# The EllipticGL(n) Dynamical Quantum Group as an 𝔥-Hopf Algebroid

Authors: Jonas T. Hartwig
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

Using the language of𝔥-Hopf algebroids which was
introduced by Etingof and Varchenko,
we construct a dynamical quantum group, ℱell(GL(n)),
from the elliptic solution of the quantum dynamical Yang-Baxter
equation with spectral parameter associated to the
Lie algebra 𝔰𝔩n. We apply the generalized FRST construction
and obtain an 𝔥-bialgebroid ℱell(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 ℱell(GL(n)).


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## Body

## 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 An(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 𝔤𝔩n. 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 an𝔥-Hopf algebroid associated to the elliptic R-matrix from [2] with both dynamical and spectral parameters for 𝔤=𝔰𝔩n. 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 ellipticGL(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 Section2.1, 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 ℱell(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 overℂn are defined in Section 5 in a similar way as in [12]. They are certain comodule algebras over ℱell(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 Section7 we show that the 𝔥-bialgebroid ℱell(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 Section7.4 we define ℱell(GL(n)) to be the localization of ℱell(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

Letp,q∈ℝ, 0<p,q<1. We assume p,q are generic in the sense that if paqb=1 for some a,b∈ℤ, then a=b=0.Denote byθ the normalized Jacobi theta function:
(2.1)θ(z)=θ(z;p)=∏j=0∞(1-zpj)(1-pj+1z).
It is holomorphic on ℂ×:=ℂ∖{0} with zero set {pk:k∈ℤ} and satisfies
(2.2)θ(z-1)=θ(pz)=-z-1θ(z)
and the addition formula
(2.3)θ(xy,xy,zw,zw)=θ(xw,xw,zy,zy)+(zy)θ(xz,xz,yw,yw),
where we use the notation
(2.4)θ(z1,…,zn)=θ(z1)⋯θ(zn).
Recall also the Jacobi triple product identity, which can be written
(2.5)∑k∈ℤ(-z)kpk(k-1)/2=θ(z)∏j=1∞(1-pj).
It will sometimes be convenient to use the auxiliary function E given by
(2.6)E:ℂ→ℂ,E(s)=qsθ(q-2s).
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

Let𝔥 be a complex vector space, viewed as an abelian Lie algebra, 𝔥* its dual space, and let V=⊕λ∈𝔥*Vλ a diagonalizable 𝔥-module. A dynamical R-matrix is by definition a meromorphic function
(2.7)R:𝔥*×ℂ×→End𝔥(V⊗V)
satisfying the quantum dynamical Yang-Baxter equation with spectral parameter (QDYBE):
(2.8)R(λ,z2z3)(23)R(λ-h2,z1z3)(13)R(λ,z1z2)(12)=R(λ-h3,z1z2)(12)R(λ,z1z3)(13)R(λ-h1,z2z3)(23).
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,
(2.9)R(λ-h3,z1z2)(12)(u⊗v⊗w)=R(λ-α,z1z2)(u⊗v)⊗w,ifw∈Vα.
An R-matrix R is called unitary if
(2.10)R(λ,z)R(λ,z-1)(21)=IdV⊗V
as meromorphic functions on 𝔥*×ℂ× with values in End𝔥(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 ℂn with standard basis e1,…,en. Define ω(i)∈𝔥* (i=1,…,n) by
(2.11)ω(i)(h)=hi,ifh=diag(h1,…,hn)∈𝔥.
We have V=⊕i=1nVω(i) and Vω(i)=ℂei. Define
(2.12)R:𝔥*×ℂ×→End(V⊗V)
by
(2.13)R(λ,z)=∑i=1nEii⊗Eii+∑i≠jα(λij,z)Eii⊗Ejj+∑i≠jβ(λij,z)Eij⊗Eji,
where Eij∈End(V) are the matrix units, λij (λ∈𝔥*) is an abbrevation for λ(Eii-Ejj), and
(2.14)α,β:ℂ×ℂ×→ℂ
are given by
(2.15)α(λ,z)=α(λ,z;p,q)=θ(z)θ(q2(λ+1))θ(q2z)θ(q2λ),(2.16)β(λ,z)=β(λ,z;p,q)=θ(q2)θ(q-2λz)θ(q2z)θ(q-2λ).Proposition 2.1  (see [2]).
The mapR 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:𝔥1*×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,
(2.17)α1(λ,x)=α1(λ,x;τ,γ)=θ1(x;τ)θ1(λ+γ;τ)θ1(x-γ;τ)θ1(λ;τ),β1(λ,x)=β1(λ,x;τ,γ)=-θ1(x+λ;τ)θ1(γ;τ)θ1(x-γ;τ)θ1(λ;τ).
Here τ,γ∈ℂ with Imτ>0, and θ1 is the first Jacobi theta function:
(2.18)θ1(x;τ)=-∑j∈ℤ+1/2eπij2τ+2πij(x+1/2).
As proved in [2], R1 satisfies the following version of the QDYBE:
(2.19)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)
and the unitarity condition
(2.20)R1(λ,x)R121(λ,-x)=IdV⊗V.
We can identify 𝔥*≃𝔥1*/ℂtr where tr∈𝔥1* is the trace. Since R1 has the form (2.13), it is constant, as a function of λ∈𝔥1*, 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).Letτ,γ∈ℂ with Imτ>0 be such that p=eπiτ, q=eπiγ. Then, as meromorphic functions of (λ,x)∈𝔥*×ℂ,
(2.21)R1(γλ,-x;τ2,γ)=R(λ,z;p,q),
where z=e2πix. Indeed, using the Jacobi triple product identity (2.5) we have
(2.22)θ1(x;τ2)=ieπi(τ/2-x)θ(z)∏j=1∞(1-pj),
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 replacingλ, xi in (2.19) by γλ, -xi and using (2.21) we obtain (2.8) with zi=e2πixi. 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 ofα,β in (2.15). It is immediate that
(2.23)α(λ,q2)=β(-λ,q2).
By induction, one generalizes (2.2) to
(2.24)θ(psz)=(-1)s(ps(s-1)/2zs)-1θ(z),fors∈ℤ.
Applying (2.24) to the definitions of α, β we get
(2.25)α(λ,pkz)=q2kα(λ,z),β(λ,pkz)=q2k(λ+1)β(λ,z),
and, using also θ(z-1)=-z-1θ(z),
(2.26)limz→p-kq-2q-1θ(q2z)qθ(q-2z)α(λ,z)=α(λ,pkq2),limz→p-kq-2q-1θ(q2z)qθ(q-2z)β(λ,z)=-β(-λ,pkq2),
for λ∈ℂ, z∈ℂ×, and k∈ℤ. By the addition formula (2.3) with
(2.27)(x,y,z,w)=(z1/2q-λ+1,z1/2qλ-1,z1/2qλ+1,z1/2q-λ-1),
we have
(2.28)α(λ,z)α(-λ,z)-β(λ,z)β(-λ,z)=q2θ(q-2z)θ(q2z).


## 2.1. Notation

Letp,q∈ℝ, 0<p,q<1. We assume p,q are generic in the sense that if paqb=1 for some a,b∈ℤ, then a=b=0.Denote byθ the normalized Jacobi theta function:
(2.1)θ(z)=θ(z;p)=∏j=0∞(1-zpj)(1-pj+1z).
It is holomorphic on ℂ×:=ℂ∖{0} with zero set {pk:k∈ℤ} and satisfies
(2.2)θ(z-1)=θ(pz)=-z-1θ(z)
and the addition formula
(2.3)θ(xy,xy,zw,zw)=θ(xw,xw,zy,zy)+(zy)θ(xz,xz,yw,yw),
where we use the notation
(2.4)θ(z1,…,zn)=θ(z1)⋯θ(zn).
Recall also the Jacobi triple product identity, which can be written
(2.5)∑k∈ℤ(-z)kpk(k-1)/2=θ(z)∏j=1∞(1-pj).
It will sometimes be convenient to use the auxiliary function E given by
(2.6)E:ℂ→ℂ,E(s)=qsθ(q-2s).
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

