥𝑦(𝜆,𝑤) has a pole. Since these poles are simple and occur only when 𝑧∈𝑞⁻²𝑝^ℤ, the function 𝜑 must have a zero of multiplicity one there. We can assume without loss of generality that 𝜑 has the specific form 𝑞𝜑(𝑤)=⁻¹𝜃(𝑞²𝑤)𝑞𝜃(𝑞⁻².𝑤) (4.6)
Then, if 𝑎≠𝑐 and 𝑏≠𝑑, (3.8) becomes the residual RLL relation (4.5).
If instead 𝑐=𝑎, 𝑏≠𝑑, and we take 𝑧₁=𝑧, 𝑧₂=𝑝^𝑘𝑞²𝑧 in (3.8), we get, using (2.26), 0=𝛼(𝜌_𝑏𝑑,𝑝^𝑘𝑞²)𝑒_𝑎𝑑(𝑝^𝑘𝑞²𝑧)𝑒_𝑎𝑏(𝑧)−𝛽(𝜌_𝑏𝑑,𝑝^𝑘𝑞²)𝑒_𝑎𝑏(𝑝^𝑘𝑞²𝑧)𝑒_𝑎𝑑(𝑧), (4.7) or, rewritten, 𝑒_𝑎𝑑(𝑝^𝑘𝑞²𝑧)𝑒_𝑎𝑏(𝑧)=𝑞^{2𝑘𝜌𝑏𝑑}𝐸(𝜌_𝑏𝑑−1)𝐸(𝜌_𝑏𝑑𝑒+1)^𝑎𝑏(𝑝^𝑘𝑞²𝑧)𝑒_𝑎𝑑(𝑧). (4.8) However this relation is already derivable from (4.4b) as follows. Take 𝑧₁=𝑝^𝑘𝑞²𝑧 and 𝑧₂=𝑧 in (4.4b) multiply both sides by 𝑞^{2𝑘𝜌𝑏𝑑}(𝐸(𝜌_𝑏𝑑−1)/𝐸(𝜌_𝑏𝑑+1)), and then use (4.4b) on the right-hand side.
Similarly, if 𝑎≠𝑐, 𝑑=𝑏, 𝑧₁=𝑧, 𝑧₂=𝑝^𝑘𝑞²𝑧, 𝜑(𝑤)=𝑞⁻¹𝜃(𝑞²𝑤)/𝑞𝜃(𝑞⁻²𝑤) in (3.8), and using (2.26) we get 𝛼(𝜆_𝑎𝑐,𝑝^𝑘𝑞²)𝑒_𝑎𝑏(𝑧)𝑒_𝑐𝑏(𝑝^𝑘𝑞²𝑧)−𝛽(𝜆_𝑐𝑎,𝑝^𝑘𝑞²)𝑒_𝑐𝑏(𝑧)𝑒_𝑎𝑏(𝑝^𝑘𝑞²𝑧)=0, (4.9) or 𝑒_𝑎𝑏(𝑧)𝑒_𝑐𝑏(𝑝^𝑘𝑞²𝑧)=𝑞^{2𝑘𝜆𝑐𝑎}𝑒_𝑐𝑏(𝑧)𝑒_𝑎𝑏(𝑝^𝑘𝑞²𝑧). (4.10) Similarly to the previous case, this identity follows already from (4.4c).

5. Left and Right Elliptic Exterior Algebras

5.1. Corepresentations of 𝔥-Bialgebroids

We recall the definition of corepresentations of an 𝔥-bialgebroid given in [13].

Definition 5.1. An 𝔥-space 𝑉 is an 𝔥^∗-graded vector space over 𝑀_𝔥^∗, 𝑉=⊕_𝛼∈𝔥^∗𝑉_𝛼, where each 𝑉_𝛼 is 𝑀_𝔥^∗-invariant. A morphism of 𝔥-spaces is a degree-preserving 𝑀_𝔥^∗-linear map.

Given an 𝔥-space 𝑉 and an 𝔥-bialgebroid 𝐴, we define 𝐴⊗𝑉 to be the 𝔥^∗-graded space with (𝐴⊗𝑉)_𝛼=⊕_𝛽∈𝔥^∗(𝐴_𝛼𝛽⊗_𝑀𝔥∗𝑉_𝛽), where ⊗_𝑀𝔥∗ denotes ⊗_ℂ modulo the relations 𝜇_𝑟(𝑓)𝑎⊗𝑣=𝑎⊗𝑓𝑣, (5.1) for 𝑓∈𝑀_𝔥^∗, 𝑎∈𝐴, 𝑣∈𝑉. 𝐴⊗𝑉 becomes an 𝔥-space with the 𝑀_𝔥^∗-action 𝑓(𝑎⊗𝑣)=𝜇_𝑙(𝑓)𝑎⊗𝑣. Similarly we define 𝑉⊗𝐴 as an 𝔥-space by (𝑉⊗𝐴)_𝛽=⊕_𝛼𝑉_𝛼⊗_𝑀𝔥∗𝐴_𝛼𝛽, where ⊗_𝑀𝔥∗ here means ⊗_ℂ modulo the relation 𝑣⊗𝜇_𝑙(𝑓)𝑎=𝑓𝑣⊗𝑎 and 𝑀_𝔥^∗-action given by 𝑓(𝑣⊗𝑎)=𝑣⊗𝜇_𝑟(𝑓)𝑎.

For any 𝔥-space 𝑉 we have isomorphisms 𝐷_𝔥⊗𝑉≃𝑉≃𝑉⊗𝐷_𝔥 given by 𝑇_−𝛼⊗𝑣≃𝑣≃𝑣⊗𝑇_𝛼,for𝑣∈𝑉_𝛼, (5.2) extended to 𝔥-space morphisms.

Definition 5.2. A left corepresentation 𝑉 of an 𝔥-bialgebroid 𝐴 is an 𝔥-space equipped with an 𝔥-space morphism Δ_𝑉∶𝑉→𝐴⊗𝑉 such that (Δ_𝑉⊗1)∘Δ_𝑉=(1⊗Δ)∘Δ_𝑉 and (𝜀⊗1)∘Δ_𝑉=Id_𝑉 (under the identification (5.2).

Definition 5.3. A left 𝔥-comodule algebra 𝑉 over an 𝔥-bialgebroid 𝐴 is a left corepresentation Δ_𝑉∶𝑉→𝐴⊗𝑉 and in addition a ℂ-algebra such that 𝑉_𝛼𝑉_𝛽⊆𝑉_𝛼+𝛽 and such that Δ_𝑉 is an algebra morphism, when 𝐴⊗𝑉 is given the natural algebra structure.

Right corepresentations and comodule algebras are defined analogously.

5.2. The Comodule Algebras Λ and Λ^′.

We define in this section an elliptic analog of the exterior algebra, following [12], where it was carried out in the trigonometric nonspectral case. It will lead to natural definitions of elliptic minors as certain elements of ℱ_ell(𝑀(𝑛)). One difference between this approach and the one in [5] is that the elliptic exterior algebra in our setting is really an algebra and not just a vector space. Another one is that the commutation relations in our elliptic exterior algebras are completely determined by requiring the natural relations (5.3a), (5.3b), and (5.5) and that the coaction is an algebra homomorphism. This fact can be seen from the proof of Proposition 5.4. Since the proof does not depend on the particular form of 𝛼 and 𝛽, we can obtain exterior algebras for any 𝔥-bialgebroid obtained through the generalized FRST construction from an R-matrix in the same manner. In particular the method is independent of the gauge equivalence class of 𝑅.

Let Λ be the unital associative ℂ-algebra generated by 𝑣_𝑖(𝑧), 1≤𝑖≤𝑛, 𝑧∈ℂ^× and a copy of 𝑀_𝔥^∗ embedded as a subalgebra subject to the relations 𝑓(𝜁)𝑣_𝑖(𝑧)=𝑣_𝑖(𝑧)𝑓(𝜁+𝜔(𝑖)),𝑣_𝑖(𝑧)𝑣_𝑖(𝑤)=0,𝛼(𝜁_𝑘𝑗,𝑧𝑤)𝑣_𝑘(𝑧)𝑣_𝑗(𝑤)+𝛽(𝜁_𝑘𝑗,𝑧𝑤)𝑣_𝑗(𝑧)𝑣_𝑘(𝑤)=0, (5.3a)(5.3b)(5.3c) for 𝑖,𝑗,𝑘∈[1,𝑛], 𝑗≠𝑘, 𝑧,𝑤∈ℂ^×, 𝑧/𝑤∉{𝑝^𝑘𝑞⁻²∶𝑘∈ℤ} and 𝑓∈𝑀_𝔥^∗. We require also the residual relation of (5.3c) obtained by multiplying by 𝜑(𝑧/𝑤)=𝑞⁻¹𝜃(𝑞²𝑧/𝑤)/𝑞𝜃(𝑞⁻²𝑧/𝑤) and letting 𝑧/𝑤→𝑝^−𝑘𝑞⁻². After simplification using (2.26), we get 𝑣_𝑘(𝑧)𝑣_𝑗(𝑝^𝑘𝑞²𝑧)=𝑞^{2𝑘𝜁𝑗𝑘}𝑣_𝑗(𝑧)𝑣_𝑘(𝑝^𝑘𝑞²𝑧). (5.3d)

Λ becomes an 𝔥-space by 𝜇_Λ(𝑓)𝑣=𝑓(𝜁)𝑣 (5.4) and requiring 𝑣_𝑖(𝑧)∈Λ_𝜔(𝑖) for each 𝑖,𝑧.

Proposition 5.4. Λ is a left comodule algebra over ℱ_ell(𝑀(𝑛)) with left coaction Δ_Λ∶Λ→ℱ_ell(𝑀(𝑛))⊗Λ satisfying Δ_Λ(𝑣_𝑖(𝑧))=_𝑗𝑒_𝑖𝑗(𝑧)⊗𝑣_𝑗(𝑧),Δ_Λ(𝑓(𝜁))=𝑓(𝜆)⊗1. (5.5)(5.6)

Proof. We have Δ_Λ(𝑣_𝑖(𝑧))Δ_Λ(𝑣_𝑖(𝑤))=_𝑗𝑘𝑒_𝑖𝑗(𝑧)𝑒_𝑖𝑘(𝑤)⊗𝑣_𝑗(𝑧)𝑣_𝑘(𝑤)=_𝑗≠𝑘(𝛼(𝜇_𝑗𝑘,𝑧𝑤)𝑒_𝑖𝑘(𝑤)𝑒_𝑖𝑗(𝑧)+𝛽(𝜇_𝑘𝑗,𝑧𝑤)𝑒_𝑖𝑗(𝑤)𝑒_𝑖𝑘(𝑧))⊗𝑣_𝑗(𝑧)𝑣_𝑘=(𝑤)_𝑗≠𝑘𝑒_𝑖𝑗(𝑤)𝑒_𝑖𝑘(𝑧)⊗(𝛼(𝜁_𝑘𝑗,𝑧𝑤)𝑣_𝑘(𝑧)𝑣_𝑗(𝑤)+𝛽(𝜁_𝑘𝑗,𝑧𝑤)𝑣_𝑗(𝑧)𝑣_𝑘(𝑤))=0. (5.7) Similarly one proves that (5.3c), (5.3d) are preserved.

Relation (5.3c) is not symmetric under interchange of 𝑗 and 𝑘. We now derive a more explicit, independent, set of relations for Λ. We will use the function 𝐸, defined in (2.6).

Proposition 5.5. (i) The following is a complete set of relations for Λ: 𝑓(𝜁)𝑣_𝑖(𝑧)=𝑣_𝑖(𝑧)𝑓(𝜁+𝜔(𝑖)),𝑣_𝑘(𝑝^𝑠𝑞²𝑧)𝑣_𝑗(𝑧)=−𝑞^{2𝑠𝜁𝑘𝑗}𝐸(𝜁_𝑘𝑗−1)𝐸(𝜁_𝑘𝑗𝑣+1)^𝑗(𝑝^𝑠𝑞²𝑧)𝑣_𝑘(𝑧),∀𝑠∈ℤ,𝑘≠𝑗,𝑣_𝑘(𝑧)𝑣_𝑗(𝑝^𝑠𝑞²𝑧)=𝑞^{2𝑠𝜁𝑗𝑘}𝑣_𝑗(𝑧)𝑣_𝑘(𝑝^𝑠𝑞²𝑧),𝑣_𝑘(𝑧)𝑣_𝑗(𝑤)=0if𝑧𝑤∉{𝑝^𝑠𝑞^±2∣𝑠∈ℤ}orif𝑘=𝑗. (5.8a)(5.8b)(5.8c)(5.8d)
(ii) The set {𝑣_𝑖𝑑(𝑧_𝑑)⋯𝑣_𝑖1(𝑧₁)∶1≤𝑖₁<⋯<𝑖_𝑑𝑧≤𝑛,^𝑖+1𝑧_𝑖∈𝑝^ℤ𝑞^±2} (5.9) is a basis for Λ over 𝑀_𝔥^∗.

Proof. (i) Elimination of the 𝑣_𝑗(𝑧)𝑣_𝑘(𝑤)-term in (5.3c) yields (𝛼(𝜁_𝑗𝑘,𝑧𝑤)𝛼(𝜁_𝑘𝑗,𝑧𝑤)−𝛽(𝜁_𝑘𝑗,𝑧𝑤)𝛽(𝜁_𝑗𝑘,𝑧𝑤))𝑣_𝑘(𝑧)𝑣_𝑗(𝑤)=0. (5.10) Combining (5.10), (2.28), and the fact that the 𝜃(𝑧) is zero precisely for 𝑧∈{𝑝^𝑘∣𝑘∈ℤ} we deduce that in Λ, 𝑣_𝑘(𝑧)𝑣_𝑗𝑧(𝑤)≠0⇒𝑤=𝑝^𝑠𝑞²forsome𝑠∈ℤ. (5.11) Using (2.25) we obtain from (5.11), (5.3b), and (5.3c) that relations (5.8b), (5.8d) hold in the left elliptic exterior algebra Λ. Relations (5.8a), (5.8c) are just repetitions of (5.3a), (5.3d).
(ii) It follows from the relations that each monomial in Λ can be uniquely written as 𝑓(𝜁)𝑣_𝑖𝑑(𝑧_𝑑)⋯𝑣_𝑖1(𝑧₁), where 1≤𝑖₁<⋯<𝑖_𝑑≤𝑛 and 𝑓∈𝑀_𝔥^∗. It remains to show that the set (5.9) is linearly independent over 𝑀_𝔥^∗. Assume that a linear combination of basis elements is zero and that the sum has minimal number of terms. By multiplying from the right or left by 𝑣_𝑗(𝑤) for appropriate 𝑗, 𝑤 we can assume that the sum is of the form 𝑓₁(𝜁)𝑣_𝑖𝑑(𝑧¹_𝑑)⋯𝑣_𝑖1(𝑧¹₁)+⋯+𝑓_𝑟(𝜁)𝑣_𝑖𝑑(𝑧^𝑟_𝑑)⋯v_𝑖1(𝑧^𝑟₁)=0, (5.12) for some fixed set {𝑖₁,…,𝑖_𝑑}. By the relations, a monomial 𝑣_𝑖𝑑(𝑧_𝑑)⋯𝑣_𝑖1(𝑧₁) can be given the "degree” ∑^𝑑_𝑖=1𝑧_𝑖𝑡^𝑖−1∈ℂ[𝑡], where 𝑡 is an indeterminate. Formally, consider ℂ(𝑡)⊗Λ, the tensor product (over ℂ) of Λ by the field of rational functions in 𝑡. We identify Λ with its image under Λ∋𝑣↦1⊗𝑣∈ℂ(𝑡)⊗Λ and view ℂ(𝑡)⊗Λ naturally as a vector space over ℂ(𝑡). By relations (5.8a)–(5.8d), there is a ℂ-algebra automorphism 𝑇 of ℂ(𝑡)⊗Λ satisfying 𝑇(𝑣_𝑗(𝑧))=𝑡𝑣_𝑗(𝑧), 𝑇(𝑓(𝜁))=𝑓(𝜁), and 𝑇(𝑝⊗1)=𝑝⊗1. Define 𝐷(𝑣_𝑖(𝑧))=𝑧𝑣_𝑖(𝑧),𝐷(𝑓(𝜁))=0,𝐷(𝑝⊗1)=0, (5.13) for 𝑓∈𝑀_𝔥^∗, 𝑝∈ℂ(𝑡) and 𝑖∈[1,𝑛], 𝑧∈ℂ^×, and extend 𝐷 to a ℂ-linear map 𝐷∶ℂ(𝑡)⊗Λ→ℂ(𝑡)⊗Λ by requiring 𝐷(𝑎𝑏)=𝐷(𝑎)𝑇(𝑏)+𝑎𝐷(𝑏), (5.14) for 𝑎,𝑏∈ℂ(𝑡)⊗Λ. The point is that the requirement (5.14) respects relations (5.8a)–(5.8d), making 𝐷 well defined. Write 𝑢_𝑗=𝑓_𝑗(𝜁)𝑣_𝑖𝑑(𝑧^𝑗_𝑑)⋯𝑣_𝑖1(𝑧^𝑗₁). Then one checks that 𝐷(𝑢_𝑗)=𝑝_𝑗(𝑡)𝑢_𝑗, where 𝑝_𝑗(𝑡)=∑^𝑑_𝑖=1𝑧^𝑗_𝑖𝑡^𝑖−1. By applying 𝐷 repeatedly we get 𝑢₁(𝑧¹)+⋯+𝑢_𝑟(𝑧^𝑟𝑝)=0,₁(𝑡)𝑢₁(𝑧¹)+⋯+𝑝_𝑟(𝑡)𝑢_𝑟(𝑧^𝑟⋮𝑝)=0,₁(𝑡)^𝑟−1𝑢₁(𝑧¹)+⋯+𝑝_𝑟(𝑡)^𝑟−1𝑢_𝑟(𝑧^𝑟)=0. (5.15) Inverting the Vandermonde matrix (𝑝_𝑗(𝑡)^𝑖−1)_𝑖𝑗 we obtain 𝑢_𝑗(𝑧^𝑗)=0 for each 𝑗, that is, 𝑓_𝑗(𝜁)=0 for each 𝑗. This proves linear independence of (5.9).