Let𝔥 be a complex vector space, viewed as an abelian Lie algebra, 𝔥* its dual space, and let V=⊕λ∈𝔥*Vλ a diagonalizable 𝔥-module. A dynamical R-matrix is by definition a meromorphic function
(2.7)R:𝔥*×ℂ×→End𝔥(V⊗V)
satisfying the quantum dynamical Yang-Baxter equation with spectral parameter (QDYBE):
(2.8)R(λ,z2z3)(23)R(λ-h2,z1z3)(13)R(λ,z1z2)(12)=R(λ-h3,z1z2)(12)R(λ,z1z3)(13)R(λ-h1,z2z3)(23).
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,
(2.9)R(λ-h3,z1z2)(12)(u⊗v⊗w)=R(λ-α,z1z2)(u⊗v)⊗w,ifw∈Vα.
An R-matrix R is called unitary if
(2.10)R(λ,z)R(λ,z-1)(21)=IdV⊗V
as meromorphic functions on 𝔥*×ℂ× with values in End𝔥(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 ℂn with standard basis e1,…,en. Define ω(i)∈𝔥* (i=1,…,n) by
(2.11)ω(i)(h)=hi,ifh=diag(h1,…,hn)∈𝔥.
We have V=⊕i=1nVω(i) and Vω(i)=ℂei. Define
(2.12)R:𝔥*×ℂ×→End(V⊗V)
by
(2.13)R(λ,z)=∑i=1nEii⊗Eii+∑i≠jα(λij,z)Eii⊗Ejj+∑i≠jβ(λij,z)Eij⊗Eji,
where Eij∈End(V) are the matrix units, λij (λ∈𝔥*) is an abbrevation for λ(Eii-Ejj), and
(2.14)α,β:ℂ×ℂ×→ℂ
are given by
(2.15)α(λ,z)=α(λ,z;p,q)=θ(z)θ(q2(λ+1))θ(q2z)θ(q2λ),(2.16)β(λ,z)=β(λ,z;p,q)=θ(q2)θ(q-2λz)θ(q2z)θ(q-2λ).Proposition 2.1  (see [2]).
The mapR 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:𝔥1*×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,
(2.17)α1(λ,x)=α1(λ,x;τ,γ)=θ1(x;τ)θ1(λ+γ;τ)θ1(x-γ;τ)θ1(λ;τ),β1(λ,x)=β1(λ,x;τ,γ)=-θ1(x+λ;τ)θ1(γ;τ)θ1(x-γ;τ)θ1(λ;τ).
Here τ,γ∈ℂ with Imτ>0, and θ1 is the first Jacobi theta function:
(2.18)θ1(x;τ)=-∑j∈ℤ+1/2eπij2τ+2πij(x+1/2).
As proved in [2], R1 satisfies the following version of the QDYBE:
(2.19)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)
and the unitarity condition
(2.20)R1(λ,x)R121(λ,-x)=IdV⊗V.
We can identify 𝔥*≃𝔥1*/ℂtr where tr∈𝔥1* is the trace. Since R1 has the form (2.13), it is constant, as a function of λ∈𝔥1*, 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).Letτ,γ∈ℂ with Imτ>0 be such that p=eπiτ, q=eπiγ. Then, as meromorphic functions of (λ,x)∈𝔥*×ℂ,
(2.21)R1(γλ,-x;τ2,γ)=R(λ,z;p,q),
where z=e2πix. Indeed, using the Jacobi triple product identity (2.5) we have
(2.22)θ1(x;τ2)=ieπi(τ/2-x)θ(z)∏j=1∞(1-pj),
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 replacingλ, xi in (2.19) by γλ, -xi and using (2.21) we obtain (2.8) with zi=e2πixi. 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 ofα,β in (2.15). It is immediate that
(2.23)α(λ,q2)=β(-λ,q2).
By induction, one generalizes (2.2) to
(2.24)θ(psz)=(-1)s(ps(s-1)/2zs)-1θ(z),fors∈ℤ.
Applying (2.24) to the definitions of α, β we get
(2.25)α(λ,pkz)=q2kα(λ,z),β(λ,pkz)=q2k(λ+1)β(λ,z),
and, using also θ(z-1)=-z-1θ(z),
(2.26)limz→p-kq-2q-1θ(q2z)qθ(q-2z)α(λ,z)=α(λ,pkq2),limz→p-kq-2q-1θ(q2z)qθ(q-2z)β(λ,z)=-β(-λ,pkq2),
for λ∈ℂ, z∈ℂ×, and k∈ℤ. By the addition formula (2.3) with
(2.27)(x,y,z,w)=(z1/2q-λ+1,z1/2qλ-1,z1/2qλ+1,z1/2q-λ-1),
we have
(2.28)α(λ,z)α(-λ,z)-β(λ,z)β(-λ,z)=q2θ(q-2z)θ(q2z).


## 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 M𝔥* be the field of meromorphic functions on 𝔥*.Definition 3.1.
An𝔥-algebra is a complex associative algebra A with 1 which is bigraded over 𝔥*, A=⊕α,β∈𝔥*Aαβ, and equipped with two algebra embeddings μl,μr:M𝔥*→A, called the left and right moment maps, such that
(3.1)μl(f)a=aμl(Tαf),μr(f)a=aμr(Tβf),fora∈Aαβ,f∈M𝔥*,
where Tα denotes the automorphism (Tαf)(ζ)=f(ζ+α) of M𝔥*. A morphism of 𝔥-algebras is an algebra homomorphism preserving the bigrading and the moment maps.Thematrix tensor productA⊗̃B of two 𝔥-algebras A, B is the 𝔥*-bigraded vector space with (A⊗̃B)αβ=⊕γ∈𝔥*(Aαγ⊗M𝔥*Bγβ), where ⊗M𝔥* denotes tensor product over ℂ modulo the relations:
(3.2)μrA(f)a⊗b=a⊗μlB(f)b,∀a∈A,b∈B,f∈M𝔥*.
The multiplication (a⊗b)(c⊗d)=ac⊗bd for a,c∈A and b,d∈B and the moment maps μl(f)=μlA(f)⊗1 and μr(f)=1⊗μrB(f) make A⊗̃B into an 𝔥-algebra.Example 3.2.
LetD𝔥 be the algebra of operators on M𝔥* of the form ∑ifiTαi with fi∈M𝔥* and αi∈𝔥*. It is an 𝔥-algebra with bigrading fT-α∈(D𝔥)αα, and both moment maps equal to the natural embedding.For any𝔥-algebra A, there are canonical isomorphisms A≃A⊗̃D𝔥≃D𝔥⊗̃A defined by
(3.3)x≃x⊗T-β≃T-α⊗x,forx∈Aαβ.Definition 3.3.
An𝔥-bialgebroid is an 𝔥-algebra A equipped with two 𝔥-algebra morphisms, the comultiplication Δ:A→A⊗̃A and the counit ε:A→D𝔥 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 that𝔥-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 an𝔥-bialgebroid from a quantum dynamical R-matrix with spectral parameter.Let𝔥 be a finite-dimensional abelian Lie algebra, V=⊕α∈𝔥*Vα a finite-dimensional diagonalizable 𝔥-module, and R:𝔥*×ℂ×→End𝔥(V⊗V) a meromorphic function. We attach to this data an 𝔥-bialgebroid AR as follows. Let {ex}x∈X be a homogeneous basis of V, where X is an index set. The matrix elements Rxyab:𝔥*×ℂ×→ℂ of R are given by
(3.4)R(ζ,z)(ea⊗eb)=∑x,y∈XRxyab(ζ,z)ex⊗ey.
They are meromorphic on 𝔥*×ℂ×. Define ω:X→𝔥* by ex∈Vω(x). Let ÃR be the complex associative algebra with 1 generated by {Lxy(z):x,y∈X,z∈ℂ×} and two copies of M𝔥*, 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∈M𝔥* and
(3.5)f(λ)Lxy(z)=Lxy(z)f(λ+ω(x)),f(ρ)Lxy(z)=Lxy(z)f(ρ+ω(y)),
for all x,y∈X, z∈ℂ× and f∈M𝔥*. The bigrading on ÃR is given by Lxy(z)∈(ÃR)ω(x),ω(y) for x,y∈X, z∈ℂ× and f(λ),f(ρ)∈(ÃR)00 for f∈M𝔥*. The moment maps are defined by μl(f)=f(λ), μr(f)=f(ρ). The counit and comultiplication are defined by
(3.6)ε(Lab(z))=δabT-ω(a),ε(f(λ))=ε(f(ρ))=fT0,Δ(Lab(z))=∑x∈XLax(z)⊗Lxb(z),Δ(f(λ))=f(λ)⊗1,Δ(f(ρ))=1⊗f(ρ).
This makes ÃR into an 𝔥-bialgebroid.Consider the ideal inÃR generated by the RLL relations:
(3.7)∑x,y∈XRacxy(λ,z1z2)Lxb(z1)Lyd(z2)=∑x,y∈XRxybd(ρ,z1z2)Lcy(z2)Lax(z1),
where a,b,c,d∈X, and z1,z2∈ℂ×. More precisely, to account for possible singularities of R, we let IR be the ideal in ÃR generated by all relations of the form
(3.8)∑x,y∈Xlimw→z1/z2(φ(w)Racxy(λ,w))Lxb(z1)Lyd(z2)=∑x,y∈Xlimw→z1/z2(φ(w)Rxybd(ρ,w))Lcy(z2)Lax(z1),
where a,b,c,d∈X, z1,z2∈ℂ×, and φ:ℂ×→ℂ is a meromorphic function such that the limits exist.We defineAR to be ÃR/IR. The bigrading descends to AR 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(IR)⊆ÃR⊗̃IR+IR⊗̃ÃR and ε(IR)=0. Thus AR is an 𝔥-bialgebroid with the induced maps.Remark 3.4.
Objects in Felder's tensor category associated to an R-matrixR are certain meromorphic functions L:𝔥*×ℂ×→End𝔥(ℂn⊗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 AR 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.AssumeRxyab(ζ,z) are defined, as meromorphic functions of ζ∈𝔥* for any z∈ℂ×. Define R(λ,z),R(ρ,z)∈End(V⊗V⊗AR) by
(3.9)R(λ,z)(ea⊗eb⊗u)=∑x,y∈Xex⊗ey⊗Rxyab(λ,z)u,R(ρ,z)(ea⊗eb⊗u)=∑x,y∈Xex⊗ey⊗Rxyab(ρ,z)u,
for a,b∈X, u∈AR. 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⊗AR) by
(3.10)L(z)=∑x,y∈XExy⊗Lxy(z).
Here Exy are the matrix units in End(V), and AR acts on itself by left multiplication. The RLL relation (3.7) is equivalent to
(3.11)R(λ,z1z2)L1(z1)L2(z2)=L2(z2)L1(z1)R(ρ+h1+h2,z1z2)
in End(V⊗V⊗AR), where Li(z)=L(z)(i,3)∈End(V⊗V⊗AR) for i=1,2 and the operator R(ρ+h1+h2,z1/z2)∈End(V⊗V⊗AR) is given by
(3.12)ea⊗eb⊗u↦∑x,y∈X‍ex⊗ey⊗Rxyab(ρ+ω(a)+ω(b),z1z2)u,
where Rxyab(ρ+ω(a)+ω(b),z1/z2) means the image in AR of the meromorphic function 𝔥*∋ζ↦Rxyab(ζ+ω(a)+ω(b),z1/z2) under the right moment map μr. This equivalence can be seen by acting on eb⊗ed⊗1 in both sides of (3.11) and collecting and equating terms of the form ea⊗ec⊗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).


## 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 M𝔥* be the field of meromorphic functions on 𝔥*.Definition 3.1.
An𝔥-algebra is a complex associative algebra A with 1 which is bigraded over 𝔥*, A=⊕α,β∈𝔥*Aαβ, and equipped with two algebra embeddings μl,μr:M𝔥*→A, called the left and right moment maps, such that
(3.1)μl(f)a=aμl(Tαf),μr(f)a=aμr(Tβf),fora∈Aαβ,f∈M𝔥*,
where Tα denotes the automorphism (Tαf)(ζ)=f(ζ+α) of M𝔥*. A morphism of 𝔥-algebras is an algebra homomorphism preserving the bigrading and the moment maps.Thematrix tensor productA⊗̃B of two 𝔥-algebras A, B is the 𝔥*-bigraded vector space with (A⊗̃B)αβ=⊕γ∈𝔥*(Aαγ⊗M𝔥*Bγβ), where ⊗M𝔥* denotes tensor product over ℂ modulo the relations:
(3.2)μrA(f)a⊗b=a⊗μlB(f)b,∀a∈A,b∈B,f∈M𝔥*.
The multiplication (a⊗b)(c⊗d)=ac⊗bd for a,c∈A and b,d∈B and the moment maps μl(f)=μlA(f)⊗1 and μr(f)=1⊗μrB(f) make A⊗̃B into an 𝔥-algebra.Example 3.2.
LetD𝔥 be the algebra of operators on M𝔥* of the form ∑ifiTαi with fi∈M𝔥* and αi∈𝔥*. It is an 𝔥-algebra with bigrading fT-α∈(D𝔥)αα, and both moment maps equal to the natural embedding.For any𝔥-algebra A, there are canonical isomorphisms A≃A⊗̃D𝔥≃D𝔥⊗̃A defined by
(3.3)x≃x⊗T-β≃T-α⊗x,forx∈Aαβ.Definition 3.3.
An𝔥-bialgebroid is an 𝔥-algebra A equipped with two 𝔥-algebra morphisms, the comultiplication Δ:A→A⊗̃A and the counit ε:A→D𝔥 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 that𝔥-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 an𝔥-bialgebroid from a quantum dynamical R-matrix with spectral parameter.Let𝔥 be a finite-dimensional abelian Lie algebra, V=⊕α∈𝔥*Vα a finite-dimensional diagonalizable 𝔥-module, and R:𝔥*×ℂ×→End𝔥(V⊗V) a meromorphic function. We attach to this data an 𝔥-bialgebroid AR as follows. Let {ex}x∈X be a homogeneous basis of V, where X is an index set. The matrix elements Rxyab:𝔥*×ℂ×→ℂ of R are given by
(3.4)R(ζ,z)(ea⊗eb)=∑x,y∈XRxyab(ζ,z)ex⊗ey.
They are meromorphic on 𝔥*×ℂ×. Define ω:X→𝔥* by ex∈Vω(x). Let ÃR be the complex associative algebra with 1 generated by {Lxy(z):x,y∈X,z∈ℂ×} and two copies of M𝔥*, 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∈M𝔥* and
(3.5)f(λ)Lxy(z)=Lxy(z)f(λ+ω(x)),f(ρ)Lxy(z)=Lxy(z)f(ρ+ω(y)),
for all x,y∈X, z∈ℂ× and f∈M𝔥*. The bigrading on ÃR is given by Lxy(z)∈(ÃR)ω(x),ω(y) for x,y∈X, z∈ℂ× and f(λ),f(ρ)∈(ÃR)00 for f∈M𝔥*. The moment maps are defined by μl(f)=f(λ), μr(f)=f(ρ). The counit and comultiplication are defined by
(3.6)ε(Lab(z))=δabT-ω(a),ε(f(λ))=ε(f(ρ))=fT0,Δ(Lab(z))=∑x∈XLax(z)⊗Lxb(z),Δ(f(λ))=f(λ)⊗1,Δ(f(ρ))=1⊗f(ρ).
This makes ÃR into an 𝔥-bialgebroid.Consider the ideal inÃR generated by the RLL relations:
(3.7)∑x,y∈XRacxy(λ,z1z2)Lxb(z1)Lyd(z2)=∑x,y∈XRxybd(ρ,z1z2)Lcy(z2)Lax(z1),
where a,b,c,d∈X, and z1,z2∈ℂ×. More precisely, to account for possible singularities of R, we let IR be the ideal in ÃR generated by all relations of the form
(3.8)∑x,y∈Xlimw→z1/z2(φ(w)Racxy(λ,w))Lxb(z1)Lyd(z2)=∑x,y∈Xlimw→z1/z2(φ(w)Rxybd(ρ,w))Lcy(z2)Lax(z1),
where a,b,c,d∈X, z1,z2∈ℂ×, and φ:ℂ×→ℂ is a meromorphic function such that the limits exist.We defineAR to be ÃR/IR. The bigrading descends to AR 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(IR)⊆ÃR⊗̃IR+IR⊗̃ÃR and ε(IR)=0. Thus AR is an 𝔥-bialgebroid with the induced maps.Remark 3.4.
Objects in Felder's tensor category associated to an R-matrixR are certain meromorphic functions L:𝔥*×ℂ×→End𝔥(ℂn⊗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 AR 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.AssumeRxyab(ζ,z) are defined, as meromorphic functions of ζ∈𝔥* for any z∈ℂ×. Define R(λ,z),R(ρ,z)∈End(V⊗V⊗AR) by
(3.9)R(λ,z)(ea⊗eb⊗u)=∑x,y∈Xex⊗ey⊗Rxyab(λ,z)u,R(ρ,z)(ea⊗eb⊗u)=∑x,y∈Xex⊗ey⊗Rxyab(ρ,z)u,
for a,b∈X, u∈AR. 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⊗AR) by
(3.10)L(z)=∑x,y∈XExy⊗Lxy(z).
Here Exy are the matrix units in End(V), and AR acts on itself by left multiplication. The RLL relation (3.7) is equivalent to
(3.11)R(λ,z1z2)L1(z1)L2(z2)=L2(z2)L1(z1)R(ρ+h1+h2,z1z2)
in End(V⊗V⊗AR), where Li(z)=L(z)(i,3)∈End(V⊗V⊗AR) for i=1,2 and the operator R(ρ+h1+h2,z1/z2)∈End(V⊗V⊗AR) is given by
(3.12)ea⊗eb⊗u↦∑x,y∈X‍ex⊗ey⊗Rxyab(ρ+ω(a)+ω(b),z1z2)u,
where Rxyab(ρ+ω(a)+ω(b),z1/z2) means the image in AR of the meromorphic function 𝔥*∋ζ↦Rxyab(ζ+ω(a)+ω(b),z1/z2) under the right moment map μr. This equivalence can be seen by acting on eb⊗ed⊗1 in both sides of (3.11) and collecting and equating terms of the form ea⊗ec⊗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 Algebraℱell(M(n))