Analogously one defines a right comodule algebra Λ^′ with generators 𝑤^𝑖(𝑧) and 𝑓(𝜁)∈𝑀_𝔥^∗. The following relations will be used: 𝑤^𝑘(𝑧)𝑤^𝑗(𝑝^𝑠𝑞²𝑧)=−𝑞^{2𝑠𝜁𝑘𝑗}𝑤^𝑗(𝑧)𝑤^𝑘(𝑝^𝑠𝑞²𝑤𝑧),∀𝑠∈ℤ,𝑘≠𝑗,_𝑘(𝑧₁)𝑤^𝑗(𝑧₂)=0,if𝑧₂𝑧₁∉{𝑝^𝑠𝑞^±2∣𝑠∈ℤ},orif𝑘=𝑗. (5.16)Λ^′ has also 𝑀_𝔥^∗-basis of the form (5.9). In fact Λ and Λ^′ are isomorphic as algebras.

5.3. Action of the Symmetric Group

From (4.4a)–(4.4d), and (4.5) we see that 𝑆_𝑛×𝑆_𝑛 acts by ℂ-algebra automorphisms on ℱ_ell(𝑀(𝑛)) as follows: (𝜎,𝜏)(𝑓(𝜆))=𝑓(𝜆∘𝐿_𝜎),(𝜎,𝜏)(𝑓(𝜇))=𝑓(𝜇∘𝐿_𝜏),(𝜎,𝜏)(𝑒_𝑖𝑗(𝑧))=𝑒_{𝜎(𝑖)𝜏(𝑗)}(𝑧), (5.17) where 𝐿_𝜎∶𝔥→𝔥 (𝜎∈𝑆_𝑛) is given by permutation of coordinates: 𝐿_𝜎(diag(ℎ₁,…,ℎ_𝑛))=diag(ℎ_𝜎(1),…,ℎ_𝜎(𝑛)). (5.18) Also, 𝑆_𝑛 acts on Λ by ℂ-algebra automorphisms via 𝜎(𝑓(𝜁))=𝑓(𝜁∘𝐿_𝜎),𝜎(𝑣_𝑖(𝑧))=𝑣_𝜎(𝑖)(𝑧). (5.19) Similarly we define an 𝑆_𝑛 action on Λ^′.

Lemma 5.6. For each 𝑣∈Λ, 𝑤∈Λ^′, and any 𝜎,𝜏∈𝑆_𝑛 we have Δ_Λ(𝜎(𝑣))=((𝜎,𝜏)⊗𝜏)(Δ_Λ(𝑣)),Δ_Λ′(𝜏(𝑤))=(𝜎⊗(𝜎,𝜏))(Δ_Λ′(𝑤)). (5.20)(5.21)

Proof. By multiplicativity, it is enough to prove these claims on the generators, which is easy.

6. Elliptic Quantum Minors

6.1. Definition

For 𝐼⊆[1,𝑛] we set 𝐹_𝐼(𝜁)=_{𝑖,𝑗∈𝐼,𝑖<𝑗}𝐸(𝜁_𝑖𝑗+1),𝐹^𝐼(𝜁)=_{𝑖,𝑗∈𝐼,𝑖<𝑗}𝐸(𝜁_𝑖𝑗), (6.1) and define the left and right elliptic sign functions: sgn_𝐼(𝜎;𝜁)=𝜎(𝐹_𝐼(𝜁))𝐹_𝜎(𝐼)=(𝜁)_{𝑖,𝑗∈𝐼,𝑖<𝑗,𝜎(𝑖)>𝜎(𝑗)}𝐸(𝜁_{𝜎(𝑖)𝜎(𝑗)}+1)𝐸(𝜁_{𝜎(𝑗)𝜎(𝑖)},+1)sgn^𝐼𝐹(𝜎;𝜁)=^𝜎(𝐼)(𝜁)𝜎(𝐹^𝐼(𝜁))=_{𝑖,𝑗∈𝐼,𝑖<𝑗,𝜎(𝑖)>𝜎(𝑗)}𝐸(𝜁_{𝜎(𝑗)𝜎(𝑖)})𝐸(𝜁_{𝜎(𝑖)𝜎(𝑗)}), (6.2) for 𝜎∈𝑆_𝑛. In fact, 𝐸(𝜁_𝑖𝑗)/𝐸(𝜁_𝑗𝑖)=−1 so sgn^[1,𝑛](𝜎;𝜁) is just the usual sign sgn(𝜎). However we view this as a “coincidence” depending on the particular choice of R-matrix from its gauge equivalence class. We keep our notation to emphasize that the methods do not depend on this choice of R-matrix.

We will denote the elements of a subset 𝐼⊆[1,𝑛] by 𝑖₁<𝑖₂<⋯.

Proposition 6.1. Let 𝐼⊆[1,𝑛], 𝑑=#𝐼, 𝜎∈𝑆_𝑛, and 𝐽=𝜎(𝐼). Then for 𝑧∈ℂ^×, 𝑣_{𝜎(𝑖𝑑)}(𝑞^2(𝑑−1)𝑧)⋯𝑣_𝜎(𝑖1)(𝑧)=sgn_𝐼(𝜎;𝜁)𝑣_𝑗𝑑(𝑞^2(𝑑−1)𝑧)⋯𝑣_𝑗1(𝑧),𝑤^𝜎(𝑖1)(𝑧)⋯𝑤^{𝜎(𝑖𝑑)}(𝑞^2(𝑑−1)𝑧)=sgn^𝐼(𝜎;𝜁)𝑤^𝑗1(𝑧)⋯𝑤^𝑗𝑑(𝑞^2(𝑑−1)𝑧). (6.3)(6.4)

Proof. We prove (6.3). The proof of (6.4) is analogous. We proceed by induction on #𝐼=𝑑, the case 𝑑=1 being clear. If 𝑑>1, set 𝐼^′={𝑖₁,…,𝑖_𝑑−1},𝐽^′=𝜎(𝐼^′). Let 1≤𝑗^′₁<⋯<𝑗^′_𝑑−1≤𝑛 be the elements of 𝐽^′. By the induction hypothesis, the left hand side of (6.3) equals 𝑣_{𝜎(𝑖𝑑)}(𝑞^2(𝑑−1)𝑧)sgn_𝐼′(𝜎,𝜁)𝑣_{𝑗′𝑑−1}(𝑞^2(𝑑−2)𝑧)⋯𝑣_𝑗′1(𝑧). (6.5) Now 𝑣_{𝜎(𝑖𝑑)}(𝑞^2(𝑑−1)𝑧) commutes with sgn_𝐼′(𝜎,𝜁) since the latter only involves 𝜁_𝑖𝑗 with 𝑖,𝑗≠𝜎(𝑖_𝑑). Using the commutation relations (5.8b) we obtain sgn_𝐼′(𝜎,𝜁)⋅_{𝑗∈𝐽′,𝑗>𝜎(𝑖𝑑)}𝐸(𝜁_{𝑗𝜎(𝑖𝑑)}+1)𝐸(𝜁_{𝜎(𝑖𝑑)𝑗}+1)⋅𝑣_𝑗𝑑(𝑞^2(𝑑−1)𝑧)⋯𝑣_𝑗1(𝑧). (6.6) Replace 𝑗∈𝐽^′ such that 𝑗>𝜎(𝑖_𝑑) by 𝜎(𝑖), where 𝑖∈𝐼,𝑖<𝑖_𝑑,𝜎(𝑖)>𝜎(𝑖_𝑑).

Introduce the normalized monomials 𝑣_𝐼(𝑧)=𝐹_𝐼(𝜁)⁻¹𝑣_𝑖𝑟(𝑞^2(𝑑−1)𝑧)𝑣_𝑖𝑟−1(𝑞^2(𝑑−2)𝑧)⋯𝑣_𝑖1(𝑧)∈Λ,𝑤^𝐼(𝑧)=𝐹^𝐼(𝜁)𝑤^𝑖1(𝑧)𝑤^𝑖2(𝑞²𝑧)⋯𝑤^𝑖𝑑(𝑞^2(𝑑−1)𝑧)∈Λ^′. (6.7)(6.8)

Corollary 6.2. Let 𝐼⊆[1,𝑛]. For any permutation 𝜎∈𝑆_𝑛, 𝜎(𝑣_𝐼(𝑧))=𝑣_𝜎(𝐼)(𝑧),𝜎(𝑤^𝐼(𝑧))=𝑤^𝜎(𝐼)(𝑧) (6.9) for any 𝑧∈ℂ^×. In particular 𝑣_𝐼(𝑧) and 𝑤^𝐼(𝑧) are fixed by any permutation which preserves the subset 𝐼.

Proof. Let 𝐽=𝜎(𝐼). Then 𝜎(𝑣_𝐼(𝑧))=𝜎(𝐹_𝐼(𝜁)⁻¹)𝑣_{𝜎(𝑖𝑑)}(𝑞^2(𝑑−1)𝑧)⋯𝑣_𝜎(𝑖1)(𝑧)=𝜎(𝐹_𝐼(𝜁))⁻¹sgn_𝐼(𝜎;𝜁)𝑣_𝑗𝑑(𝑞^2(𝑑−1)𝑧)⋯𝑣_𝑗1(𝑧)=𝑣_𝜎(𝐼)(𝑧). (6.10) The proof for 𝑤^𝐼(𝑧) is analogous.

For any 𝐼⊆[1,𝑛], let 𝑆_𝐼 denote the group of all permutations of the set 𝐼. We are now ready to define certain elements of the 𝔥-bialgebroid ℱ_ell(𝑀(𝑛)) which are analogs of minors.

Proposition 6.3. For 𝐼,𝐽⊆[1,𝑛] and 𝑧∈ℂ^×, the left and right elliptic minors, ^⟵𝜉^𝐽_𝐼(𝑧) and 𝜉^𝐽_𝐼(𝑧), respectively, can be defined by Δ_Λ(𝑣_𝐼(𝑧))=_𝐽^⟵𝜉^𝐽_𝐼(𝑧)⊗𝑣_𝐽(𝑧),Δ_Λ′(𝑤^𝐽(𝑧))=_𝐼𝑤^𝐼(𝑧)⊗𝜉^𝐽_𝐼(𝑧), (6.11)(6.12) where the sums are taken over all subsets of [1,𝑛].
If #𝐼≠#𝐽, then ^⟵𝜉^𝐽_𝐼(𝑧)=0=𝜉^𝐽_𝐼(𝑧), for all 𝑧. If #𝐼=#𝐽=𝑑, they are explicitly given by ^⟵𝜉^𝐽_𝐼𝐹(𝑧)=^𝐽(𝜌)𝐹_𝐼(𝜆)_{𝜏∈𝑆𝐽}sgn_𝐽(𝜏;𝜌)sgn_𝐼𝑒(𝜎;𝜆)^{𝜎(𝑖𝑑)𝜏(𝑗𝑑)}(𝑞^2(𝑑−1)𝑧)𝑒_{𝜎(𝑖𝑑−1)𝜏(𝑗𝑑−1)}(𝑞^2(𝑑−2)𝑧)⋯𝑒_{𝜎(𝑖1)𝜏(𝑗1)}(𝑧) (6.13) for any 𝜎∈𝑆_𝐼, and 𝜉^𝐽_𝐼𝐹(𝑧)=^𝐽(𝜌)𝐹^𝐼(𝜆)_{𝜎∈𝑆𝐼}sgn^𝐽(𝜏;𝜌)sgn^𝐼𝑒(𝜎;𝜆)^{𝜎(𝑖1)𝜏(𝑗1)}(𝑧)𝑒_{𝜎(𝑖2)𝜏(𝑗2)}(𝑞²𝑧)⋯𝑒_{𝜎(𝑖𝑑)𝜏(𝑗𝑑)}(𝑞^2(𝑑−1)𝑧) (6.14) for any 𝜏∈𝑆_𝐽. Moreover, (𝜎,𝜏)(^⟵𝜉^𝐽_𝐼(𝑧))=^⟵𝜉^𝜏(𝐽)_𝜎(𝐼)(𝑧),(𝜎,𝜏)(𝜉^𝐽_𝐼(𝑧))=𝜉^𝜏(𝐽)_𝜎(𝐼)(𝑧) (6.15) for any (𝜎,𝜏)∈𝑆_𝑛×𝑆_𝑛 and 𝑧∈ℂ^×.

Remark 6.4. In Theorem 6.10 we will prove that, in fact, ^⟵𝜉^𝐽_𝐼(𝑧)=𝜉^𝐽_𝐼(𝑧).