We now specialize to the case where𝔥 is the Cartan subalgebra of 𝔰𝔩(n), V=ℂn, 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 an𝔥-bialgebroid which we denote by ℱell(M(n)). The generators Lij(z) will be denoted by eij(z). Thus ℱell(M(n)) is the unital associative ℂ-algebra generated by eij(z), i,j∈[1,n], z∈ℂ×, and two copies of M𝔥*, whose elements are denoted by f(λ) and f(ρ) for f∈M𝔥*, subject to the following relations:
(4.1)f(λ)eij(z)=eij(z)f(λ+ω(i)),f(ρ)eij(z)=eij(z)f(ρ+ω(j)),
for all f∈M𝔥*, i,j∈[1,n], and z∈ℂ×, and
(4.2)∑x,y=1nRacxy(λ,z1z2)exb(z1)eyd(z2)=∑x,y=1nRxybd(ρ,z1z2)ecy(z2)eax(z1),
for all a,b,c,d∈[1,n]. More explicitly, from (2.13) we have
(4.3)Rxyab(ζ,z)={1,a=b=x=y,α(ζxy,z),a≠b,x=a,y=b,β(ζxy,z),a≠b,x=b,y=a,0,otherwise,
which substituted into (4.2) yields four families of relations:
(4.4a)eab(z1)eab(z2)=eab(z2)eab(z1),(4.4b)eab(z1)ead(z2)=α(ρbd,z1z2)ead(z2)eab(z1)+β(ρdb,z1z2)eab(z2)ead(z1),(4.4c)α(λac,z1z2)eab(z1)ecb(z2)+β(λac,z1z2)ecb(z1)eab(z2)=ecb(z2)eab(z1),(4.4d)α(λac,z1z2)eab(z1)ecd(z2)+β(λac,z1z2)ecb(z1)ead(z2)=α(ρbd,z1z2)ecd(z2)eab(z1)+β(ρdb,z1z2)ecb(z2)ead(z1),
where a,b,c,d∈[1,n], a≠c, and b≠d. Since θ has zeros precisely at pk,k∈ℤ, α and β have poles at z=q-2pk,k∈ℤ. Thus (4.4b)–(4.4d) are to hold for z1,z2∈ℂ× with z1/z2∉{pkq-2:k∈ℤ}.In (3.8), assuming a≠c, b≠d, taking z1=z, z2=pkq2z, φ(w)=q-1θ(q2w)/qθ(q-2w), and using the limit formulas (2.26), we obtain the relation
(4.5)α(λac,q2)(eab(z)ecd(pkq2z)-q2kλcaecb(z)ead(pkq2z))=α(ρbd,q2)ecd(pkq2z)eab(z)-q2kρbdβ(ρbd,q2)ecb(pkq2z)ead(z).
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 IR. Hence (4.1), (4.4a)–(4.4d), and (4.5) consitute the defining relations of the algebra ℱell(M(n)).Proof.
Assume that we have a relation of the form (3.8) and that a limit in one of the terms, limw→zφ(w)Rxyab(λ,w), say, exists and is nonzero. Then one of the following cases occurs.
(1)
Atw=z, φ(w) and Rxyab(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)
Atw=z, φ(w) has a pole while Rxyab(λ,w) is regular. Then Rxyab(λ,w) must vanish identically at w=z. The only case where this is possible is when x≠y and Rxyab(λ,w)=α(λxy,w), and z=pk. 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)
Atw=z, φ(w) is regular while Rxyab(λ,w) has a pole. Since these poles are simple and occur only when z∈q-2pℤ, the function φ must have a zero of multiplicity one there. We can assume without loss of generality that φ has the specific form
(4.6)φ(w)=q-1θ(q2w)qθ(q-2w).
Then, ifa≠c and b≠d, (3.8) becomes the residual RLL relation (4.5).
If insteadc=a, b≠d, and we take z1=z, z2=pkq2z in (3.8), we get, using (2.26),
(4.7)0=α(ρbd,pkq2)ead(pkq2z)eab(z)-β(ρbd,pkq2)eab(pkq2z)ead(z),
or, rewritten,
(4.8)ead(pkq2z)eab(z)=q2kρbdE(ρbd-1)E(ρbd+1)eab(pkq2z)ead(z).
However this relation is already derivable from (4.4b) as follows. Take z1=pkq2z and z2=z in (4.4b) multiply both sides by q2kρbd(E(ρbd-1)/E(ρbd+1)), and then use (4.4b) on the right-hand side.
Similarly, ifa≠c, d=b, z1=z, z2=pkq2z, φ(w)=q-1θ(q2w)/qθ(q-2w) in (3.8), and using (2.26) we get
(4.9)α(λac,pkq2)eab(z)ecb(pkq2z)-β(λca,pkq2)ecb(z)eab(pkq2z)=0,
or
(4.10)eab(z)ecb(pkq2z)=q2kλcaecb(z)eab(pkq2z).
Similarly to the previous case, this identity follows already from (4.4c).


## 5. Left and Right Elliptic Exterior Algebras

### 5.1. Corepresentations of𝔥-Bialgebroids

We recall the definition of corepresentations of an𝔥-bialgebroid given in [13].Definition 5.1.
An𝔥-space V is an 𝔥*-graded vector space over M𝔥*, V=⊕α∈𝔥*Vα, where each Vα is M𝔥*-invariant. A morphism of 𝔥-spaces is a degree-preserving M𝔥*-linear map.Given an𝔥-space V and an 𝔥-bialgebroid A, we define A⊗̃V to be the 𝔥*-graded space with (A⊗̃V)α=⊕β∈𝔥*(Aαβ⊗M𝔥*Vβ), where ⊗M𝔥* denotes ⊗ℂ modulo the relations
(5.1)μr(f)a⊗v=a⊗fv,
for f∈M𝔥*, a∈A, v∈V. A⊗̃V becomes an 𝔥-space with the M𝔥*-action f(a⊗v)=μl(f)a⊗v. Similarly we define V⊗̃A as an 𝔥-space by (V⊗̃A)β=⊕αVα⊗M𝔥*Aαβ, where ⊗M𝔥* here means ⊗ℂ modulo the relation v⊗μl(f)a=fv⊗a and M𝔥*-action given by f(v⊗a)=v⊗μr(f)a.For any𝔥-space V we have isomorphisms D𝔥⊗̃V≃V≃V⊗̃D𝔥 given by
(5.2)T-α⊗v≃v≃v⊗Tα,forv∈Vα,
extended to 𝔥-space morphisms.Definition 5.2.
A left corepresentationV of an 𝔥-bialgebroid A is an 𝔥-space equipped with an 𝔥-space morphism ΔV:V→A⊗̃V such that (ΔV⊗1)∘ΔV=(1⊗Δ)∘ΔV and (ε⊗1)∘ΔV=IdV (under the identification (5.2).Definition 5.3.
A left𝔥-comodule algebra V over an 𝔥-bialgebroid A is a left corepresentation ΔV:V→A⊗̃V and in addition a ℂ-algebra such that VαVβ⊆Vα+β and such that ΔV 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 AlgebrasΛ 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 ℱell(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.LetΛ be the unital associative ℂ-algebra generated by vi(z), 1≤i≤n, z∈ℂ× and a copy of M𝔥* embedded as a subalgebra subject to the relations
(5.3a)f(ζ)vi(z)=vi(z)f(ζ+ω(i)),(5.3b)vi(z)vi(w)=0,(5.3c)α(ζkj,zw)vk(z)vj(w)+β(ζkj,zw)vj(z)vk(w)=0,
for i,j,k∈[1,n], j≠k, z,w∈ℂ×, z/w∉{pkq-2:k∈ℤ} and f∈M𝔥*. 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
(5.3d)vk(z)vj(pkq2z)=q2kζjkvj(z)vk(pkq2z).Λ becomes an 𝔥-space by
(5.4)μΛ(f)v=f(ζ)v
and requiring vi(z)∈Λω(i) for each i,z.Proposition 5.4.
Λ is a left comodule algebra over ℱell(M(n)) with left coaction ΔΛ:Λ→ℱell(M(n))⊗̃Λ satisfying
(5.5)ΔΛ(vi(z))=∑jeij(z)⊗vj(z),(5.6)ΔΛ(f(ζ))=f(λ)⊗1.Proof.
We have(5.7)ΔΛ(vi(z))ΔΛ(vi(w))=∑jkeij(z)eik(w)⊗vj(z)vk(w)=∑j≠k(α(μjk,zw)eik(w)eij(z)+β(μkj,zw)eij(w)eik(z))⊗vj(z)vk(w)=∑j≠keij(w)eik(z)⊗(α(ζkj,zw)vk(z)vj(w)+β(ζkj,zw)vj(z)vk(w))=0.
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 forΛ:
(5.8a)f(ζ)vi(z)=vi(z)f(ζ+ω(i)),(5.8b)vk(psq2z)vj(z)=-q2sζkjE(ζkj-1)E(ζkj+1)vj(psq2z)vk(z),∀s∈ℤ,k≠j,(5.8c)vk(z)vj(psq2z)=q2sζjkvj(z)vk(psq2z),(5.8d)vk(z)vj(w)=0ifzw∉{psq±2∣s∈ℤ}orifk=j.
(ii) The set(5.9){vid(zd)⋯vi1(z1):1≤i1<⋯<id≤n,zi+1zi∈pℤq±2}
is a basis for Λ over M𝔥*.Proof.
(i) Elimination of thevj(z)vk(w)-term in (5.3c) yields
(5.10)(α(ζjk,zw)α(ζkj,zw)-β(ζkj,zw)β(ζjk,zw))vk(z)vj(w)=0.
Combining (5.10), (2.28), and the fact that the θ(z) is zero precisely for z∈{pk∣k∈ℤ} we deduce that in Λ,
(5.11)vk(z)vj(w)≠0⇒zw=psq2forsomes∈ℤ.
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 inΛ can be uniquely written as f(ζ)vid(zd)⋯vi1(z1), where 1≤i1<⋯<id≤n and f∈M𝔥*. It remains to show that the set (5.9) is linearly independent over M𝔥*. 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 vj(w) for appropriate j, w we can assume that  the sum is of the form
(5.12)f1(ζ)vid(zd1)⋯vi1(z11)+⋯+fr(ζ)vid(zdr)⋯vi1(z1r)=0,
for some fixed set {i1,…,id}. By the relations, a monomial vid(zd)⋯vi1(z1) can be given the "degree” ∑i=1dziti-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(vj(z))=tvj(z), T(f(ζ))=f(ζ), and T(p⊗1)=p⊗1. Define
(5.13)D(vi(z))=zvi(z),D(f(ζ))=0,D(p⊗1)=0,
for f∈M𝔥*, 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
(5.14)D(ab)=D(a)T(b)+aD(b),
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 uj=fj(ζ)vid(zdj)⋯vi1(z1j). Then one checks that D(uj)=pj(t)uj, where pj(t)=∑i=1dzijti-1. By applying D repeatedly we get
(5.15)u1(z1)+⋯+ur(zr)=0,p1(t)u1(z1)+⋯+pr(t)ur(zr)=0,⋮p1(t)r-1u1(z1)+⋯+pr(t)r-1ur(zr)=0.
Inverting the Vandermonde matrix (pj(t)i-1)ij we obtain uj(zj)=0 for each j, that is, fj(ζ)=0 for each j. This proves linear independence of (5.9).Analogously one defines a right comodule algebraΛ′ with generators wi(z) and f(ζ)∈M𝔥*. The following relations will be used:
(5.16)wk(z)wj(psq2z)=-q2sζkjwj(z)wk(psq2z),∀s∈ℤ,k≠j,wk(z1)wj(z2)=0,ifz2z1∉{psq±2∣s∈ℤ},orifk=j.Λ′ has also M𝔥*-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 Sn×Sn acts by ℂ-algebra automorphisms on ℱell(M(n)) as follows:
(5.17)(σ,τ)(f(λ))=f(λ∘Lσ),(σ,τ)(f(μ))=f(μ∘Lτ),(σ,τ)(eij(z))=eσ(i)τ(j)(z),
where Lσ:𝔥→𝔥 (σ∈Sn) is given by permutation of coordinates:
(5.18)Lσ(diag(h1,…,hn))=diag(hσ(1),…,hσ(n)).
Also, Sn acts on Λ by ℂ-algebra automorphisms via
(5.19)σ(f(ζ))=f(ζ∘Lσ),σ(vi(z))=vσ(i)(z).
Similarly we define an Sn action on Λ′.Lemma 5.6.
For eachv∈Λ, w∈Λ′, and any σ,τ∈Sn we have
(5.20)ΔΛ(σ(v))=((σ,τ)⊗τ)(ΔΛ(v)),(5.21)ΔΛ′(τ(w))=(σ⊗(σ,τ))(ΔΛ′(w)).Proof.
By multiplicativity, it is enough to prove these claims on the generators, which is easy.