Proof. We prove the statements concerning the left elliptic minor ^⟵𝜉^𝐽_𝐼(𝑧). We have Δ_Λ(𝑣_𝐼(𝑧))=_{1≤𝑘1,…,𝑘𝑑≤𝑛}𝐹_𝐼(𝜆)⁻¹𝑒_{𝑖𝑑𝑘𝑑}(𝑞^2(𝑑−1)𝑧)⋯𝑒_𝑖1𝑘1(𝑧)⊗𝑣_𝑘𝑑(𝑞^2(𝑑−1)𝑧)⋯𝑣_𝑘1(𝑧)=_{𝐽,#𝐽=𝑑𝜏∈𝑆𝐽}𝐹_𝐼(𝜆)⁻¹𝑒_{𝑖𝑑𝜏(𝑗𝑑)}(𝑞^2(𝑑−1)𝑧)⋯𝑒_{𝑖1𝜏(𝑗1)}(𝑧)⊗𝑣_{𝜏(𝑗𝑑)}(𝑞^2(𝑑−1)𝑧)⋯𝑣_𝜏(𝑗1)(𝑧)=_{𝐽,#𝐽=𝑑}(_{𝜏∈𝑆𝐽}𝜏(𝐹_𝐽(𝜌))𝐹_𝐼𝑒(𝜆)^{𝑖𝑑𝜏(𝑗𝑑)}(𝑞^2(𝑑−1)𝑧)⋯𝑒_{𝑖1𝜏(𝑗1)}(𝑧))⊗𝑣_𝐽(𝑧). (6.16) Thus (6.11) holds when ^⟵𝜉^𝐽_𝐼(𝑧) is defined by (6.13) with 𝜎=Id. Then the right hand side of (6.13) equals (𝜎,Id)(^⟵𝜉^𝐽_𝐼(𝑧)) for any 𝜎∈𝑆_𝐼. Thus only (6.15) remains. Using (5.20) and Corollary 6.2 we have Δ_Λ(𝜎(𝑣_𝐼(𝑧)))=((𝜎,𝜏)⊗𝜏)(Δ_Λ(𝑣_𝐼(𝑧)))=_𝐽(𝜎,𝜏)((^⟵𝜉^𝐽_𝐼(𝑧))⊗𝑣_𝜏(𝐽)(𝑧)). (6.17) On the other hand, again by Corollary 6.2, Δ_Λ(𝜎(𝑣_𝐼(𝑧)))=Δ_Λ(𝑣_𝜎(𝐼)(𝑧))=_𝐽(^⟵𝜉^𝜏(𝐽)_𝜎(𝐼)(𝑧)⊗𝑣_𝜏(𝐽)(𝑧)), (6.18) where we made the substitution 𝐽↦𝜏(𝐽). This proves the first equality in (6.15). The statements concerning the right elliptic minors are proved analogously.

6.2. Equality of Left and Right Minors

The goal of this section is to prove Theorem 6.10 stating that the left and right elliptic minors coincide. We use ideas from Section 3 of [5], where the authors study the objects of Felder's tensor category [2] and associate a linear operator (product of R-matrices) on 𝑉^⊗𝑛 to each diagram of a certain form, a kind of braid group representation. Then they consider the operator associated to the longest permutation, in [7] called the Cherednik operator. Instead of working with representations, we proceed inside the 𝔥-bialgebroid ℱ_ell(𝑀(𝑛)) and consider certain operators on 𝑉^⊗𝑛⊗ℱ_ell(𝑀(𝑛)) depending on 𝑛 spectral parameters. Using the analog of the Cherednik operator we prove an extended RLL relation (6.38). Theorem 6.10 then follows by extracting matrix elements from both sides of this matrix equation.

In this section, we set ℱ=ℱ_ell(𝑀(𝑛)). Recall the operators from Section 3.3, defined for any 𝔥-bialgebroid 𝐴_𝑅 obtained from the FRST construction. When specializing to ℱ we get operators 𝖱(𝜆,𝑧), 𝖱(𝜌,𝑧)∈End(𝑉⊗𝑉⊗ℱ), where 𝑉=ℂ^𝑛. For 𝑧∈ℂ^×, define the following linear operators on 𝑉^⊗𝑛⊗ℱ: 𝖱^𝑖𝑗(𝜆,𝑧)∶=lim_𝑤→𝑧𝜃(𝑞²𝑤)𝖱(𝜆,𝑤)^{(𝑖,𝑗,𝑛+1)},𝖱^𝑖𝑗(𝜌,𝑧)∶=lim_𝑤→𝑧𝜃(𝑞²𝑤)𝖱(𝜌,𝑤)^{(𝑖,𝑗,𝑛+1)}. (6.19) The upper indices in parenthesis are tensor leg numbering and indicate the tensor factors the operator should act on. The limits are taken in the sense of taking limits of each matrix element. These operators are well defined for any 𝑧, since we multiply away the singularities in 𝑧 of 𝛼 and 𝛽 (2.15), (2.16).

Let ℰ_𝑛 denote the algebra of all functions 𝐹∶(ℂ^×)^𝑛→End(𝑉^⊗𝑛⊗ℱ). (6.20) The symmetric group 𝑆_𝑛 acts on ℰ_𝑛 by 𝜎(𝐹(𝑧))=(𝜎⊗Id_ℱ)∘𝐹(𝜎(𝑧))∘(𝜎⁻¹⊗Id_ℱ), (6.21) for 𝐹∈ℰ_𝑛 and 𝜎∈𝑆_𝑛. In the right hand side of (6.21), 𝜎 acts on (ℂ^×)^𝑛 by permuting coordinates, and on 𝑉^⊗𝑛 by permuting the tensor factors. For example, we have (23)(𝖱¹²𝑧(𝜆,¹𝑧₂))=𝖱¹³𝑧(𝜆,¹𝑧₃). (6.22) Consider the skew group algebra ℰ_𝑛∗𝑆_𝑛, defined as the algebra with underlying space ℰ_𝑛⊗ℂ𝑆_𝑛, where ℂ𝑆_𝑛 is the group algebra, with the multiplication (𝐹(𝑧)⊗𝜎)(𝐺(𝑧)⊗𝜏)=𝐹(𝑧)𝜎(𝐺(𝑧))⊗𝜎𝜏, (6.23) for 𝜎,𝜏∈𝑆_𝑛, 𝐹,𝐺∈ℰ_𝑛. Since 𝜎 acts on ℰ_𝑛 by automorphisms, ℰ_𝑛∗𝑆_𝑛 is an associative algebra. The constant function 𝑧↦Id_𝑉^⊗𝑛_⊗ℱ⊗(1) is the unit element. Let 𝐵_𝑛 be the monoid (set with unital associative multiplication) generated by {𝑠₁,…,𝑠_𝑛−1} and relations 𝑠_𝑖𝑠_𝑖+1𝑠_𝑖=𝑠_𝑖+1𝑠_𝑖𝑠_𝑖+1,for𝑠1≤𝑖≤𝑛−2,_𝑖𝑠_𝑗=𝑠_𝑗𝑠_𝑖,if|𝑖−𝑗|>1. (6.24) Let 𝜎_𝑖=(𝑖𝑖+1)∈𝑆_𝑛. We have an epimorphism 𝜋∶𝐵_𝑛→𝑆_𝑛 given by 𝜋(𝑠_𝑖)=𝜎_𝑖, 𝜋(1)=(1). Define 𝑊(1)=Id_𝑉^⊗𝑛_⊗ℱ⊗(1),𝑊(𝑠_𝑖)=𝖱^𝑖,𝑖+1(𝜆−ℎ^≥𝑖+2,𝑧_𝑖𝑧_𝑖+1)⊗𝜎_𝑖. (6.25) Here and below we use ℎ^≥𝑘 to denote the expression ∑^𝑛_𝑗=𝑘ℎ^𝑗, and operators involving shifts ℎ^𝑖 such as 𝖱^{𝑛−2,𝑛−1}(𝜆−ℎ^𝑛,𝑧_𝑖/𝑧_𝑖+1) are defined as in Section 3.3.

Proposition 6.5. 𝑊 extends to a well-defined morphism of monoids, that is, a map 𝑊∶𝐵_𝑛→ℰ_𝑛∗𝑆_𝑛 (6.26) satisfying 𝑊(𝑏₁𝑏₂)=𝑊(𝑏₁)𝑊(𝑏₂) for any 𝑏₁,𝑏₂∈𝐵_𝑛.

Proof. We have to show the relations 𝑊(𝑠_𝑖)𝑊(𝑠_𝑖+1)𝑊(𝑠_𝑖)=𝑊(𝑠_𝑖+1)𝑊(𝑠_𝑖)𝑊(𝑠_𝑖+1),𝑊(𝑠_𝑖)𝑊(𝑠_𝑗)=𝑊(𝑠_𝑗)𝑊(𝑠_𝑖)if|𝑖−𝑗|>1. (6.27)(6.28) Relation (6.27) follows from the QDYBE (2.8). For example, 𝑊(𝑠_𝑖)𝑊(𝑠_𝑖+1)𝑊(𝑠_𝑖) equals 𝖱^𝑖,𝑖+1(𝜆−ℎ^≥𝑖+2,𝑧_𝑖𝑧_𝑖+1)𝖱^𝑖,𝑖+2(𝜆−ℎ^≥𝑖+3,𝑧_𝑖𝑧_𝑖+2)𝖱^{𝑖+1,𝑖+2}(𝜆−ℎ^𝑖−ℎ^≥𝑖+3,𝑧_𝑖+1𝑧_𝑖+2)⊗𝜎_𝑖𝜎_𝑖+1𝜎_𝑖. (6.29) Relation (6.28) is easy to check, using the 𝔥-invariance of 𝑅.

For 𝑏∈𝐵_𝑛 we define 𝑊_𝑏(𝜆,𝑧)∈ℰ_𝑛 by 𝑊(𝑏)=𝑊_𝑏(𝜆,𝑧)⊗𝜋(𝑏). (6.30) From this and the product rule (6.23) it follows that 𝑊_𝑏1𝑏2(𝜆,𝑧)=𝑊_𝑏1(𝜆,𝑧)⋅𝜋(𝑏₁)(𝑊_𝑏2(𝜆,𝑧)), (6.31) for 𝑏₁,𝑏₂∈𝐵_𝑛. By replacing 𝜆 by 𝜌 we get similarly operators 𝑊_𝑏(𝜌,𝑧).

Recall the operators 𝖫(𝑧)∈End(𝑉⊗ℱ) from Section 3.3. Define for 𝑧∈ℂ^×, 𝑖∈[1,𝑛], 𝖫^𝑖(𝑧)=𝖫(𝑧)^{(𝑖,𝑛+1)}∈End(𝑉^⊗𝑛⊗ℱ). (6.32)

If 𝑖,𝑗,𝑘 are distinct, then one can check that 𝖱^𝑖𝑗(𝜆−ℎ^𝑘,𝑧)𝖫^𝑘(𝑤)=𝖫^𝑘(𝑤)𝖱^𝑖𝑗(𝜆,𝑧),𝖱^𝑖𝑗(𝜌,𝑧)𝖫^𝑘(𝑤)=𝖫^𝑘(𝑤)𝖱^𝑖𝑗(𝜌+ℎ^𝑘,𝑧). (6.33)(6.34)

Due to the RLL relations (3.8) we have 𝖱¹²𝑧(𝜆,¹𝑧₂)𝖫¹(𝑧₁)𝖫²(𝑧₂)=𝖫²(𝑧₂)𝖫¹(𝑧₁)𝖱¹²(𝜌+ℎ¹+ℎ²,𝑧₁𝑧₂) (6.35) for any 𝑧₁,𝑧₂∈ℂ^×.

Define 𝑡_𝑑∈𝐵_𝑛, 𝑑∈[1,𝑛], recursively by 𝑡_𝑑=𝑡_𝑑−1𝑠_𝑑−1𝑠_𝑑−2⋯𝑠₁,𝑑>11,𝑑=1. (6.36) Let 𝜏_𝑑 be the image of 𝑡_𝑑 in 𝑆_𝑛: 𝜏_𝑑∶=𝜋(𝑡_𝑑)=12⋯𝑑𝑑+1⋯𝑛𝑑𝑑−1⋯1𝑑+1⋯𝑛∈𝑆_𝑛. (6.37)

Proposition 6.6. Let 1≤𝑑≤𝑛. For any 𝑧=(𝑧₁,…,𝑧_𝑑)∈(ℂ^×)^𝑑 we have 𝑊_𝑡𝑑(𝜆,𝑧)𝖫¹(𝑧₁)⋯𝖫^𝑑(𝑧_𝑑)=𝖫^𝑑(𝑧_𝑑)⋯𝖫¹(𝑧₁)𝑊_𝑡𝑑(𝜌+ℎ^≤𝑑,𝑧). (6.38)

Proof. We use induction on 𝑑. The case 𝑑=1 is trivial, while 𝑑=2 is the RLL relation (6.35). If 𝑑>2, write 𝑡_𝑑=𝑡_𝑑−1𝑢_𝑑, where 𝑢_𝑑=𝑠_𝑑−1𝑠_𝑑−2⋯𝑠₁. Thus, by (6.31), 𝑊_𝑡𝑑(𝜆,𝑧)=𝑊_𝑡𝑑−1(𝜆,𝑧)⋅𝜏_𝑑−1(𝑊_𝑢𝑑(𝜆,𝑧)). (6.39) We claim that 𝜏_𝑑−1(𝑊_𝑢𝑑(𝜆,𝑧))𝐿¹(𝑧₁)⋯𝐿^𝑑(𝑧_𝑑)=𝐿^𝑑(𝑧_𝑑)𝐿¹(𝑧₁)⋯𝐿^𝑑−1(𝑧_𝑑−1)𝜏_𝑑−1(𝑊_𝑢𝑑(𝜌+ℎ^≤𝑑,𝑧)). (6.40) For notational simplicity, set 𝜆^′=𝜆−ℎ^>𝑑. A calculation using (6.30) shows that, compare the proof of Proposition 6.5, 𝑊_𝑢𝑑(𝜆,𝑧)=𝖱^{𝑑−1,𝑑}(𝜆^′,𝑧_𝑑−1𝑧_𝑑)𝖱^{𝑑−2,𝑑}(𝜆^′−ℎ^𝑑−1,𝑧_𝑑−2𝑧_𝑑)⋯𝖱^1,𝑑(𝜆^′−ℎ^[2,𝑑−1],𝑧₁𝑧_𝑑), (6.41) where ℎ^[𝑎,𝑏] means ∑_{𝑎≤𝑗≤𝑏}ℎ^𝑗. Thus 𝜏_𝑑−1(𝑊_𝑢𝑑(𝜆,𝑧))=𝖱^1,𝑑(𝜆^′,𝑧₁𝑧_𝑑)𝖱^2,𝑑(𝜆^′−ℎ¹,𝑧₂𝑧_𝑑)⋯𝖱^{𝑑−1,𝑑}(𝜆^′−ℎ^≤𝑑−2,𝑧_𝑑−1𝑧_𝑑). (6.42) Using (6.33) and the RLL relation (6.35) repeatedly, we obtain (6.40). Now the proposition follows by induction on 𝑑, using that 𝑊_𝑡𝑑−1(𝜆,𝑧)𝐿^𝑑(𝑧_𝑑)=𝐿^𝑑(𝑧_𝑑)𝑊_𝑡𝑑−1(𝜆+ℎ^𝑑,𝑧) (6.43) which follows from (6.33).

The operator 𝖢(𝜆,𝑧)∶=𝑊_𝑡𝑛(𝜆,𝑧) is called the Cherednik operator. For an operator 𝐹(𝑧)∈ℰ_𝑛 we define its matrix elements 𝐹(𝑧)^{𝑎1,…,𝑎𝑛}_{𝑥1,…,𝑥𝑛}∈ℱ by 𝐹(𝑧)(𝑒_𝑎1⊗⋯⊗𝑒_𝑎𝑛⊗1)=_{𝑥1,…,𝑥𝑛}𝑒_𝑥1⊗⋯⊗𝑒_𝑥𝑛⊗𝐹(𝑧)^{𝑎1,…,𝑎𝑛}_{𝑥1,…,𝑥𝑛}. (6.44)

Proposition 6.7. Let 𝛼(𝜆,𝑧)=lim_𝑤→𝑧𝜃(𝑞²𝑤)𝛼(𝜆,𝑤)=𝜃(𝑧)𝜃(𝑞^2(𝜆+1))𝜃(𝑞^2𝜆). (6.45) Then 𝖢(𝜆,𝑧)^1,…,𝑛_1,…,𝑛=_𝑖<𝑗𝛼(𝜆_𝑖𝑗,𝑧_𝑖𝑧_𝑗)=_𝑖<𝑗𝑧𝑞𝜃(^𝑖𝑧_𝑗𝐹)⋅^[1,𝑛](𝜆)𝐹^[1,𝑛].(𝜆) (6.46)

Proof. The second equality follows from the definition (6.1) of 𝐹_𝐼 and 𝐹^𝐼. We prove by induction on 𝑑 that 𝑊_𝑡𝑑(𝜆,𝑧)^1,…,𝑛_1,…,𝑛=∏_{𝑖<𝑗≤𝑑}𝛼(𝜆_𝑖𝑗,𝑧_𝑖/𝑧_𝑗). For 𝑑=2 we have 𝑡_𝑑=𝑠₁ and 𝑊_𝑠1(𝜆,𝑧)^1,…,𝑛_1,…,𝑛=𝖱¹²(𝜆−ℎ^>2,𝑧₁/𝑧₂)^1,…,𝑛_1,…,𝑛=𝛼(𝜆₁₂,𝑧₁/𝑧₂) as claimed. For 𝑑>2, using factorization (6.39) we have 𝑊_𝑡𝑑(𝜆,𝑧)^1,…,𝑛_1,…,𝑛=_{𝑥1,…,𝑥𝑛}𝑊_𝑡𝑑−1(𝜆,𝑧)^{𝑥1,…,𝑥𝑛}_1,…,𝑛𝜏_𝑑−1(𝑊_𝑢𝑑(𝜆,𝑧))_𝑥^1,…,𝑛_{1,…,𝑥𝑛}. (6.47)
Since 𝑊_𝑡𝑑−1(𝜆,𝑧) is a product of operators of the form 𝜎(𝖱^𝑖𝑖+1(𝜆,𝑧_𝑖/𝑧_𝑖+1)) where 1≤𝑖≤𝑑−2 and 𝜎∈𝑆_𝑛, 𝜎(𝑗)=𝑗,𝑗>𝑑−1, and each of these operators preserve the subspace spanned by 𝑒_𝜏(1)⊗⋯⊗𝑒_{𝜏(𝑑−1)}⊗𝑒_𝑑⊗⋯⊗𝑒_𝑛⊗𝑎, where 𝜏∈𝑆_𝑑−1 and 𝑎∈ℱ; the operator 𝑊_𝑡𝑑−1(𝜆,𝑧) also preserves this subspace. This means that 𝑊_𝑡𝑑−1(𝜆,𝑧)^{𝑥1,…,𝑥𝑛}_1,…,𝑛=0 unless 𝑥_𝑗=𝑗 for 𝑗≥𝑑 and {𝑥₁,…,𝑥_𝑑−1}={1,…,𝑑−1}. Furthermore, by (6.42), 𝜏_𝑑−1(𝑊_𝑢𝑑(𝜆,𝑧))_𝑥^1,…,𝑛_{1,…,𝑥𝑑−1,𝑑,…,𝑛}=_{𝑦2,…,𝑦𝑑−1}𝑅^1𝑦2_𝑥1𝑑𝑧(𝜆,¹𝑧_𝑑)𝑅^2𝑦3_𝑥2𝑦2𝑧(𝜆−𝜔(1),²𝑧_𝑑)⋯𝑅_𝑥^{𝑑−1,𝑑}_{𝑑−1𝑦𝑑−1}(𝜆−_{𝑘≤𝑑−2}𝑧𝜔(𝑘),^𝑑−1𝑧_𝑑). (6.48) Here 𝑅^𝑎𝑏_𝑥𝑦(𝜆,𝑧)=lim_𝑤→𝑧𝜃(𝑞²𝑤)𝑅^𝑎𝑏_𝑥𝑦(𝜆,𝑤). Since 𝑅^𝑎𝑏_𝑥𝑦(𝜆,𝑧)=0 unless {𝑥,𝑦}={𝑎,𝑏}, we deduce that, when {𝑥₁,…,𝑥_𝑑−1}={1,…,𝑑−1}, the terms in the sum (6.48) are zero unless 𝑥_𝑖=𝑖 for all 𝑖 and 𝑦_𝑗=𝑑 for all 𝑗. Substituting into (6.47) the claim follows by induction.

Lemma 6.8. Fix 2≤𝑑≤𝑛 and 𝑖<𝑑. Then there are elements 𝑏,𝑐∈𝐵_𝑛 such that 𝑡_𝑑=𝑠_𝑖𝑏 and 𝑡_𝑑=𝑐𝑠_𝑖.

Proof. Since 𝑡₂=𝑠₁ and 𝑡₃=𝑠₁𝑠₂𝑠₁=𝑠₂𝑠₁𝑠₂, the statement clearly holds for 𝑑=2,3. Assuming 𝑑>3, we first prove the existence of 𝑏. If 𝑖<𝑑−1 then by induction there is a 𝑏^′∈𝐵_𝑛 such that 𝑡_𝑑−1=𝑠_𝑖𝑏^′. Hence 𝑡_𝑑=𝑡_𝑑−1𝑠_𝑑−1⋯𝑠₁=𝑠_𝑖𝑏^′𝑠_𝑑−1⋯𝑠₁. Thus we can take 𝑏=𝑏^′𝑠_𝑑−1⋯𝑠₁. If 𝑖=𝑑−1, write 𝑡_𝑑=𝑡_𝑑−2𝑠_𝑑−2⋯𝑠₁𝑠_𝑑−1⋯𝑠₁. Then move each of the 𝑑−1 rightmost factors 𝑠_𝑑−1,…,𝑠₁ as far to the left as possible, using that 𝑠_𝑗𝑠_𝑘=𝑠_𝑘𝑠_𝑗 when |𝑗−𝑘|>1. This gives 𝑡_𝑑=𝑡_𝑑−2𝑠_𝑑−2𝑠_𝑑−1𝑠_𝑑−3𝑠_𝑑−2𝑠_𝑑−4⋯𝑠₂𝑠₃𝑠₁𝑠₂𝑠₁. (6.49) Then use 𝑠_𝑗𝑠_𝑗+1𝑠_𝑗=𝑠_𝑗+1𝑠_𝑗𝑠_𝑗+1 repeatedly, working from right to left, to obtain 𝑡_𝑑=𝑡_𝑑−2𝑠_𝑑−1𝑠_𝑑−2𝑠_𝑑−1𝑠_𝑑−3𝑠_𝑑−2⋯𝑠₄𝑠₂𝑠₃𝑠₁𝑠₂. (6.50) Finally, 𝑠_𝑑−1 can be moved to the left of 𝑡_𝑑−2 since the latter is a product of 𝑠_𝑗's with 𝑗≤𝑑−3.
To prove the existence of 𝑐 we note that 𝐵_𝑛 carries an involution ∗∶𝐵_𝑛→𝐵_𝑛 satisfying (𝑎₁𝑎₂)^∗=𝑎^∗₂𝑎^∗₁ for any 𝑎₁,𝑎₂∈𝐵_𝑛, defined by 𝑠^∗_𝑗=𝑠_𝑗 for 𝑗∈[1,𝑛] and 1^∗=1. Thus it suffices to show that 𝑡^∗_𝑑=𝑡_𝑑 for any 𝑑. This is trivial for 𝑑=2,3. When 𝑑>3 we have, by induction on 𝑑,
𝑡^∗_𝑑=(𝑡_𝑑−1𝑠_𝑑−1⋯𝑠₁)^∗=𝑠₁⋯𝑠_𝑑−1𝑡_𝑑−1=𝑠₁⋯𝑠_𝑑−1𝑡_𝑑−2𝑠_𝑑−2⋯𝑠₁=𝑠₁⋯𝑠_𝑑−2𝑡_𝑑−2𝑠_𝑑−1𝑠_𝑑−2⋯𝑠₁=𝑡^∗_𝑑−1𝑠_𝑑−1⋯𝑠₁=𝑡_𝑑(since𝑠_𝑑−1commuteswith𝑡_𝑑−2). (6.51)

Proposition 6.9. Let 𝑤=(𝑧₀,𝑞²𝑧₀,…,𝑞^2(𝑛−1)𝑧₀), where 𝑧₀≠0 is arbitrary, and let 𝜎,𝜏∈𝑆_𝑛. Then 𝖢(𝜆,𝑤)^{𝜏(1),…,𝜏(𝑛)}_{𝜎(1),…,𝜎(𝑛)}=sgn_[1,𝑛](𝜎;𝜆)sgn^[1,𝑛](𝜏;𝜆)𝖢(𝜆,𝑤)^1,…,𝑛_1,…,𝑛. (6.52)

Proof. First we claim that for all 𝜎,𝜏∈𝑆_𝑛 and each 𝑖∈[1,𝑛], 𝑊_𝑠𝑖(𝜆,𝑤)^{𝜏(1),…,𝜏(𝑛)}_{𝜎𝜎𝑖(1),…,𝜎𝜎𝑖(𝑛)}=𝜎(sgn_[1,𝑛](𝜎_𝑖;𝜆))𝑊_𝑠𝑖(𝜆,𝑤)^{𝜏(1),…,𝜏(𝑛)}_{𝜎(1),…,𝜎(𝑛)},𝑊_𝑠𝑖(𝜆,𝑤)^{𝜏𝜎𝑖(1),…,𝜏𝜎𝑖(𝑛)}_{𝜎(1),…,𝜎(𝑛)}=𝜏(sgn^[1,𝑛](𝜎_𝑖;𝜆))𝑊_𝑠𝑖(𝜆,𝑤)^{𝜏(1),…,𝜏(𝑛)}_{𝜎(1),…,𝜎(𝑛)}=−𝑊_𝑠𝑖(𝜆,𝑤)^{𝜏(1),…,𝜏(𝑛)}_{𝜎(1),…,𝜎(𝑛)}. (6.53)(6.54) Indeed, assume that 𝑧_𝑖/𝑧_𝑖+1=𝑞⁻² and that {𝑎₁,…,𝑎_𝑛}={𝑏₁,…,𝑏_𝑛}=[1,𝑛]. Then 𝑊_𝑠𝑖(𝜆,𝑧)^{𝑏1,…,𝑏𝑛}_{𝑎1,…,𝑎𝑛}≠0 if and only if {𝑎_𝑖,𝑎_𝑖+1}={𝑏_𝑖,𝑏_𝑖+1} in which case 𝑊_𝑠𝑖(𝜆,𝑧)^{𝑏1,…,𝑏𝑛}_{𝑎1,…,𝑎𝑛}=𝐸(1)𝐸(𝜆_{𝑎𝑖𝑎𝑖+1}+1)𝐸(𝜆_{𝑏𝑖+1𝑏𝑖}). (6.55) From this and the definitions of the sign functions, (6.2), the claims follow. Next, we prove (6.52) by induction on the sum ℓ of the lengths of 𝜎 and 𝜏. If ℓ=0, it is trivial. Assuming (6.52) holds for (𝜎,𝜏) we prove it holds for (𝜎𝜎_𝑖,𝜏) and (𝜎,𝜏𝜎_𝑖) where 𝑖 is arbitrary.
Let 𝑖∈[1,𝑛]. By Lemma 6.8 we have 𝑡_𝑛=𝑠_𝑖𝑏 for some 𝑏∈𝐵_𝑛. We have 𝑊_𝑡𝑛(𝜆,𝑤)^{𝜏(1),…,𝜏(𝑛)}_{𝜎𝜎𝑖(1),…,𝜎𝜎𝑖(𝑛)}=(𝑊_𝑠𝑖(𝜆,𝑤)𝜎_𝑖(𝑊_𝑏(𝜆,𝑤)))^{𝜏(1),…,𝜏(𝑛)}_{𝜎𝜎𝑖(1),…,𝜎𝜎𝑖(𝑛)}=_{𝑥1,…,𝑥𝑛}𝑊_𝑠𝑖(𝜆,𝑤)^{𝑥1,…,𝑥𝑛}_{𝜎𝜎𝑖(1),…,𝜎𝜎𝑖(𝑛)}𝜎_𝑖(𝑊_𝑏(𝜆,𝑤))_𝑥^{𝜏(1),…,𝜏(𝑛)}_{1,…,𝑥𝑛}. (6.56) As in the proof of Proposition 6.7, 𝑊_𝑠𝑖(𝜆,𝑤)^{𝑥1,…,𝑥𝑛}_{𝜎𝜎𝑖(1),…,𝜎𝜎𝑖(𝑛)} is zero, if 𝑥₁,…,𝑥_𝑛 is not a permutation of 1,…,𝑛. Using (6.53) we obtain 𝜎(sgn_[1,𝑛](𝜎_𝑖;𝜆))_{𝑥1,…,𝑥𝑛}𝑊_𝑠𝑖(𝜆,𝑤)^{𝑥1,…,𝑥𝑛}_{𝜎(1),…,𝜎(𝑛)}𝜎_𝑖(𝑊_𝑏(𝜆,𝑤))_𝑥^{𝜏(1),…,𝜏(𝑛)}_{1,…,𝑥𝑛}=𝜎(sgn_[1,𝑛](𝜎_𝑖;𝜆))𝑊_𝑡𝑛(𝜆,𝑤)^{𝜏(1),…,𝜏(𝑛)}_{𝜎(1),…,𝜎(𝑛)}. (6.57) Using the induction hypothesis and the relation sgn_[1,𝑛](𝜎;𝜆)𝜎(sgn_[1,𝑛](𝜎_𝑖;𝜆))=sgn_[1,𝑛](𝜎𝜎_𝑖;𝜆) we obtain (6.52) for (𝜎𝜎_𝑖,𝜏).
For the other case, let 𝑖 be arbitrary, and set 𝑗=𝜏_𝑛(𝑖). By Lemma 6.8 there is a 𝑐∈𝐵_𝑛 such that 𝑡_𝑛=𝑐𝑠_𝑗. Recall the surjective morphism 𝜋∶𝐵_𝑛→𝑆_𝑛 sending 𝑠_𝑖 to 𝜎_𝑖=(𝑖𝑖+1). Then 𝜎_𝑗𝜋(𝑐)=𝜋(𝑐)𝜎_𝑖. We have 𝑊_𝑡𝑛(𝜆,𝑤)^{𝜏𝜎𝑖(1),…,𝜏𝜎𝑖(𝑛)}_{𝜎(1),…,𝜎(𝑛)}=(𝑊_𝑐(𝜆,𝑤)⋅𝜋(𝑐)(𝑊_𝑠𝑗(𝜆,𝑤)))^{𝜏𝜎𝑖(1),…𝜏𝜎𝑖(𝑛)}_{𝜎(1),…,𝜎(𝑛)}=_{𝑥1,…,𝑥𝑛}𝑊_𝑐(𝜆,𝑤)^{𝑥1,…,𝑥𝑛}_{𝜎(1),…,𝜎(𝑛)}𝜋(𝑐)(𝑊_𝑠𝑗(𝜆,𝑤))^{𝜏𝜎𝑖(1),…,𝜏𝜎𝑖}_𝑥^(𝑛)_{1,…,𝑥𝑛}. (6.58) It is easy to check that 𝜎(𝐹(𝑧))^{𝑏1,…,𝑏𝑛}_{𝑎1,…,𝑎𝑛}=𝐹(𝜎(𝑧))^{𝑏𝜎(1),…,𝑏𝜎(𝑛)}_{𝑎𝜎(1),…,𝑎𝜎(𝑛)} for any 𝐹(𝑧)∈ℰ_𝑛 and 𝜎∈𝑆_𝑛. Define 𝑤_𝑖 by (𝑤₁,…,𝑤_𝑛)=𝑤=(𝑧₀,𝑞²𝑧₀,…,𝑞^2(𝑛−1)𝑧₀). Then 𝑤_𝑖/𝑤_𝑖+1=𝑞⁻² for each 𝑖. Set 𝑤^′=(𝑤_{𝜋(𝑐)(1)},…,𝑤_{𝜋(𝑐)(𝑛)}). For each 𝑖, 𝑤_{𝜋(𝑐)(𝑖)}/𝑤_{𝜋(𝑐)(𝑖+1)}=𝑤_{𝜏𝑛(𝑖+1)}/𝑤_{𝜏𝑛(𝑖)}=𝑞⁻² also. Therefore 𝑊_𝑡𝑛(𝜆,𝑤)^{𝜏𝜎𝑖(1),…,𝜏𝜎𝑖(𝑛)}_{𝜎(1),…,𝜎(𝑛)}=_{𝑥1,…,𝑥𝑛}𝑊_𝑐(𝜆,𝑤)^{𝑥1,…,𝑥𝑛}_{𝜎(1),…,𝜎(𝑛)}𝑊_𝑠𝑗(𝜆,𝑤^′)^{𝜏𝜎𝑖𝜋(𝑐)(1),…,𝜏𝜎𝑖}_𝑥^{𝜋(𝑐)(𝑛)}_{𝜋(𝑐)(1),…,𝑥𝜋(𝑐)(𝑛)}=_{𝑥1,…,𝑥𝑛}𝑊_𝑐(𝜆,𝑤)^{𝑥1,…,𝑥𝑛}_{𝜎(1),…,𝜎(𝑛)}𝑊_𝑠𝑗(𝜆,𝑤^′)^{𝜏𝜋(𝑐)𝜎𝑗(1),…,𝜏𝜋(𝑐)𝜎𝑗}_𝑥^(𝑛)_{𝜋(𝑐)(1),…,𝑥𝜋(𝑐)(𝑛)}=_{𝑥1,…,𝑥𝑛}𝑊_𝑐(𝜆,𝑤)^{𝑥1,…,𝑥𝑛}_{𝜎(1),…,𝜎(𝑛)}(sgn𝜎_𝑗)𝑊_𝑠𝑗(𝜆,𝑤^′)_𝑥^{𝜏𝜋(𝑐)(1),…,𝜏𝜋(𝑐)(𝑛)}_{𝜋(𝑐)(1),…,𝑥𝜋(𝑐)(𝑛)}=_{𝑥1,…,𝑥𝑛}𝑊_𝑐(𝜆,𝑤)^{𝑥1,…,𝑥𝑛}_{𝜎(1),…,𝜎(𝑛)}(−1)𝜋(𝑐)(𝑊_𝑠𝑗(𝜆,𝑤))^{𝜏(1),…,𝜏(𝑛)}_{x1,…,𝑥𝑛}=−𝑊_𝑡𝑛(𝜆,𝑤)^{𝜏(1),…,𝜏(𝑛)}_{𝜎(1),…,𝜎(𝑛)}. (6.59) By the induction hypothesis it follows that (6.52) holds for (𝜎,𝜏𝜎_𝑖). This proves the formula (6.52).