## 5.1. Corepresentations of𝔥-Bialgebroids

We recall the definition of corepresentations of an𝔥-bialgebroid given in [13].Definition 5.1.
An𝔥-space V is an 𝔥*-graded vector space over M𝔥*, V=⊕α∈𝔥*Vα, where each Vα is M𝔥*-invariant. A morphism of 𝔥-spaces is a degree-preserving M𝔥*-linear map.Given an𝔥-space V and an 𝔥-bialgebroid A, we define A⊗̃V to be the 𝔥*-graded space with (A⊗̃V)α=⊕β∈𝔥*(Aαβ⊗M𝔥*Vβ), where ⊗M𝔥* denotes ⊗ℂ modulo the relations
(5.1)μr(f)a⊗v=a⊗fv,
for f∈M𝔥*, a∈A, v∈V. A⊗̃V becomes an 𝔥-space with the M𝔥*-action f(a⊗v)=μl(f)a⊗v. Similarly we define V⊗̃A as an 𝔥-space by (V⊗̃A)β=⊕αVα⊗M𝔥*Aαβ, where ⊗M𝔥* here means ⊗ℂ modulo the relation v⊗μl(f)a=fv⊗a and M𝔥*-action given by f(v⊗a)=v⊗μr(f)a.For any𝔥-space V we have isomorphisms D𝔥⊗̃V≃V≃V⊗̃D𝔥 given by
(5.2)T-α⊗v≃v≃v⊗Tα,forv∈Vα,
extended to 𝔥-space morphisms.Definition 5.2.
A left corepresentationV of an 𝔥-bialgebroid A is an 𝔥-space equipped with an 𝔥-space morphism ΔV:V→A⊗̃V such that (ΔV⊗1)∘ΔV=(1⊗Δ)∘ΔV and (ε⊗1)∘ΔV=IdV (under the identification (5.2).Definition 5.3.
A left𝔥-comodule algebra V over an 𝔥-bialgebroid A is a left corepresentation ΔV:V→A⊗̃V and in addition a ℂ-algebra such that VαVβ⊆Vα+β and such that ΔV 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 AlgebrasΛ 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 ℱell(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.LetΛ be the unital associative ℂ-algebra generated by vi(z), 1≤i≤n, z∈ℂ× and a copy of M𝔥* embedded as a subalgebra subject to the relations
(5.3a)f(ζ)vi(z)=vi(z)f(ζ+ω(i)),(5.3b)vi(z)vi(w)=0,(5.3c)α(ζkj,zw)vk(z)vj(w)+β(ζkj,zw)vj(z)vk(w)=0,
for i,j,k∈[1,n], j≠k, z,w∈ℂ×, z/w∉{pkq-2:k∈ℤ} and f∈M𝔥*. 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
(5.3d)vk(z)vj(pkq2z)=q2kζjkvj(z)vk(pkq2z).Λ becomes an 𝔥-space by
(5.4)μΛ(f)v=f(ζ)v
and requiring vi(z)∈Λω(i) for each i,z.Proposition 5.4.
Λ is a left comodule algebra over ℱell(M(n)) with left coaction ΔΛ:Λ→ℱell(M(n))⊗̃Λ satisfying
(5.5)ΔΛ(vi(z))=∑jeij(z)⊗vj(z),(5.6)ΔΛ(f(ζ))=f(λ)⊗1.Proof.
We have(5.7)ΔΛ(vi(z))ΔΛ(vi(w))=∑jkeij(z)eik(w)⊗vj(z)vk(w)=∑j≠k(α(μjk,zw)eik(w)eij(z)+β(μkj,zw)eij(w)eik(z))⊗vj(z)vk(w)=∑j≠keij(w)eik(z)⊗(α(ζkj,zw)vk(z)vj(w)+β(ζkj,zw)vj(z)vk(w))=0.
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 forΛ:
(5.8a)f(ζ)vi(z)=vi(z)f(ζ+ω(i)),(5.8b)vk(psq2z)vj(z)=-q2sζkjE(ζkj-1)E(ζkj+1)vj(psq2z)vk(z),∀s∈ℤ,k≠j,(5.8c)vk(z)vj(psq2z)=q2sζjkvj(z)vk(psq2z),(5.8d)vk(z)vj(w)=0ifzw∉{psq±2∣s∈ℤ}orifk=j.
(ii) The set(5.9){vid(zd)⋯vi1(z1):1≤i1<⋯<id≤n,zi+1zi∈pℤq±2}
is a basis for Λ over M𝔥*.Proof.
(i) Elimination of thevj(z)vk(w)-term in (5.3c) yields
(5.10)(α(ζjk,zw)α(ζkj,zw)-β(ζkj,zw)β(ζjk,zw))vk(z)vj(w)=0.
Combining (5.10), (2.28), and the fact that the θ(z) is zero precisely for z∈{pk∣k∈ℤ} we deduce that in Λ,
(5.11)vk(z)vj(w)≠0⇒zw=psq2forsomes∈ℤ.
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 inΛ can be uniquely written as f(ζ)vid(zd)⋯vi1(z1), where 1≤i1<⋯<id≤n and f∈M𝔥*. It remains to show that the set (5.9) is linearly independent over M𝔥*. 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 vj(w) for appropriate j, w we can assume that  the sum is of the form
(5.12)f1(ζ)vid(zd1)⋯vi1(z11)+⋯+fr(ζ)vid(zdr)⋯vi1(z1r)=0,
for some fixed set {i1,…,id}. By the relations, a monomial vid(zd)⋯vi1(z1) can be given the "degree” ∑i=1dziti-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(vj(z))=tvj(z), T(f(ζ))=f(ζ), and T(p⊗1)=p⊗1. Define
(5.13)D(vi(z))=zvi(z),D(f(ζ))=0,D(p⊗1)=0,
for f∈M𝔥*, 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
(5.14)D(ab)=D(a)T(b)+aD(b),
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 uj=fj(ζ)vid(zdj)⋯vi1(z1j). Then one checks that D(uj)=pj(t)uj, where pj(t)=∑i=1dzijti-1. By applying D repeatedly we get
(5.15)u1(z1)+⋯+ur(zr)=0,p1(t)u1(z1)+⋯+pr(t)ur(zr)=0,⋮p1(t)r-1u1(z1)+⋯+pr(t)r-1ur(zr)=0.
Inverting the Vandermonde matrix (pj(t)i-1)ij we obtain uj(zj)=0 for each j, that is, fj(ζ)=0 for each j. This proves linear independence of (5.9).Analogously one defines a right comodule algebraΛ′ with generators wi(z) and f(ζ)∈M𝔥*. The following relations will be used:
(5.16)wk(z)wj(psq2z)=-q2sζkjwj(z)wk(psq2z),∀s∈ℤ,k≠j,wk(z1)wj(z2)=0,ifz2z1∉{psq±2∣s∈ℤ},orifk=j.Λ′ has also M𝔥*-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 Sn×Sn acts by ℂ-algebra automorphisms on ℱell(M(n)) as follows:
(5.17)(σ,τ)(f(λ))=f(λ∘Lσ),(σ,τ)(f(μ))=f(μ∘Lτ),(σ,τ)(eij(z))=eσ(i)τ(j)(z),
where Lσ:𝔥→𝔥 (σ∈Sn) is given by permutation of coordinates:
(5.18)Lσ(diag(h1,…,hn))=diag(hσ(1),…,hσ(n)).
Also, Sn acts on Λ by ℂ-algebra automorphisms via
(5.19)σ(f(ζ))=f(ζ∘Lσ),σ(vi(z))=vσ(i)(z).
Similarly we define an Sn action on Λ′.Lemma 5.6.
For eachv∈Λ, w∈Λ′, and any σ,τ∈Sn we have
(5.20)ΔΛ(σ(v))=((σ,τ)⊗τ)(ΔΛ(v)),(5.21)ΔΛ′(τ(w))=(σ⊗(σ,τ))(ΔΛ′(w)).Proof.
By multiplicativity, it is enough to prove these claims on the generators, which is easy.


## 6. Elliptic Quantum Minors

### 6.1. Definition

ForI⊆[1,n] we set
(6.1)FI(ζ)=∏i,j∈I,i<jE(ζij+1),FI(ζ)=∏i,j∈I,i<jE(ζij),
and define the left and right elliptic sign functions:
(6.2)sgnI(σ;ζ)=σ(FI(ζ))Fσ(I)(ζ)=∏i,j∈I,i<j,σ(i)>σ(j)E(ζσ(i)σ(j)+1)E(ζσ(j)σ(i)+1),sgnI(σ;ζ)=Fσ(I)(ζ)σ(FI(ζ))=∏i,j∈I,i<j,σ(i)>σ(j)E(ζσ(j)σ(i))E(ζσ(i)σ(j)),
for σ∈Sn. In fact, E(ζij)/E(ζji)=-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 subsetI⊆[1,n] by i1<i2<⋯.Proposition 6.1.
LetI⊆[1,n], d=#I, σ∈Sn, and J=σ(I). Then for z∈ℂ×,
(6.3)vσ(id)(q2(d-1)z)⋯vσ(i1)(z)=sgnI(σ;ζ)vjd(q2(d-1)z)⋯vj1(z),(6.4)wσ(i1)(z)⋯wσ(id)(q2(d-1)z)=sgnI(σ;ζ)wj1(z)⋯wjd(q2(d-1)z).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,…,id-1},J′=σ(I′). Let 1≤j1′<⋯<jd-1′≤n be the elements of J′. By the induction hypothesis, the left hand side of (6.3) equals
(6.5)vσ(id)(q2(d-1)z)sgnI′(σ,ζ)vjd-1′(q2(d-2)z)⋯vj1′(z).
Now vσ(id)(q2(d-1)z) commutes with sgnI′(σ,ζ) since the latter only involves ζij with i,j≠σ(id). Using the commutation relations (5.8b) we obtain
(6.6)sgnI′(σ,ζ)·∏j∈J′,j>σ(id)E(ζjσ(id)+1)E(ζσ(id)j+1)·vjd(q2(d-1)z)⋯vj1(z).
Replace j∈J′ such that j>σ(id) by σ(i), where i∈I,i<id,σ(i)>σ(id).Introduce the normalized monomials(6.7)vI(z)=FI(ζ)-1vir(q2(d-1)z)vir-1(q2(d-2)z)⋯vi1(z)∈Λ,(6.8)wI(z)=FI(ζ)wi1(z)wi2(q2z)⋯wid(q2(d-1)z)∈Λ′.Corollary 6.2.
LetI⊆[1,n]. For any permutation σ∈Sn,
(6.9)σ(vI(z))=vσ(I)(z),σ(wI(z))=wσ(I)(z)
for any z∈ℂ×. In particular vI(z) and wI(z) are fixed by any permutation which preserves the subset I.Proof.
LetJ=σ(I). Then
(6.10)σ(vI(z))=σ(FI(ζ)-1)vσ(id)(q2(d-1)z)⋯vσ(i1)(z)=σ(FI(ζ))-1sgnI(σ;ζ)vjd(q2(d-1)z)⋯vj1(z)=vσ(I)(z).
The proof for wI(z) is analogous.For anyI⊆[1,n], let SI denote the group of all permutations of the set I. We are now ready to define certain elements of the 𝔥-bialgebroid ℱell(M(n)) which are analogs of minors.Proposition 6.3.
ForI,J⊆[1,n] and z∈ℂ×, the left and right elliptic minors, ξ⟵IJ(z) and ξ⃗IJ(z), respectively, can be defined by
(6.11)ΔΛ(vI(z))=∑Jξ⟵IJ(z)⊗vJ(z),(6.12)ΔΛ′(wJ(z))=∑IwI(z)⊗ξ⃗IJ(z),
where the sums are taken over all subsets of [1,n].
If#I≠#J, then ξ⟵IJ(z)=0=ξ⃗IJ(z), for all z. If #I=#J=d, they are explicitly given by
(6.13)ξ⟵IJ(z)=FJ(ρ)FI(λ)∑τ∈SJ‍sgnJ(τ;ρ)sgnI(σ;λ)eσ(id)τ(jd)(q2(d-1)z)eσ(id-1)τ(jd-1)(q2(d-2)z)⋯eσ(i1)τ(j1)(z)
for any σ∈SI, and
(6.14)ξ⃗IJ(z)=FJ(ρ)FI(λ)∑σ∈SIsgnJ(τ;ρ)sgnI(σ;λ)eσ(i1)τ(j1)(z)eσ(i2)τ(j2)(q2z)⋯eσ(id)τ(jd)(q2(d-1)z)
for any τ∈SJ. Moreover,
(6.15)(σ,τ)(ξ⟵IJ(z))=ξ⟵σ(I)τ(J)(z),(σ,τ)(ξ⃗IJ(z))=ξ⃗σ(I)τ(J)(z)
for any (σ,τ)∈Sn×Sn and z∈ℂ×.Remark 6.4.
In Theorem6.10 we will prove that, in fact, ξ⟵IJ(z)=ξ⃗IJ(z).Proof.
We prove the statements concerning the left elliptic minorξ⟵IJ(z). We have
(6.16)ΔΛ(vI(z))=∑1≤k1,…,kd≤n‍FI(λ)-1eidkd(q2(d-1)z)⋯ei1k1(z)⊗vkd(q2(d-1)z)⋯vk1(z)=∑J,#J=d‍∑τ∈SJ‍FI(λ)-1eidτ(jd)(q2(d-1)z)⋯ei1τ(j1)(z)⊗vτ(jd)(q2(d-1)z)⋯vτ(j1)(z)=∑J,#J=d(∑τ∈SJ‍τ(FJ(ρ))FI(λ)eidτ(jd)(q2(d-1)z)⋯ei1τ(j1)(z))⊗vJ(z).
Thus (6.11) holds when ξ⟵IJ(z) is defined by (6.13) with σ=Id. Then the right hand side of (6.13) equals (σ,Id)(ξ⟵IJ(z)) for any σ∈SI. Thus only (6.15) remains. Using (5.20) and Corollary 6.2 we have
(6.17)ΔΛ(σ(vI(z)))=((σ,τ)⊗τ)(ΔΛ(vI(z)))=∑J(σ,τ)((ξ⟵IJ(z))⊗vτ(J)(z)).
On the other hand, again by Corollary 6.2,
(6.18)ΔΛ(σ(vI(z)))=ΔΛ(vσ(I)(z))=∑J(ξ⟵σ(I)τ(J)(z)⊗vτ(J)(z)),
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 Theorem6.10 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 ℱell(M(n)) and consider certain operators on V⊗n⊗ℱell(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 setℱ=ℱell(M(n)). Recall the operators from Section 3.3, defined for any 𝔥-bialgebroid AR obtained from the FRST construction. When specializing to ℱ we get operators R(λ,z), R(ρ,z)∈End(V⊗V⊗ℱ), where V=ℂn. For z∈ℂ×, define the following linear operators on V⊗n⊗ℱ:
(6.19)Rij(λ,z):=limw→zθ(q2w)R(λ,w)(i,j,n+1),Rij(ρ,z):=limw→zθ(q2w)R(ρ,w)(i,j,n+1).
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).Letℰn denote the algebra of all functions
(6.20)F:(ℂ×)n→End(V⊗n⊗ℱ).
The symmetric group Sn acts on ℰn by
(6.21)σ(F(z))=(σ⊗Idℱ)∘F(σ(z))∘(σ-1⊗Idℱ),
for F∈ℰn and σ∈Sn. 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
(6.22)(23)(R12(λ,z1z2))=R13(λ,z1z3).
Consider the skew group algebra ℰn*Sn, defined as the algebra with underlying space ℰn⊗ℂSn, where ℂSn is the group algebra, with the multiplication
(6.23)(F(z)⊗σ)(G(z)⊗τ)=F(z)σ(G(z))⊗στ,
for σ,τ∈Sn, F,G∈ℰn. Since σ acts on ℰn by automorphisms, ℰn*Sn is an associative algebra. The constant function z↦IdV⊗n⊗ℱ⊗(1) is the unit element. Let Bn be the monoid (set with unital associative multiplication) generated by {s1,…,sn-1} and relations
(6.24)sisi+1si=si+1sisi+1,for1≤i≤n-2,sisj=sjsi,if|i-j|>1.
Let σi=(ii+1)∈Sn. We have an epimorphism π:Bn→Sn given by π(si)=σi,  π(1)=(1). Define
(6.25)W(1)=IdV⊗n⊗ℱ⊗(1),W(si)=Ri,i+1(λ-h≥i+2,zizi+1)⊗σi.
Here and below we use h≥k to denote the expression ∑j=knhj, and operators involving shifts hi such as Rn-2,n-1(λ-hn,zi/zi+1) are defined as in Section 3.3.Proposition 6.5.
W extends to a well-defined morphism of monoids, that is, a map
(6.26)W:Bn→ℰn*Sn
satisfying W(b1b2)=W(b1)W(b2) for any b1,b2∈Bn.Proof.
We have to show the relations(6.27)W(si)W(si+1)W(si)=W(si+1)W(si)W(si+1),(6.28)W(si)W(sj)=W(sj)W(si)if|i-j|>1.
Relation (6.27) follows from the QDYBE (2.8). For example, W(si)W(si+1)W(si) equals
(6.29)Ri,i+1(λ-h≥i+2,zizi+1)Ri,i+2(λ-h≥i+3,zizi+2)Ri+1,i+2(λ-hi-h≥i+3,zi+1zi+2)⊗σiσi+1σi.
Relation (6.28) is easy to check, using the 𝔥-invariance of R.Forb∈Bn we define Wb(λ,z)∈ℰn by
(6.30)W(b)=Wb(λ,z)⊗π(b).
From this and the product rule (6.23) it  follows that
(6.31)Wb1b2(λ,z)=Wb1(λ,z)·π(b1)(Wb2(λ,z)),
for b1,b2∈Bn. By replacing λ by ρ we get similarly operators Wb(ρ,z).Recall the operatorsL(z)∈End(V⊗ℱ) from Section 3.3. Define for z∈ℂ×, i∈[1,n],
(6.32)Li(z)=L(z)(i,n+1)∈End(V⊗n⊗ℱ).Ifi,j,k are distinct, then one can check that
(6.33)Rij(λ-hk,z)Lk(w)=Lk(w)Rij(λ,z),(6.34)Rij(ρ,z)Lk(w)=Lk(w)Rij(ρ+hk,z).Due to the RLL relations (3.8) we have
(6.35)R12(λ,z1z2)L1(z1)L2(z2)=L2(z2)L1(z1)R12(ρ+h1+h2,z1z2)
for any z1,z2∈ℂ×.Definetd∈Bn, d∈[1,n], recursively by
(6.36)td={td-1sd-1sd-2⋯s1,d>11,d=1.
Let τd be the image of td in Sn:
(6.37)τd:=π(td)=(12⋯dd+1⋯ndd-1⋯1d+1⋯n)∈Sn.Proposition 6.6.
Let1≤d≤n. For any z=(z1,…,zd)∈(ℂ×)d we have
(6.38)Wtd(λ,z)L1(z1)⋯Ld(zd)=Ld(zd)⋯L1(z1)Wtd(ρ+h≤d,z).Proof.
We use induction ond. The case d=1 is trivial, while d=2 is the RLL relation (6.35). If d>2, write td=td-1ud, where ud=sd-1sd-2⋯s1. Thus, by (6.31),
(6.39)Wtd(λ,z)=Wtd-1(λ,z)·τd-1(Wud(λ,z)).
We claim that
(6.40)τd-1(Wud(λ,z))L1(z1)⋯Ld(zd)=Ld(zd)L1(z1)⋯Ld-1(zd-1)τd-1(Wud(ρ+h≤d,z)).