Theorem 6.10. For any subsets 𝐼,𝐽⊆[1,𝑛] and 𝑧∈ℂ^×, the left and right elliptic minors coincide: ^⟵𝜉^𝐽_𝐼(𝑧)=𝜉^𝐽_𝐼(𝑧). (6.60) We denote this common element by 𝜉^𝐽_𝐼(𝑧).

Proof. If #𝐼≠#𝐽 then both sides are zero. Suppose #𝐼=#𝐽=𝑑. By relation (6.15) we can, after applying a suitable automorphism, assume that 𝐼=𝐽=[1,𝑑]. Since the subalgebra of ℱ generated by 𝑒_𝑖𝑗(𝑧), 𝑖,𝑗∈[1,𝑑], 𝑧∈ℂ^× and 𝑓(𝜆),𝑓(𝜌) with 𝑓∈𝑀_𝔥^∗_𝑑⊆𝑀_𝔥^∗, 𝔥_𝑑 being the Cartan subalgebra of 𝔰𝔩(𝑑), is isomorphic to ℱ_ell(𝑀(𝑑)), we can also assume 𝑑=𝑛. Identifying the matrix element 1,…,𝑛1,…,𝑛 on both sides of (6.38) we get _{𝑥1,…,𝑥𝑛}C(𝜆,𝑧)^{𝑥1,…,𝑥𝑛}_1,…,𝑛𝑒_𝑥1,1(𝑧₁)⋯𝑒_{𝑥𝑛,𝑛}(𝑧_𝑛)=_{𝑥1,…,𝑥𝑛}𝑒_{𝑛,𝑥𝑛}(𝑧_𝑛)⋯𝑒_1,𝑥1(𝑧₁)𝖢(𝜌+ℎ^≤𝑛,𝑧)_𝑥^1,…,𝑛_{1,…,𝑥𝑛}. (6.61) As in the proof of Proposition 6.7, 𝖢(𝜆,𝑧)^{𝑥1,…,𝑥𝑛}_1,…,𝑛 is zero if 𝑥₁,…,𝑥_𝑛 is not a permutation of 1,…,𝑛. Taking 𝑧=𝑤=(𝑧₀,𝑞²𝑧₀,…,𝑞^2(𝑛−1)𝑧₀) and dividing both sides by ∏_𝑖<𝑗𝑞𝜃(𝑤_𝑖/𝑤_𝑗)=∏_𝑖<𝑗𝑞𝜃(𝑞^{2(𝑖−𝑗)}) we get 𝐹_[1,𝑛](𝜆)𝐹^[1,𝑛](𝜆)_{𝜎∈𝑆𝑛}sgn^[1,𝑛](𝜎;𝜆)⁻¹𝑒_𝜎(1)1(𝑧₀)⋯𝑒_{𝜎(𝑛)𝑛}(𝑞^2(𝑛−1)𝑧₀)=𝐹_[1,𝑛](𝜌)𝐹^[1,𝑛](𝜌)_{𝜏∈𝑆𝑛}sgn_[1,𝑛](𝜏;𝜌)𝑒_{𝑛𝜏(𝑛)}(𝑞^2(𝑛−1)𝑧₀)⋯𝑒_1𝜎(1)(𝑧₀). (6.62) Multiplying by 𝐹^[1,𝑛](𝜌)/𝐹_[1,𝑛](𝜆) and comparing with (6.13) and (6.14), we deduce that 𝜉^[1,𝑛]_[1,𝑛](𝑧₀)=^⟵𝜉^[1,𝑛]_[1,𝑛](𝑧₀), as desired.

6.3. Laplace Expansions

Using the left (right) ℱ_ell(𝑀(𝑛))-comodule algebra structure of Λ(Λ^′) it is straightforward to prove Laplace expansion formulas for the elliptic minors. For subsets 𝐼,𝐽⊆[1,𝑛] we define 𝑆_𝑙(𝐼,𝐽;𝜁),𝑆_𝑟(𝐼,𝐽;𝜁)∈𝑀_𝔥^∗ by 𝑣_𝐼(𝑞^2#𝐽𝑧)𝑣_𝐽(𝑧)=𝑆_𝑙(𝐼,𝐽;𝜁)𝑣_𝐼∪𝐽𝑤(𝑧),_𝐼(𝑧)𝑤^𝐽(𝑞^2#𝐼𝑧)=𝑆_𝑟(𝐼,𝐽;𝜁)𝑤^𝐼∪𝐽(𝑧). (6.63) That this is possible follows from the definitions (6.7) and (6.8) of 𝑣_𝐼(𝑧),𝑤^𝐼(𝑧), and the commutation relations (5.8b)–(5.8d), (5.16). In particular 𝑆_𝑙(𝐼,𝐽;𝜁)=0=𝑆_𝑟(𝐼,𝐽;𝜁), if 𝐼∩𝐽≠∅.

Theorem 6.11. (i) Let 𝐼₁,𝐼₂,𝐽⊆[1,𝑛], and set 𝐼=𝐼₁∪𝐼₂. Then 𝑆_𝑙(𝐼₁,𝐼₂;𝜆)𝜉^𝐽_𝐼(𝑧)=_{𝐽1∪𝐽2=𝐽}𝑆_𝑙(𝐽₁,𝐽₂;𝜌)𝜉^𝐽1_𝐼1(𝑞^2#𝐼2𝑧)𝜉^𝐽2_𝐼2(𝑧). (6.64)
(ii) Let 𝐽₁,𝐽₂,𝐼⊆[1,𝑛] and set 𝐽=𝐽₁∪𝐽₂. Then 𝑆_𝑟(𝐽₁,𝐽₂;𝜌)𝜉^𝐽_𝐼(𝑧)=_{𝐼1∪𝐼2=𝐼}𝑆_𝑟(𝐼₁,𝐼₂;𝜆)𝜉^𝐽1_𝐼1(𝑧)𝜉^𝐽2_𝐼2(𝑞^2#𝐽1𝑧). (6.65)

Proof. We have Δ_Λ(𝑣_𝐼1(𝑞^2#𝐼2𝑧))Δ_Λ(𝑣_𝐼2(𝑧))=_𝐽1,𝐽2𝜉^𝐽1_𝐼1(𝑞^2#𝐼2𝑧)𝜉^𝐽2_𝐼2(𝑧)⊗𝑣_𝐽1(𝑞^2#𝐼2𝑧)𝑣_𝐽2(𝑧)=_𝐽1,𝐽2𝜉^𝐽1_𝐼1(𝑞^2#𝐼2𝑧)𝜉^𝐽2_𝐼2(𝑧)⊗𝑆_𝑙(𝐽₁,𝐽₂;𝜁)𝑣_𝐽(𝑧)=_𝐽(_{𝐽1∪𝐽2=𝐽}𝑆_𝑙(𝐽₁,𝐽₂;𝜌)𝜉^𝐽1_𝐼1(𝑞^2#𝐼2𝑧)𝜉^𝐽2_𝐼2(𝑧))⊗𝑣_𝐽(𝑧). (6.66) On the other hand, Δ_Λ(𝑣_𝐼1(𝑞^2#𝐼2𝑧))Δ_Λ(𝑣_𝐼2(𝑧))=Δ_Λ(𝑣_𝐼1(𝑞^2#𝐼2𝑧)𝑣_𝐼2(𝑧))=Δ_Λ(𝑆_𝑙(𝐼₁,𝐼₂;𝜁)𝑣_𝐼(𝑧))=_𝐽𝑆_𝑙(𝐼₁,𝐼₂;𝜆)𝜉^𝐽_𝐼(𝑧)⊗𝑣_𝐽(𝑧). (6.67) Equating these expressions proves (6.64) since, by Proposition 5.5, the set {𝑣_𝐽(𝑧)∶𝐽⊆[1,𝑛]} is linearly independent over 𝑀_𝔥^∗. The second part is completely analogous, using the right comodule algebra Λ^′ in place of Λ.

In Section 7.4 we will need the following lemma, relating the left and right signums 𝑆_𝑙(𝐼,𝐽;𝜁) and 𝑆_𝑟(𝐼,𝐽;𝜁), defined in (6.63). In the nonspectral trigonometric case the corresponding identity was proved in [15, proof of Proposition 4.1.22].

Lemma 6.12. Let 𝐼,𝐽 be two disjoint subsets of [1,𝑛]. Then 𝑆_𝑙(𝐼,𝐽;𝜁+𝜔(𝐼))=𝑆_𝑟(𝐽,𝐼;𝜁)⁻¹, (6.68) where 𝜔(𝐼)=∑_𝑖∈𝐼𝜔(𝑖).

Proof. First we claim that, we have the following explicit formulas: 𝑆_𝑙(𝐼,𝐽;𝜁)=_{𝑖∈𝐼,𝑗∈𝐽}𝐸(𝜁_𝑗𝑖+1),𝑆_𝑟(𝐼,𝐽;𝜁)=_{𝑖∈𝐼,𝑗∈𝐽}𝐸(𝜁_𝑖𝑗)⁻¹. (6.69)(6.70) Recall the definition (6.7) of 𝑣_𝐼(𝑧). Since 𝐸 is odd, relation (5.8b) implies that 𝑣_𝑖(𝑞²𝑧)𝑣_𝑗(𝑧)=𝐸(𝜁_𝑗𝑖+1)𝐸(𝜁_𝑖𝑗𝑣+1)^𝑗(𝑞²𝑧)𝑣_𝑖(𝑧). (6.71) Also, 𝐹_𝐽(𝜁) only involves 𝜁_𝑖𝑗 with 𝑖,𝑗∈𝐽 so it commutes with any 𝑣_𝑘(𝑧) with 𝑘∈𝐼 (since 𝐼∩𝐽=∅). From these facts we obtain 𝑣_𝐼(𝑞^2#𝐽𝑧)𝑣_𝐽𝐹(𝑧)=^𝐼(𝜁)⁻¹𝐹_𝐽(𝜁)⁻¹𝐹_𝐼∪𝐽(𝜁)⁻¹_{i∈𝐼,𝑗∈𝐽𝑖<𝑗}𝐸(𝜁_𝑗𝑖+1)𝐸(𝜁_𝑖𝑗𝑣+1)^𝐼∪𝐽(𝑧)=_{(𝑖,𝑗)∈𝐾𝑖<𝑗}𝐸(𝜁_𝑖𝑗+1)_{𝑖∈𝐼,𝑗∈𝐽𝑖<𝑗}𝐸(𝜁_𝑗𝑖+1)𝐸(𝜁_𝑖𝑗𝑣+1)^𝐼∪𝐽(𝑧)=_{𝑖∈𝐼,𝑗∈𝐽}𝐸(𝜁_𝑗𝑖+1)𝑣_𝐼∪𝐽(𝑧), (6.72) where 𝐾=(𝐼×𝐽)∪(𝐽×𝐼). This proves (6.69). Similarly one proves (6.70). Now we have 𝑆_𝑙(𝐽,𝐼;𝜁+𝜔(𝐽))⁻¹=_{𝑖∈𝐼,𝑗∈𝐽}𝐸((𝜁+𝜔(𝐽))_𝑖𝑗+1)⁻¹=_{𝑖∈𝐼,𝑗∈𝐽}𝐸(𝜁_𝑖𝑗)⁻¹=𝑆_𝑟(𝐼,𝐽;𝜁). (6.73) Here we used that for any 𝑖∈𝐼,𝑗∈𝐽 we have 𝜔(𝐽)(𝐸_𝑖𝑖)=0, 𝜔(𝐽)(𝐸_𝑗𝑗)=1, and hence (𝜔(𝐽))_𝑖𝑗=−1.

7. The Cobraiding and the Elliptic Determinant

7.1. Cobraidings for 𝔥-Bialgebroids

The following definition of a cobraiding was given in [14] and studied further in [10]. When 𝔥=0 the notion reduces to ordinary cobraidings for bialgebras.

Definition 7.1. A cobraiding on an 𝔥-bialgebroid 𝐴 is a ℂ-bilinear map ⟨⋅,⋅⟩∶𝐴×𝐴→𝐷_𝔥 such that, for any 𝑎,𝑏,𝑐∈𝐴 and 𝑓∈𝑀_𝔥^∗, ⟨𝐴_𝛼𝛽,𝐴_𝛾𝛿⟩⊆(𝐷_𝔥)_{𝛼+𝛾,𝛽+𝛿},⟨𝜇_𝑟(𝑓)𝑎,𝑏⟩=⟨𝑎,𝜇_𝑙(𝑓)𝑏⟩=𝑓𝑇₀∘⟨𝑎,𝑏⟩,⟨𝑎𝜇_𝑙(𝑓),𝑏⟩=⟨𝑎,𝑏𝜇_𝑟(𝑓)⟩=⟨𝑎,𝑏⟩∘𝑓𝑇₀,⟨𝑎𝑏,𝑐⟩=_𝑖⟨𝑎,𝑐^′_𝑖⟩𝑇_𝛽𝑖⟨𝑏,𝑐_𝑖^′′⟩,Δ(𝑐)=_𝑖𝑐^′_𝑖⊗𝑐_𝑖^′′,𝑐_𝑖^′′∈𝐴_𝛽𝑖𝛾,⟨𝑎,𝑏𝑐⟩=_𝑖⟨𝑎_𝑖^′′,𝑏⟩𝑇_𝛽𝑖⟨𝑎^′_𝑖,𝑐⟩,Δ(𝑎)=_𝑖𝑎^′_𝑖⊗𝑎_𝑖^′′,𝑎_𝑖^′′∈𝐴_𝛽𝑖𝛾,⟨𝑎,1⟩=⟨1,𝑎⟩=𝜀(𝑎),_𝑖𝑗𝜇_𝑙(⟨𝑎^′_𝑖,𝑏^′_𝑗⟩1)𝑎_𝑖^′′𝑏_𝑗^′′=_𝑖𝑗𝜇_𝑟(⟨𝑎_𝑖^′′,𝑏_𝑗^′′⟩1)𝑏^′_𝑗𝑎^′_𝑖. (7.1a)(7.1b)(7.1c)(7.1d)(7.1e)(7.1f)(7.1g)
The following definition was given in unpublished notes by Rosengren [16]. The terminology is motivated by Proposition 7.6 concerning FRST algebras 𝐴_𝑅, but it makes sense for arbitrary 𝔥-bialgebroids.