For notational simplicity, set λ′=λ-h>d. A calculation using (6.30) shows that, compare  the proof of Proposition 6.5,
(6.41)Wud(λ,z)=Rd-1,d(λ′,zd-1zd)Rd-2,d(λ′-hd-1,zd-2zd)⋯R1,d(λ′-h[2,d-1],z1zd),
where h[a,b] means ∑a≤j≤bhj. Thus
(6.42)τd-1(Wud(λ,z))=R1,d(λ′,z1zd)R2,d(λ′-h1,z2zd)⋯Rd-1,d(λ′-h≤d-2,zd-1zd).
Using (6.33) and the RLL relation (6.35) repeatedly, we obtain (6.40). Now the proposition follows by induction on d, using that
(6.43)Wtd-1(λ,z)Ld(zd)=Ld(zd)Wtd-1(λ+hd,z)
which follows from (6.33).The operatorC(λ,z):=Wtn(λ,z) is called the Cherednik operator. For an operator F(z)∈ℰn we define its matrix elements F(z)x1,…,xna1,…,an∈ℱ by
(6.44)F(z)(ea1⊗⋯⊗ean⊗1)=∑x1,…,xn‍ex1⊗⋯⊗exn⊗F(z)x1,…,xna1,…,an.Proposition 6.7.
Let(6.45)α̃(λ,z)=limw→zθ(q2w)α(λ,w)=θ(z)θ(q2(λ+1))θ(q2λ).
Then
(6.46)C(λ,z)1,…,n1,…,n=∏i<jα̃(λij,zizj)=∏i<jqθ(zizj)·F[1,n](λ)F[1,n](λ).Proof.
The second equality follows from the definition (6.1) of FI and FI. We prove by induction on d that Wtd(λ,z)1,…,n1,…,n=∏i<j≤dα̃(λij,zi/zj). For d=2 we have td=s1 and Ws1(λ,z)1,…,n1,…,n=R12(λ-h>2,z1/z2)1,…,n1,…,n=α̃(λ12,z1/z2) as claimed. For d>2, using factorization (6.39) we have
(6.47)Wtd(λ,z)1,…,n1,…,n=∑x1,…,xn‍Wtd-1(λ,z)1,…,nx1,…,xnτd-1(Wud(λ,z))x1,…,xn1,…,n.
SinceWtd-1(λ,z) is a product of operators of the form σ(Rii+1(λ,zi/zi+1)) where 1≤i≤d-2 and σ∈Sn, σ(j)=j,j>d-1, and each of these operators preserve the subspace spanned by eτ(1)⊗⋯⊗eτ(d-1)⊗ed⊗⋯⊗en⊗a, where τ∈Sd-1 and a∈ℱ; the operator Wtd-1(λ,z) also preserves this subspace. This means that Wtd-1(λ,z)1,…,nx1,…,xn=0 unless xj=j for j≥d and {x1,…,xd-1}={1,…,d-1}. Furthermore, by (6.42),
(6.48)τd-1(Wud(λ,z))x1,…,xd-1,d,…,n1,…,n=∑y2,…,yd-1‍R̃x1d1y2(λ,z1zd)R̃x2y22y3(λ-ω(1),z2zd)⋯R̃xd-1yd-1d-1,d(λ-∑k≤d-2ω(k),zd-1zd).
Here R̃xyab(λ,z)=limw→zθ(q2w)Rxyab(λ,w). Since R̃xyab(λ,z)=0 unless {x,y}={a,b}, we deduce that, when {x1,…,xd-1}={1,…,d-1}, the terms in the sum (6.48) are zero unless xi=i for all i and yj=d for all j. Substituting into (6.47) the claim follows by induction.Lemma 6.8.
Fix2≤d≤n and i<d. Then there are elements b,c∈Bn such that td=sib and td=csi.Proof.
Sincet2=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′∈Bn such that td-1=sib′. Hence td=td-1sd-1⋯s1=sib′sd-1⋯s1. Thus we can take b=b′sd-1⋯s1. If i=d-1, write td=td-2sd-2⋯s1sd-1⋯s1. Then move each of the d-1 rightmost factors sd-1,…,s1 as far to the left as possible, using that sjsk=sksj when |j-k|>1. This gives
(6.49)td=td-2sd-2sd-1sd-3sd-2sd-4⋯s2s3s1s2s1.
Then use sjsj+1sj=sj+1sjsj+1 repeatedly, working from right to left, to obtain
(6.50)td=td-2sd-1sd-2sd-1sd-3sd-2⋯s4s2s3s1s2.
Finally, sd-1 can be moved to the left of td-2 since the latter is a product of sj's with j≤d-3.
To prove the existence ofc we note that Bn carries an involution *:Bn→Bn satisfying (a1a2)*=a2*a1* for any a1,a2∈Bn, defined by sj*=sj for j∈[1,n] and 1*=1. Thus it suffices to show that td*=td for any d. This is trivial for d=2,3. When d>3 we have, by induction on d,
(6.51)td*=(td-1sd-1⋯s1)*=s1⋯sd-1td-1=s1⋯sd-1td-2sd-2⋯s1=s1⋯sd-2td-2sd-1sd-2⋯s1=td-1*sd-1⋯s1=td(sincesd-1commuteswithtd-2).Proposition 6.9.
Letw=(z0,q2z0,…,q2(n-1)z0), where z0≠0 is arbitrary, and let σ,τ∈Sn. Then
(6.52)C(λ,w)σ(1),…,σ(n)τ(1),…,τ(n)=sgn[1,n](σ;λ)sgn[1,n](τ;λ)C(λ,w)1,…,n1,…,n.Proof.
First we claim that for allσ,τ∈Sn and each i∈[1,n],
(6.53)Wsi(λ,w)σσi(1),…,σσi(n)τ(1),…,τ(n)=σ(sgn[1,n](σi;λ))Wsi(λ,w)σ(1),…,σ(n)τ(1),…,τ(n),(6.54)Wsi(λ,w)σ(1),…,σ(n)τσi(1),…,τσi(n)=τ(sgn[1,n](σi;λ))Wsi(λ,w)σ(1),…,σ(n)τ(1),…,τ(n)=-Wsi(λ,w)σ(1),…,σ(n)τ(1),…,τ(n).
Indeed, assume that zi/zi+1=q-2 and that {a1,…,an}={b1,…,bn}=[1,n]. Then Wsi(λ,z)a1,…,anb1,…,bn≠0 if and only if {ai,ai+1}={bi,bi+1} in which case
(6.55)Wsi(λ,z)a1,…,anb1,…,bn=E(1)E(λaiai+1+1)E(λbi+1bi).
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 (σσi,τ) and (σ,τσi) where i is arbitrary.
Leti∈[1,n]. By Lemma 6.8 we have tn=sib for some b∈Bn. We have
(6.56)Wtn(λ,w)σσi(1),…,σσi(n)τ(1),…,τ(n)=(Wsi(λ,w)σi(Wb(λ,w)))σσi(1),…,σσi(n)τ(1),…,τ(n)=∑x1,…,xnWsi(λ,w)σσi(1),…,σσi(n)x1,…,xnσi(Wb(λ,w))x1,…,xnτ(1),…,τ(n).
As in the proof of Proposition 6.7, Wsi(λ,w)σσi(1),…,σσi(n)x1,…,xn is zero, if x1,…,xn is not a permutation of 1,…,n. Using (6.53) we obtain
(6.57)σ(sgn[1,n](σi;λ))∑x1,…,xn‍Wsi(λ,w)σ(1),…,σ(n)x1,…,xnσi(Wb(λ,w))x1,…,xnτ(1),…,τ(n)=σ(sgn[1,n](σi;λ))Wtn(λ,w)σ(1),…,σ(n)τ(1),…,τ(n).
Using the induction hypothesis and the relation sgn[1,n](σ;λ)σ(sgn[1,n](σi;λ))=sgn[1,n](σσi;λ) we obtain (6.52) for (σσi,τ).
For the other case, leti be arbitrary, and set j=τn(i). By Lemma 6.8 there is a c∈Bn such that tn=csj. Recall the surjective morphism π:Bn→Sn sending si to σi=(ii+1). Then σjπ(c)=π(c)σi. We have
(6.58)Wtn(λ,w)σ(1),…,σ(n)τσi(1),…,τσi(n)=(Wc(λ,w)·π(c)(Wsj(λ,w)))σ(1),…,σ(n)τσi(1),…τσi(n)=∑x1,…,xnWc(λ,w)σ(1),…,σ(n)x1,…,xnπ(c)(Wsj(λ,w))x1,…,xnτσi(1),…,τσi(n).
It is easy to check that σ(F(z))a1,…,anb1,…,bn=F(σ(z))aσ(1),…,aσ(n)bσ(1),…,bσ(n) for any F(z)∈ℰn and σ∈Sn. Define wi by (w1,…,wn)=w=(z0,q2z0,…,q2(n-1)z0). Then wi/wi+1=q-2 for each i. Set w′=(wπ(c)(1),…,wπ(c)(n)). For each i, wπ(c)(i)/wπ(c)(i+1)=wτn(i+1)/wτn(i)=q-2 also. Therefore
(6.59)Wtn(λ,w)σ(1),…,σ(n)τσi(1),…,τσi(n)=∑x1,…,xn‍Wc(λ,w)σ(1),…,σ(n)x1,…,xnWsj(λ,w′)xπ(c)(1),…,xπ(c)(n)τσiπ(c)(1),…,τσiπ(c)(n)=∑x1,…,xn‍Wc(λ,w)σ(1),…,σ(n)x1,…,xnWsj(λ,w′)xπ(c)(1),…,xπ(c)(n)τπ(c)σj(1),…,τπ(c)σj(n)=∑x1,…,xn‍Wc(λ,w)σ(1),…,σ(n)x1,…,xn(sgnσj)Wsj(λ,w′)xπ(c)(1),…,xπ(c)(n)τπ(c)(1),…,τπ(c)(n)=∑x1,…,xn‍Wc(λ,w)σ(1),…,σ(n)x1,…,xn(-1)π(c)(Wsj(λ,w))x1,…,xnτ(1),…,τ(n)=-Wtn(λ,w)σ(1),…,σ(n)τ(1),…,τ(n).
By the induction hypothesis it follows that (6.52) holds for (σ,τσi). This proves the formula (6.52).Theorem 6.10.
For any subsetsI,J⊆[1,n] and z∈ℂ×, the left and right elliptic minors coincide:
(6.60)ξ⟵IJ(z)=ξ⃗IJ(z).
We denote this common element by ξIJ(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 eij(z), i,j∈[1,d], z∈ℂ× and f(λ),f(ρ) with f∈M𝔥d*⊆M𝔥*, 𝔥d being the Cartan subalgebra of 𝔰𝔩(d), is isomorphic to ℱell(M(d)), we can also assume d=n. Identifying the matrix element 1,…,n1,…,n on both sides of (6.38) we get
(6.61)∑x1,…,xn‍C(λ,z)1,…,nx1,…,xnex1,1(z1)⋯exn,n(zn)=∑x1,…,xn‍en,xn(zn)⋯e1,x1(z1)C(ρ+h≤n,z)x1,…,xn1,…,n.
As in the proof of Proposition 6.7, C(λ,z)1,…,nx1,…,xn is zero if x1,…,xn is not a permutation of 1,…,n. Taking z=w=(z0,q2z0,…,q2(n-1)z0) and dividing both sides by ∏i<jqθ(wi/wj)=∏i<jqθ(q2(i-j)) we get
(6.62)F[1,n](λ)F[1,n](λ)∑σ∈Sn‍sgn[1,n](σ;λ)-1eσ(1)1(z0)⋯eσ(n)n(q2(n-1)z0)=F[1,n](ρ)F[1,n](ρ)∑τ∈Sn‍sgn[1,n](τ;ρ)enτ(n)(q2(n-1)z0)⋯e1σ(1)(z0).
Multiplying by F[1,n](ρ)/F[1,n](λ) and comparing with (6.13) and (6.14), we deduce that ξ⃗[1,n][1,n](z0)=ξ⟵[1,n][1,n](z0), as desired.