Definition 7.2. A cobraiding ⟨⋅,⋅⟩ on an 𝔥-bialgebroid 𝐴 is called unitary if 𝜀(𝑎𝑏)=_{(𝑎),(𝑏)}⟨𝑎^′,𝑏^′⟩𝑇_{𝜔12(𝑎)+𝜔12(𝑏)}⟨𝑎^′′,𝑏^′′⟩ (7.2) for all 𝑎,𝑏∈𝐴. In such sums we always assume, without loss of generality, that all 𝑎^′,𝑎^′′,𝑏^′,𝑏^′′ are homogenous and use the notation 𝜔₁₂(𝑎)=𝛾 if 𝑎^′∈𝐴_𝛼𝛾 for some 𝛼 (or equivalently, if 𝑎^′′∈𝐴_𝛾𝛽 for some 𝛽).

7.2. Cobraidings for the FRST Algebras 𝐴_𝑅

Now let 𝑅∶𝔥^∗×ℂ^×→End_𝔥(𝑉⊗𝑉) be a meromorphic function, and let 𝐴_𝑅 be the 𝔥-bialgebroid associated to 𝑅 as in Section 3.2.

Proposition 7.3. Assume that 𝜑∶ℂ^×→ℂ is a holomorphic function, not vanishing identically, such that, for each 𝑥,𝑦,𝑎,𝑏∈𝑋, 𝑧∈ℂ^×, the limit lim_𝑤→𝑧(𝜑(𝑤)𝑅^𝑎𝑏_𝑥𝑦(𝜁,𝑤)) exists and defines a meromorphic function in 𝑀_𝔥^∗. Then the following statements are equivalent:
(i)there exists a cobraiding ⟨⋅,⋅⟩∶𝐴_𝑅×𝐴_𝑅→𝐷_𝔥 satisfying ⟨𝐿_𝑖𝑗(𝑧₁),𝐿_𝑘𝑙(𝑧₂)⟩=lim_{𝑤→𝑧1/𝑧2}(𝜑(𝑤)𝑅^𝑗𝑙_𝑖𝑘(𝜁,𝑤))𝑇_{−𝜔(𝑖)−𝜔(𝑘)}, (7.3)(ii)𝑅 satisfies the QDYBE (2.8).

Remark 7.4. (a) The identity (7.1g) is not necessary when proving that (i) implies (ii). Without assuming (7.1g), ⟨⋅,⋅⟩ is a pairing on 𝐴^cop×𝐴. See [14].
(b) Without the factor 𝜑(𝑤), the cobraiding is not well defined if 𝑅(𝜁,𝑧) has poles in the 𝑧 variable. We also remark that the residual relations (3.8) are necessary for (ii) to imply (i).

Proof. The proof is straightforward and is carried out in [15, Lemma 2.2.5], under the assumption that the R-matrix is regular in the spectral variable.

We will now generalize slightly the notion of a unitary cobraiding on 𝐴_𝑅 to account for spectral singularities in the R-matrix as follows.

Call 𝑎∈𝐴_𝑅spectrally homogenous if there exist 𝑘∈ℤ_≥0 and 𝑧₁,…,𝑧_𝑘∈ℂ^× such that 𝑎∈_{𝜎∈𝑆𝑘𝑖𝑙,𝑗𝑙∈𝑋}𝑀_𝔥^∗⊗𝑀_𝔥^∗𝐿_𝑖1𝑗1(𝑧_𝜎(1))⋯𝐿_{𝑖𝑘𝑗𝑘}(𝑧_𝜎(𝑘)). (7.4) The multiset {𝑧_𝑖}_𝑖 is called the spectral degree of 𝑎 and is denoted by sdeg(𝑎). Note that the spectral degree of a nonzero spectrally homogenous element is uniquely defined, since the RLL relations (3.8) are spectrally homogenous.

Let 𝜑∶ℂ^×→ℂ be holomorphic. For spectrally homogenous elements 𝑎,𝑏∈𝐴_𝑅, define the regularizing factor 𝜑(𝑎,𝑏) by 𝜑(𝑎,𝑏)=_{1≤𝑖≤𝑘,1≤𝑗≤𝑙}𝑧𝜑(^𝑖𝑤_𝑗), (7.5) where {𝑧_𝑖}_𝑖=sdeg(𝑎), {𝑤_𝑗}_𝑗=sdeg(𝑏).

Definition 7.5. Let 𝜑∶ℂ^×→ℂ be holomorphic. A cobraiding ⟨⋅,⋅⟩ on 𝐴_𝑅 is unitary with respect to 𝜑 if 𝜑(𝑎,𝑏)𝜑(𝑏,𝑎)𝜀(𝑎𝑏)=_{(𝑎),(𝑏)}⟨𝑎^′,𝑏^′⟩𝑇_{𝜔12(𝑎)+𝜔12(𝑏)}⟨𝑎^′′,𝑏^′′⟩ (7.6) for all spectrally homogenous 𝑎,𝑏∈𝐴_𝑅.

The following proposition was proved in [16] if the spectral variables are taken to be generic so that no regularizing factors are needed.

Proposition 7.6. Suppose that 𝑅∶𝔥^∗×ℂ^×→End_ℂ(𝑉⊗𝑉) satisfies the QDYBE and is unitary: 𝑅(𝜁,𝑧)𝑅(𝜁,𝑧⁻¹)⁽²¹⁾=Id_𝑉⊗𝑉. Suppose that 𝜑∶ℂ^×→ℂ is nonzero holomorphic such that lim_𝑤→𝑧(𝜑(𝑤)𝑅^𝑎𝑏_𝑥𝑦(𝜁,𝑤)) exists and is a holomorphic function in 𝑀_𝔥^∗. Then the cobraiding ⟨⋅,⋅⟩ on 𝐴_𝑅 given in Proposition 7.3 is unitary with respect to 𝜑.

Proof. Since both sides are holomorphic in the spectral variables, it is enough to prove it for generic values. We claim that for such values, 𝜑(𝑎,𝑏)⟨𝑎,𝑏⟩_𝑅=⟨𝑎,𝑏⟩ where ⟨⋅,⋅⟩_𝑅 is the cobraiding, defined only for generic spectral values, determined by ⟨𝐿_𝑖𝑗(𝑧),𝐿_𝑘𝑙(𝑤)⟩_𝑅=𝑅^𝑗𝑙_𝑖𝑘(𝜁,𝑧/𝑤)𝑇_{−𝜔(𝑖)−𝜔(𝑘)}. Indeed, this claim follows by induction from the identities (7.1d) and (7.1e) using that 𝜑(𝑎₁,𝑏)𝜑(𝑎₂,𝑐)=𝜑(𝑎₃,𝑏𝑐) and 𝜑(𝑐,𝑎₁)𝜑(𝑏,𝑎₂)=𝜑(𝑐𝑏,𝑎₃) for spectrally homogenous 𝑎_𝑖,𝑏,𝑐∈𝐴_𝑅, the 𝑎_𝑖 having the same spectral degree.
Since the R-matrix 𝑅 is unitary, the statement of the lemma now follows from the identity 𝜀(𝑎𝑏)=_{(𝑎),(𝑏)}⟨𝑎^′,𝑏^′⟩_𝑅𝑇_{𝜔12(𝑎)+𝜔12(𝑏)}⟨𝑎^′′,𝑏^′′⟩_𝑅 (7.7) holding for generic spectral values which was proved by Rosengren [16].

7.2.1. The Case of ℱ_ell(𝑀(𝑛))

Specializing further to the algebra of interest, ℱ_ell(𝑀(𝑛)), we obtain the following corollary.

Corollary 7.7. The 𝔥-bialgebroid ℱ_ell(𝑀(𝑛)) carries a cobraiding ⟨⋅,⋅⟩ satisfying ⟨𝑒_𝑖𝑗(𝑧),𝑒_𝑘𝑙(𝑤)⟩=𝑅^𝑗𝑙_𝑖𝑘𝑧(𝜁,𝑤)𝑇_{−𝜔(𝑖)−𝜔(𝑘)}∀𝑧,𝑤∈ℂ^×,𝑖,𝑗∈[1,𝑛], (7.8) where 𝑅^𝑗𝑙_𝑖𝑘(𝜁,𝑧)=lim_𝑤→𝑧(𝜃(𝑞²𝑤)𝑅^𝑗𝑙_𝑖𝑘(𝜁,𝑤)). (7.9) Moreover, this cobraiding is unitary with respect to 𝜑∶ℂ^×→ℂ, 𝜑(𝑧)=𝜃(𝑞²𝑧).

Proof. It suffices to notice that, by (4.3), (2.15), and (2.16), 𝑅 is regular in 𝑧 and apply Propositions 7.3 and 7.6.

7.3. Properties of the Elliptic Determinant

A common method used to study quantum minors and prove that quantum determinants are central is the fusion procedure, going back to work by Kulish and Sklyanin [17]. Another approach, using representation theory, was developed by Noumi et al. [18]. In this section we show how to prove that the elliptic determinant is central using the properties of the cobraiding on ℱ_ell(𝑀(𝑛)) and how to resolve technical issues connected with the spectral singularities of the elliptic R-matrix.

Let 𝐴=ℱ_ell(𝑀(𝑛)). When 𝐼=𝐽=[1,𝑛] we set det(𝑧)=𝜉^𝐽_𝐼(𝑧) (7.10) for 𝑧∈ℂ^×, where 𝜉^𝐽_𝐼(𝑧) is the elliptic minor given in Theorem 6.10. Thus one possible expression for det(𝑧) is det(𝑧)=_{𝜎∈𝑆𝑛}𝐹^[1,𝑛](𝜌)𝜎(𝐹^[1,𝑛]𝑒(𝜆))^𝜎(1)1(𝑧)𝑒_𝜎(2)2(𝑞²𝑧)⋯𝑒_{𝜎(𝑛)𝑛}(𝑞^2(𝑛−1)𝑧). (7.11)

Theorem 7.8. (a)det(𝑧) is a grouplike element of 𝐴 for each 𝑧∈ℂ^×, that is, Δ(det(𝑧))=det(𝑧)⊗det(𝑧),𝜀(det(𝑧))=1. (7.12)
(b)det(𝑧) is a central element in ℱ_ell(𝑀(𝑛)): [𝑒_𝑖𝑗(𝑧),det(𝑤)]=[𝑓(𝜆),det(𝑤)]=[𝑓(𝜌),det(𝑤)]=0 (7.13) for all 𝑓∈𝑀_𝔥^∗, 𝑖,𝑗∈[1,𝑛] and all 𝑧,𝑤∈ℂ^×.Proof. Let Λ^𝑛(𝑧)=𝑀_𝔥^∗𝑣_𝐼(𝑧), where 𝐼=[1,𝑛]. It is a one-dimensional subcorepresentation of the left exterior corepresentation Λ. Its matrix element is det(𝑧), that is, Δ(𝑣_𝐼(𝑧))=det(𝑧)⊗𝑣_𝐼(𝑧). (7.14) From the coassociativiy and counity axioms for a corepresentation, it follows that det(𝑧) is grouplike, proving part (a).
The rest of this section is devoted to the proof of part (b). It follows from the definition that det(𝑧)∈𝐴₀₀ and thus it commutes with 𝑓(𝜌) and 𝑓(𝜆) for any 𝑓∈𝑀_𝔥^∗. To prove that it commutes with the generators 𝑒_𝑖𝑗(𝑧) we need several lemmas which we now state and prove.
Lemma 7.9. For 𝑖,𝑗∈[1,𝑛], 𝐼,𝐽⊆[1,𝑛], #𝐼=#𝐽=2, we have ⟨𝜉^𝐽_𝐼(𝑤),𝑒_𝑖𝑗(𝑧)⟩=0,if𝑤𝑧∈𝑝^ℤ,⟨𝑒_𝑖𝑗(𝑧),𝜉^𝐽_𝐼(𝑤)⟩=0,if𝑞²𝑤𝑧∈𝑝^ℤ. (7.15)(7.16)Proof. Let 𝐼={𝑖₁,𝑖₂}, 𝑖₁<𝑖₂, 𝐽={𝑗₁,𝑗₂}, 𝑗₁<𝑗₂. Using the left expansion formula (6.13) and (7.1b), (7.1c) we have ⟨𝜉^𝐽_𝐼(𝑤),𝑒_𝑖𝑗(𝑧)⟩=⟨𝐸(𝜌_𝑗1𝑗2+1)𝐸(𝜆_𝑖1𝑖2𝑒+1)^𝑖2𝑗2(𝑞²𝑤)𝑒_𝑖1𝑗1(𝑤),𝑒_𝑖𝑗(𝑧)⟩+[𝑗₁⟷𝑗₂]=𝐸(𝜁_𝑗1𝑗2+1+1)⟨𝑒_𝑖2𝑗2(𝑞²𝑤)𝑒_𝑖1𝑗1(𝑤),𝑒_𝑖𝑗1(𝑧)⟩𝐸(𝜁_𝑖1𝑖2+1)+[𝑗₁⟷𝑗₂]. (7.17) Thus we need to prove that for 𝑤/𝑧∈𝑝^ℤ, the first term is antisymmetric in 𝑗₁,𝑗₂. By (7.1d), 𝐸(𝜁_𝑗1𝑗2+1)⟨𝑒_𝑖2𝑗2(𝑞²𝑤)𝑒_𝑖1𝑗1(𝑤),𝑒_𝑖𝑗(𝑧)⟩=𝐸(𝜁_𝑗1𝑗2+1)_𝑥⟨𝑒_𝑖2𝑗2(𝑞²𝑤),𝑒_𝑖𝑥(𝑧)⟩𝑇_𝜔(𝑥)⟨𝑒_𝑖1𝑗1(𝑤),𝑒_𝑥𝑗(𝑧)⟩=𝐸(𝜁_𝑗1𝑗2+1)_𝑥𝑅^𝑗2𝑥_𝑖2𝑖𝑞(𝜁,²𝑤𝑧)𝑅^𝑗1𝑗_𝑖1𝑥(𝜁−𝜔(𝑗₂𝑤),𝑧)𝑇_{−𝜔(𝑗1)−𝜔(𝑗2)−𝜔(𝑗)}. (7.18) Take now 𝑤=𝑝^𝑘𝑧 where 𝑘∈ℤ. One checks that 𝑅^𝑎𝑏_𝑥𝑦(𝜁,𝑝^𝑘)=𝜃(𝑞²𝑝^𝑘)𝑞^{2𝑘(𝜁𝑏𝑎+1−𝛿𝑎𝑏)}𝛿_𝑎𝑦𝛿_𝑏𝑥, (7.19) where 𝛿_𝑥𝑦 is the Kronecker delta. In particular, only the 𝑥=𝑗₁ term is nonzero. Now the antisymmetry of (7.18) in 𝑗₁,𝑗₂ follows by applying the identities 𝑅^𝑗1𝑗_𝑗𝑗1(𝜁−𝜔(𝑗₂),𝑝^𝑘)=𝑞^{2𝑘𝜁𝑗2𝑗1}⋅𝑅^𝑗2𝑗_𝑗𝑗2(𝜁−𝜔(𝑗₁),𝑝^𝑘),𝐸(𝜁_𝑗1𝑗2+1)𝑅^𝑗2𝑗1_𝑖2𝑖(𝜁,𝑞²𝑝^𝑘)=−𝑞^{2𝑘𝜁𝑗1𝑗2}⋅𝐸(𝜁_𝑗2𝑗1+1)𝑅^𝑗1𝑗2_𝑖2𝑖(𝜁,𝑞²𝑝^𝑘). (7.20)(7.21) Relation (7.20) can be proved directly from (7.19) while for (7.21) one can use that 𝑅^𝑎𝑏_𝑥𝑦(𝜁,𝑝^𝑘𝑧)𝑅^𝑏𝑎_𝑥𝑦(𝜁,𝑝^𝑘𝑧)=𝑞^{2𝑘𝜁𝑏𝑎}𝑅^𝑎𝑏_𝑥𝑦(𝜁,𝑧)𝑅^𝑏𝑎_𝑥𝑦(𝜁,𝑧) (7.22) together with the relation 𝐸(𝜁_𝑗1𝑗2+1)𝛼(𝜁_𝑗2𝑗1,𝑞²)=−𝐸(𝜁_𝑗2𝑗1+1)𝛽(𝜁_𝑗2𝑗1,𝑞²) (7.23) which holds for any 𝑗₁≠𝑗₂ which is easily proved by applying 𝜃(𝑧⁻¹)=−𝑧⁻¹𝜃(𝑧) three times.
Relation (7.16) can be proved analogously, using the right expansion formula (6.14) for 𝜉^𝐽_𝐼(𝑤) instead.Since the cobraiding depends holomorphically on the spectral variables, and all zeros of 𝜃 are simple and of the form 𝑝^𝑘, we conclude that the following limits exist for all 𝑧,𝑤∈ℂ^×, 𝑖,𝑗,𝐼,𝐽, #𝐼=#𝐽=2: ⟨𝜉^𝐽_𝐼(𝑤),𝑒_𝑖𝑗(𝑧)⟩^′∶=lim_{(𝑧1,𝑤1)→(𝑧,𝑤)}⟨𝜉^𝐽_𝐼(𝑤₁),𝑒_𝑖𝑗(𝑧₁)⟩𝜃(𝑧₁/𝑤₁),⟨𝑒_𝑖𝑗(𝑧),𝜉^𝐽_𝐼(𝑤)⟩^′∶=lim_{(𝑧1,𝑤1)→(𝑧,𝑤)}⟨𝑒_𝑖𝑗(𝑧₁),𝜉^𝐽_𝐼(𝑤₁)⟩𝜃(𝑞²𝑤₁/𝑧₁). (7.24) Taking 𝑎=𝑒_𝑖𝑗(𝑧₁),𝑏=𝜉^𝐽_𝐼(𝑤₁) in (7.6), dividing both sides by 𝜃(𝑧₁/𝑤₁)𝜃(𝑞²𝑤₁/𝑧₁) and taking the limits (𝑧₁,𝑤₁)→(𝑧,𝑤), where 𝑧,𝑤∈ℂ^× are arbitrary, we get 𝜓(𝑧,𝑤)𝜀(𝑒_𝑖𝑗(𝑧)𝜉^𝐽_𝐼(𝑤))=_𝑥,𝑋⟨𝜉^𝑋_𝐼(𝑤),𝑒_𝑖𝑥(𝑧)⟩^′𝑇_{𝜔(𝑥)+𝜔(𝑥1)+𝜔(𝑥2)}⟨𝑒_𝑥𝑗(𝑧),𝜉^𝐽_𝑋(𝑤)⟩^′, (7.25) and interchanging 𝑎 and 𝑏, 𝜓(𝑧,𝑤)𝜀(𝜉^𝐽_𝐼(𝑤)𝑒_𝑖𝑗(𝑧))=_𝑥,𝑋⟨𝑒_𝑖𝑥(𝑧),𝜉^𝑋_𝐼(𝑤)⟩^′𝑇_{𝜔(𝑥)+𝜔(𝑥1)+𝜔(𝑥2)}⟨𝜉^𝐽_𝑋(𝑤),𝑒_𝑥𝑗(𝑧)⟩^′, (7.26) for all 𝑧,𝑤∈ℂ^×, where 𝜓∶(ℂ^×)²→ℂ is given by 𝑞𝜓(𝑧,𝑤)=𝜃(²𝑧𝑤𝑞)𝜃(⁴𝑤𝑧). (7.27) We are now ready to prove the key identity.Lemma 7.10. For any 𝑖,𝑗∈[1,𝑛], 𝐼,𝐽⊆[1,𝑛], #𝐼=#𝐽=2 and any 𝑧,𝑤∈ℂ^×, 𝑞²𝑤/𝑧∉𝑝^ℤ we have 𝜓(𝑧,𝑤)_𝑥,𝑋𝜇_𝑙(⟨𝜉^𝑋_𝐼(𝑤),𝑒_𝑖𝑥(𝑧)⟩^′1)𝜉^𝐽_𝑋(𝑤)𝑒_𝑥𝑗(𝑧)=𝜓(𝑧,𝑤)_𝑥,𝑋𝜇_𝑟(⟨𝜉^𝐽_𝑋(𝑤),𝑒_𝑥𝑗(𝑧)⟩^′1)𝑒_𝑖𝑥(𝑧)𝜉^𝑋_𝐼(𝑤). (7.28)Proof. Using the counit axiom followed by (7.25) we have 𝜓(𝑧,𝑤)𝑒_𝑖𝑗(𝑧)𝜉^𝐽_𝐼(𝑤)=𝜓(𝑧,𝑤)_𝑥,𝑋𝜇_𝑙(𝜀(𝑒_𝑖𝑥(𝑧)𝜉^𝑋_𝐼(𝑤))1)𝑒_𝑥𝑗(𝑧)𝜉^𝐽_𝑋(𝑤)=_{𝑥,𝑦,𝑋,𝑌}𝜇_𝑙(⟨𝜉^𝑌_𝐼(𝑤),𝑒_𝑖𝑦(𝑧)⟩^′1)𝜇_𝑙(⟨𝑒_𝑦𝑥(𝑧),𝜉^𝑋_𝑌(𝑤)⟩^′1)𝑒_𝑥𝑗(𝑧)𝜉^𝐽_𝑋(𝑤). (7.29) Applying the identity obtained by dividing by 𝜃(𝑞²𝑤/𝑧) in both sides of the cobraiding identity (7.1g) with 𝑎=𝑒_𝑦𝑗(𝑧),𝑏=𝜉^𝐽_𝑌(𝑤) in the right hand side of (7.29) gives 𝜓(𝑧,𝑤)𝑒_𝑖𝑗(𝑧)𝜉^𝐽_𝐼(𝑤)=_{𝑥,𝑦,𝑋,𝑌}𝜇_𝑙(⟨𝜉^𝑌_𝐼(𝑤),𝑒_𝑖𝑦(𝑧)⟩^′1)𝜇_𝑟(⟨𝑒_𝑥𝑗(𝑧),𝜉^𝐽_𝑋(𝑤)⟩^′1)𝜉^𝑋_𝑌(𝑤)𝑒_𝑦𝑥(𝑧). (7.30) Now multiply both sides by 𝜇_𝑟(⟨𝜉^𝐾_𝐽(𝑤),𝑒_𝑗𝑘(𝑧)⟩^′1), and sum over 𝑗,𝐽. After applying (7.26) in the right hand side we get 𝜓(𝑧,𝑤)_𝑗,𝐽𝜇_𝑟(⟨𝜉^𝐾_𝐽(𝑤),𝑒_𝑗𝑘(𝑧)⟩^′1)𝑒_𝑖𝑗(𝑧)𝜉^𝐽_𝐼(𝑤)=𝜓(𝑧,𝑤)_{𝑥,𝑦,𝑋,𝑌}𝜇_𝑙(⟨𝜉^𝑌_𝐼(𝑤),𝑒_𝑖𝑦(𝑧)⟩^′1)𝜇_𝑟(𝜀(𝜉^𝐾_𝑋(𝑤)𝑒_𝑥𝑘(𝑧))1)𝜉^𝑋_𝑌(𝑤)𝑒_𝑦𝑥(𝑧). (7.31) By the counit axiom the last expression equals 𝜓(𝑧,𝑤)_𝑦,𝑌𝜇_𝑙(⟨𝜉^𝑌_𝐼(𝑤),𝑒_𝑖𝑦(𝑧)⟩^′1)𝜉^𝐾_𝑌(𝑤)𝑒_𝑦𝑘(𝑧). (7.32)Lemma 7.11. (a) The limit ⟨det(𝑤),𝑒_𝑖𝑗(𝑧)⟩^′∶=lim_{(𝑧1,𝑤1)→(𝑧,𝑤)}⟨det(𝑤₁),e_𝑖𝑗(𝑧₁)⟩𝜃(𝑤₁/𝑧₁)𝜃(𝑞²𝑤₁/𝑧₁)⋯𝜃(𝑞^2(𝑛−2)𝑤₁/𝑧₁) (7.33) exists for any 𝑧,𝑤∈ℂ^×.
(b) We have 𝜇_𝑙(⟨det(𝑤),𝑒₁₁(𝑧)⟩^′1)det(𝑤)𝑒₁₁(𝑧)=𝜇_𝑟(⟨det(𝑤),𝑒₁₁(𝑧)⟩^′1)𝑒₁₁(𝑧)det(𝑤) (7.34) for any 𝑧,𝑤∈ℂ^×.Proof. (a) We must show that ⟨det(𝑤),𝑒_𝑖𝑗(𝑧)⟩ vanishes for 𝑞^2𝑘𝑤/𝑧∈𝑝^ℤ, where 𝑘∈{0,1,…,𝑛−2}. Applying the Laplace expansion (6.65) twice we get det(𝑤)=_{𝐼1∪𝐼2∪𝐼3=[1,𝑛]}𝑆_𝑟(𝐼₁,𝐼₂,𝐼₃;𝜆)𝜉^𝐽1_𝐼1(𝑤)𝜉^𝐽2_𝐼2(𝑞^2#𝐽1𝑤)𝜉^𝐽3_𝐼3(𝑞^2(#𝐽1+2)𝑤), (7.35) where 𝐽₁={1,…,𝑘}, 𝐽₂={𝑘+1,𝑘+2}, 𝐽₃={𝑘+3,…,𝑛}. Substituting this in the pairing and applying the multiplication-comultiplication relation (7.1d) we see that each term contains a factor of the form ⟨𝜉^𝑌_𝑋(𝑞^2𝑘𝑤),𝑒_𝑥𝑦(𝑧)⟩, where #𝑋=#𝑌=2, which indeed vanishes for 𝑞^2𝑘𝑤/𝑧∈𝑝^ℤ by Lemma 7.9.
(b) By (7.1a), ⟨det(𝑤),𝑒_𝑥𝑦(𝑧)⟩=0, if 𝑥≠𝑦. Thus (7.34) can be written _𝑥𝜇_𝑙(⟨det(𝑤),𝑒_1𝑥(𝑧)⟩^′1)det(𝑤)𝑒_𝑥1(𝑧)=_𝑥𝜇_𝑟(⟨det(𝑤),𝑒_𝑥1(𝑧)⟩^′1)𝑒_1𝑥(𝑧)det(𝑤). (7.36) If 𝑞^2𝑘𝑤/𝑧∉𝑝^ℤ for any 𝑘∈{0,1,…,𝑛−2}, this follows from the cobraiding identity (7.1g) with 𝑎=det(𝑤),𝑏=𝑒₁₁(𝑧) by dividing by the nonzero number ∏^𝑛−2_𝑘=0𝜃(𝑞^𝑘𝑤/𝑧).
So assume 𝑞^2𝑘𝑤/𝑧∈𝑝^ℤ for some 𝑘∈{0,1,…,𝑛−2}. We again use the iterated Laplace expansion (7.35). For simplicity of notation, we write it as det(𝑤)=∑𝑎₁𝑎₂𝑎₃ where 𝑎₂ is the 2×2 minor. Put 𝑏=𝑒₁₁(𝑧). Substituting this, and expanding ⟨𝑎₁𝑎₂𝑎₃,𝑏^′⟩ using (7.1d), we get after simplification _𝑥𝜇_𝑙(⟨det(𝑤),𝑒_1𝑥(𝑧)⟩^′1)det(𝑤)𝑒_𝑥1=1(𝑧)∏^𝑛−2_{𝑚=0,𝑚≠𝑘}𝜃(𝑞^2𝑚𝑤/𝑧)_{(𝑎),(𝑏)}𝜇_𝑙(⟨𝑎^′₁,𝑏^′⟩1)𝑎₁^′′𝜇_𝑙(⟨𝑎^′₂,𝑏^′′⟩^′1)𝑎₂^′′𝜇_𝑙(⟨𝑎^′₃,𝑏⁽³⁾⟩1)𝑎₃^′′𝑏⁽⁴⁾. (7.37) Now using the cobraiding identity (7.1g) and its primed version for quadratic minors (7.28), we can move the 𝑏 all the way to the left. Doing the steps backwards the claim follows.It remains to calculate ⟨det(𝑤),𝑒₁₁(𝑧)⟩^′1. Lemma 7.12. We have ⟨det(𝑤),𝑒₁₁(𝑧)⟩^′1=𝑞^𝑛−1𝑞𝜃(^2𝑛𝑤𝑧). (7.38)Proof. Expanding det(𝑤) using the left expansion formula (6.13) with 𝜎=,12⋯𝑛𝑛𝑛−1⋯1 (7.39) the longest element in 𝑆_𝑛, and applying (7.1d) repeatedly we have (putting 𝐼=[1,𝑛]) ⟨det(𝑤),𝑒₁₁(𝑧)⟩=_{𝜏∈𝑆𝑛}⟨𝜏(𝐹_𝐼(𝜌))𝜎(𝐹_𝐼𝑒(𝜆))^1𝜏(𝑛)(𝑞^2(𝑛−1)𝑤)⋯𝑒_𝑛𝜏(1)(𝑤),𝑒₁₁(𝑧)⟩=_{𝜏∈𝑆𝑛𝑥1,…,𝑥𝑛−1}𝜏(𝐹_𝐼(𝜁))⟨𝑒_1𝜏(𝑛)(𝑞^2(𝑛−1)𝑤),𝑒_1𝑥1(𝑧)⟩𝑇_𝜔(𝑥1)⋯𝑇_{𝜔(𝑥𝑛−1)}⟨𝑒_𝑛𝜏(1)(𝑤),𝑒_{𝑥𝑛−11}(𝑧)⟩𝜎(𝐹_𝐼(𝜁)⁻¹). (7.40)
One proves inductively that in all nonzero terms we have 𝜏(𝑗)=𝑛+1−𝑗 and 𝑥_𝑛−𝑗=1 for all 1≤𝑗≤𝑛−1 by looking from right to left: ⟨𝑒_𝑛𝜏(1)(𝑤),𝑒_{𝑥𝑛−11}(𝑧)⟩=𝑅^𝜏(1)1_{𝑛𝑥𝑛−1}(𝜁,𝑤/𝑧)𝑇_{−𝜔(𝑛)−𝜔(𝑥𝑛−1)} which, if 1≠𝑛, is nonzero only for 𝜏(1)=𝑛 and 𝑥_𝑛−1=1 by (4.3). Then looking at the second pairing from the right we see that 𝜏(2)=𝑛−1 and 𝑥_𝑛−2=1 if it is nonzero, and so on. Thus only the term 𝜏=𝜎 and 𝑥₁=⋯=𝑥_𝑛−1=1 survives and it equals 𝜎(𝐹_𝐼(𝜁))𝑅¹¹₁₁𝑞(𝜁,^2(𝑛−1)𝑤𝑧)𝑇_−𝜔(1)𝑅²¹₂₁𝑞(𝜁,^2(𝑛−2)𝑤𝑧)𝑇_−𝜔(2)⋯𝑇_{−𝜔(𝑛−1)}𝑅^𝑛1_𝑛1𝑤(𝜁,𝑧)𝑇_{−𝜔(𝑛)−𝜔(1)}𝜎(𝐹_𝐼(𝜁))⁻¹. (7.41) Using that 𝜎(𝐹_𝐼(𝜁))=∏_𝑖<𝑗𝐸(𝜁_𝑗𝑖+1) and that 𝑅^𝑗1_𝑗1(𝜁−𝜔(1),𝑧)=𝛼(𝜁_𝑗1+1,𝑧)=𝑞𝜃(𝑧)(𝐸(𝜁_𝑗1+2)/𝐸(𝜁_𝑗1+1)) we get 𝑞^𝑛−1𝑞𝜃(^2𝑛𝑤𝑧𝑞)⋅𝜃(^2(𝑛−2)𝑤𝑧𝑞)𝜃(^2(𝑛−3)𝑤𝑧𝑤)⋯𝜃(𝑧)⋅_𝑖<𝑗𝐸(𝜁_𝑗𝑖+1)_1<𝑗𝐸(𝜁_𝑗1+2)𝐸(𝜁_𝑗1𝑇+1)^−𝜔(1)_𝑖<𝑗𝐸(𝜁_𝑗𝑖+1)⁻¹. (7.42) The factors involving the dynamical variable 𝜁 cancel and the claim follows.
By Lemmas 7.11(b) and 7.12 we conclude that det(𝑤) commutes with 𝑒₁₁(𝑧) if 𝑞^2𝑛𝑤/𝑧∉𝑝^ℤ. By applying an automorphism from the 𝑆_𝑛×𝑆_𝑛-action on 𝐴 as defined in Section 5.3 and using that det(𝑧) is fixed by those, by relation (6.15), we conclude that det(𝑤) commutes with any 𝑒_𝑖𝑗(𝑧) as long as 𝑞^2𝑛𝑤/𝑧∉𝑝^ℤ.
For the remaining case we can note that relations (4.4a)–(4.4d) and (4.5) imply that there is a ℂ-linear map 𝑇∶ℱ_ell(𝑀(𝑛))→ℱ_ell(𝑀(𝑛)) such that 𝑇(𝑎𝑏)=𝑇(𝑏)𝑇(𝑎) for all 𝑎,𝑏∈ℱ_ell(𝑀(𝑛)), given by 𝑇(𝑒_𝑖𝑗(𝑧))=𝑒_i𝑗(𝑧⁻¹),𝑇(𝑓(𝜆))=𝑓(−𝜆),𝑇(𝑓(𝜌))=𝑓(−𝜌), (7.43) for all 𝑓∈𝑀_𝔥^∗, 𝑖,𝑗∈[1,𝑛], and 𝑧∈ℂ^×. One verifies that 𝑇(det(𝑧))=det(𝑞^{−2(𝑛−1)}𝑧⁻¹).
We have proved that [det(𝑤),𝑒_𝑖𝑗(𝑧)]=0 if 𝑞^2𝑛𝑤/𝑧∉𝑝^ℤ. Assume 𝑞^2𝑛𝑤/𝑧∈𝑝^ℤ. Then 𝑇([det(𝑤),𝑒_i𝑗(𝑧)])=[𝑒_𝑖𝑗(𝑧⁻¹),det(𝑞^{−2(𝑛−1)}𝑤⁻¹)]=0 (7.44) since 𝑞^{−2(𝑛−1)}𝑤⁻¹/𝑧⁻¹=𝑞²(𝑞^2𝑛𝑤/𝑧)⁻¹∉𝑝^ℤ. This finishes the proof of Theorem(b).

7.4. The Antipode

We use the following definition for the antipode, given in [13].

Definition 7.13. An 𝔥-Hopf algebroid is an 𝔥-bialgebroid 𝐴 equipped with a ℂ-linear map 𝑆∶𝐴→𝐴, called the antipode, such that 𝑆(𝜇_𝑟(𝑓)𝑎)=𝑆(𝑎)𝜇_𝑙(𝑓),𝑆(𝑎𝜇_𝑙(𝑓))=𝜇_𝑟(𝑓)𝑆(𝑎),𝑎∈𝐴,𝑓∈𝑀_𝔥^∗,𝑚∘(Id⊗𝑆)∘Δ(𝑎)=𝜇_𝑙(𝜀(𝑎)1),𝑎∈𝐴,𝑚∘(𝑆⊗Id)∘Δ(𝑎)=𝜇_𝑟(𝑇_𝛼(𝜀(𝑎)1)),𝑎∈𝐴_𝛼𝛽, (7.45)(7.46) where 𝑚 denotes the multiplication, and 𝜀(𝑎)1 is the result of applying the difference operator 𝜀(𝑎) to the constant function 1∈𝑀_𝔥^∗.

Let ℱ_ell(𝑀(𝑛))[det(𝑧)⁻¹∶𝑧∈ℂ^×] be the polynomial algebra in uncountably many variables det(𝑧)⁻¹, 𝑧∈ℂ^×, with coefficients in ℱ_ell(𝑀(𝑛)). We define ℱ_ell(𝐺𝐿(𝑛)) to be ℱ_ell(𝑀(𝑛))[det(𝑧)⁻¹∶𝑧∈ℂ^×]𝐽, (7.47) where 𝐽 is the ideal generated by the relations det(𝑧)det(𝑧)⁻¹=1=det(𝑧)⁻¹det(𝑧) for each 𝑧∈ℂ^×. We extend the bigrading of ℱ_ell(𝑀(𝑛)) to ℱ_ell(𝑀(𝑛))[det(𝑧)⁻¹∶𝑧∈ℂ^×] by requiring that det(𝑧)⁻¹ has bidegree 0,0 for each 𝑧∈ℂ^×. Then 𝐽 is homogenous and the bigrading descends to ℱ_ell(𝐺𝐿(𝑛)). We extend the comultiplication and counit by requiring that det(𝑧)⁻¹ is grouplike for each 𝑧∈ℂ^×, that is, Δ(det(𝑧)⁻¹)=det(𝑧)⁻¹⊗det(𝑧)⁻¹,𝜀(det(𝑧)⁻¹)=1. (7.48) Here 1 denotes the identity operator in 𝐷_𝔥. One verifies that 𝐽 is a coideal and that 𝜀(𝐽)=0, which induces operations Δ,𝜀 on ℱ_ell(𝐺𝐿(𝑛)). In this way ℱ_ell(𝐺𝐿(𝑛)) becomes an 𝔥-bialgebroid. This algebra is nontrivial since 𝜀(𝐽)=0 implies that 𝐽 is a proper ideal.

For 𝑖∈[1,𝑛] we set ={1,…,𝑖−1,𝑖+1,…,𝑛}. For a meromorphic function 𝑓 on 𝔥^∗, we denote the images of 𝑓 under the left and right moment maps in ℱ_ell(𝐺𝐿(𝑛)) also by 𝑓(𝜆) and 𝑓(𝜌) respectively.

Theorem 7.14. ℱ_ell(𝐺𝐿(𝑛)) is an 𝔥-Hopf algebroid with antipode 𝑆 given by 𝑆(𝑓(𝜆))=𝑓(𝜌),𝑆(𝑓(𝜌))=𝑓(𝜆),𝑆(𝑒_𝑖𝑗𝑆(𝑧))=^𝑟(𝚥,̂{𝑗};𝜆)𝑆_𝑟(,{𝑖};𝜌)det(𝑞^{−2(𝑛−1)}𝑧)⁻¹𝜉_̂𝚥(𝑞^{−2(𝑛−1)}𝑧),𝑆(det(𝑧)⁻¹)=det(𝑧), (7.49)(7.50)(7.51) for all 𝑓∈𝑀_𝔥^∗, 𝑖,𝑗∈[1,𝑛] and 𝑧∈ℂ^×.

Proof. We proceed in steps.
Step 1. Define 𝑆 on the generators of ℱ_ell(𝑀(𝑛)) by (7.49), (7.50). We show that the antipode axiom (7.46) holds if 𝑎 is a generator. Indeed for 𝑎=𝑓(𝜆) or 𝑎=𝑓(𝜌), 𝑓∈𝑀_𝔥^∗, this is easily checked. Let 𝑎=𝑒_𝑖𝑗(𝑧). Using the right Laplace expansion (6.65) with 𝐽₁=, 𝐽₂={𝑗}, 𝐼=[1,𝑛] and 𝑧 replaced by 𝑞^{−2(𝑛−1)}𝑧 we obtain ^𝑛_𝑥=1𝑆(𝑒_𝑖𝑥(𝑧))𝑒_𝑥𝑗(𝑧)=𝛿_𝑖𝑗. (7.52) Similarly, using the left Laplace expansion (6.64) with 𝐼₁={𝑖}, 𝐼₂=𝚥̂, 𝐽=[1,𝑛], and 𝑧 replaced by 𝑞^{−2(𝑛−1)}𝑧, together with the identity (6.68), we get ^𝑛_𝑥=1𝑒_𝑖𝑥(𝑧)𝑆(𝑒_𝑥𝑗(𝑧))=𝛿_𝑖𝑗, (7.53) using also the crucial fact that, by Theorem 7.8, 𝑒_𝑖𝑗(𝑧) commutes in ℱ_ell(𝑀(𝑛)) with det(𝑞^{−2(𝑛−1)}𝑧) and hence in ℱ_ell(𝐺𝐿(𝑛)) with det(𝑞^{−2(𝑛−1)}𝑧)⁻¹. This proves that the antipode axiom (7.46) is satisfied for 𝑎=𝑒_𝑖𝑗(𝑧).
Step 2. We show that 𝑆 extends to a ℂ-linear map 𝑆∶ℱ_ell(𝑀(𝑛))→ℱ_ell(𝐺𝐿(𝑛)) satisfying 𝑆(𝑎𝑏)=𝑆(𝑏)𝑆(𝑎). For this we must verify that 𝑆 preserves the relations, (4.1), (4.2), (4.5) of ℱ_ell(𝑀(𝑛)). Since 𝑆(𝑒_𝑖𝑗(𝑧))∈ℱ_ell(𝐺𝐿(𝑛))_{̂𝜔(𝚥),𝜔()} and 𝜔(𝑖)+𝜔()=0, we have 𝑆(𝑒_𝑖𝑗(𝑧))𝑆(𝑓(𝜆))=𝑆(𝑒_𝑖𝑗(𝑧))𝑓(𝜌)=𝑓(𝜌−𝜔())𝑆(𝑒_𝑖𝑗(𝑧))=𝑓(𝜌+𝜔(𝑖))𝑆(𝑒_𝑖𝑗(𝑧))=𝑆(𝑓(𝜆+𝜔(𝑖)))𝑆(𝑒_𝑖𝑗(𝑧)). (7.54) Similarly, 𝑆(𝑒_𝑖𝑗(𝑧))𝑆(𝑓(𝜌))=𝑆(𝑓(𝜌+𝜔(𝑗)))𝑆(𝑒_𝑖𝑗(𝑧)) so relations (4.1) are preserved. Next, consider the RLL relation ^𝑛_𝑥,𝑦=1𝑅^𝑥𝑦_𝑎𝑐𝑧(𝜆,¹𝑧₂)𝑒_𝑥𝑏(𝑧₁)𝑒_𝑦𝑑(𝑧₂)=^𝑛_𝑥,𝑦=1𝑅^𝑏𝑑_𝑥𝑦𝑧(𝜌,¹𝑧₂)𝑒_𝑐𝑦(𝑧₂)𝑒_𝑎𝑥(𝑧₁). (7.55) Multiply (7.55) from the left by 𝑆(𝑒_𝑖𝑐(𝑧₂)) and from the right by 𝑆(𝑒_𝑑𝑘(𝑧₂)), sum over 𝑐,𝑑, and use (7.52), (7.53) to obtain _𝑥,𝑐𝑅^𝑥𝑘_𝑎𝑐̂𝑧(𝜆−𝜔(𝑐),¹𝑧₂)𝑆(𝑒_𝑖𝑐(𝑧₂))𝑒_𝑥𝑏(𝑧₁)=_𝑥,𝑑𝑅^𝑏𝑑_𝑥𝑖𝑧(𝜌−𝜔(),¹𝑧₂)𝑒_𝑎𝑥(𝑧₁)𝑆(𝑒_𝑑𝑘(𝑧₂)). (7.56) Then multiply from the left by 𝑆(𝑒_𝑗𝑎(𝑧₁)) and from the right by 𝑆(𝑒_𝑏𝑙(𝑧₁)), sum over 𝑎,𝑏, and use (7.52), (7.53) again to get _𝑎,𝑐𝑅^𝑙𝑘_𝑎𝑐̂𝑧(𝜆−𝜔(𝑎)−𝜔(𝑐),¹𝑧₂)𝑆(𝑒_𝑗𝑎(𝑧₁))𝑆(𝑒_𝑖𝑐(𝑧₂))=_𝑏,𝑑𝑅^𝑏𝑑_j𝑖̂𝑧(𝜌−𝜔(𝚥)−𝜔(),¹𝑧₂)𝑆(𝑒_𝑑𝑘(𝑧₂))𝑆(𝑒_𝑏𝑙(𝑧₁)). (7.57) Since 𝑆(𝑒_𝑖𝑗(𝑧))∈ℱ_ell(𝐺𝐿(𝑛))_̂𝚥, and 𝑅^𝑏𝑑_𝑗𝑖(𝜌−𝜔(𝚥)̂−𝜔(),𝑧₁/𝑧₂)=𝑅^𝑏𝑑_𝑗𝑖(𝜌−𝜔(𝑏̂)−𝜔(𝑑),𝑧₁/𝑧₂) by the 𝔥-invariance of 𝑅, (7.57) can be rewritten _𝑎,𝑐𝑆(𝑒_𝑗𝑎(𝑧₁))𝑆(𝑒_𝑖𝑐(𝑧₂))𝑅^𝑙𝑘_𝑎𝑐𝑧(𝜆,¹𝑧₂)=_𝑏,𝑑𝑆(𝑒_𝑑𝑘(𝑧₂))𝑆(𝑒_𝑏𝑙(𝑧₁))𝑅^𝑏𝑑_𝑗𝑖𝑧(𝜌,¹𝑧₂). (7.58) This is the result of formally applying 𝑆 to the RLL relations, proving that 𝑆 preserves (4.2). Similarly (4.5) is preserved.
Step 3. Since, by the above steps, (7.46) holds on the generators of ℱ_ell(𝑀(𝑛)) and 𝑆(𝑎𝑏)=𝑆(𝑏)𝑆(𝑎) for all 𝑎,𝑏∈ℱ_ell(𝑀(𝑛)), it follows that (7.46) holds for any 𝑎∈ℱ_ell(𝑀(𝑛)). By taking in particular 𝑎=det(𝑧) we get det(𝑧)𝑆(det(𝑧))=1,𝑆(det(𝑧))det(𝑧)=1, (7.59) respectively. Thus, definining 𝑆 on det(𝑧)⁻¹ by (7.51), the relations det(𝑧)det(𝑧)⁻¹=1=det(𝑧)⁻¹det(𝑧) are preserved by 𝑆. Hence 𝑆 extends to an antimultiplicative ℂ-linear map 𝑆∶ℱ_ell(𝐺𝐿(𝑛))→ℱ_ell(𝐺𝐿(𝑛)) satisfying the antipode axiom (7.46) on ℱ_ell(𝑀(𝑛)) and on det(𝑧)⁻¹. Hence (7.46) holds for any 𝑎∈ℱ_ell(𝐺𝐿(𝑛)).

8. Concluding Remarks

To define the antipode we only needed that 𝑒_𝑖𝑗(𝑧) commutes with det(𝑞^{−2(𝑛−1)}𝑧). This can also be proved using the Laplace expansions.

Perhaps one could avoid problems with spectral poles and zeros of the R-matrix by thinking of the algebra as generated by meromorphic sections of a 𝑀_𝔥^∗_⊕𝔥^∗-line bundle over the elliptic curve ℂ^×/{𝑧∼𝑝𝑧}. In this direction we found that the relation 𝑒_𝑖𝑗(𝑝𝑧)=𝑞^{𝜆𝑖−𝜌𝑗}𝑒_𝑖𝑗(𝑧) respects the RLL relation (here 𝔥 should be the Cartan subalgebra of 𝔤𝔩_𝑛). This relation should then most likely be added to the algebra.

Acknowledgments

The author was supported by the Netherlands Organization for Scientific Research (NWO) in the VIDI-project “Symmetry and modularity in exactly solvable models”. The author is greatly indebted to H. Rosengren for many inspiring and helpful discussions, and to J. Stokman for his support and helpful comments. The author is also grateful to an anonymous referee of an earlier version of this paper for detailed comments which have led to improvement of the section on centrality of the determinant.