### 6.3. Laplace Expansions

Using the left (right)ℱell(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 Sl(I,J;ζ),Sr(I,J;ζ)∈M𝔥* by
(6.63)vI(q2#Jz)vJ(z)=Sl(I,J;ζ)vI∪J(z),wI(z)wJ(q2#Iz)=Sr(I,J;ζ)wI∪J(z).
That this is possible follows from the definitions (6.7) and (6.8) of vI(z),wI(z), and the commutation relations (5.8b)–(5.8d), (5.16). In particular Sl(I,J;ζ)=0=Sr(I,J;ζ), if I∩J≠∅.Theorem 6.11.
(i) LetI1,I2,J⊆[1,n], and set I=I1∪I2. Then
(6.64)Sl(I1,I2;λ)ξIJ(z)=∑J1∪J2=J‍Sl(J1,J2;ρ)ξI1J1(q2#I2z)ξI2J2(z).
(ii) LetJ1,J2,I⊆[1,n] and set J=J1∪J2. Then
(6.65)Sr(J1,J2;ρ)ξIJ(z)=∑I1∪I2=I‍Sr(I1,I2;λ)ξI1J1(z)ξI2J2(q2#J1z).Proof.
We have(6.66)ΔΛ(vI1(q2#I2z))ΔΛ(vI2(z))=∑J1,J2‍ξI1J1(q2#I2z)ξI2J2(z)⊗vJ1(q2#I2z)vJ2(z)=∑J1,J2‍ξI1J1(q2#I2z)ξI2J2(z)⊗Sl(J1,J2;ζ)vJ(z)=∑J(∑J1∪J2=J‍Sl(J1,J2;ρ)ξI1J1(q2#I2z)ξI2J2(z))⊗vJ(z).
On the other hand,
(6.67)ΔΛ(vI1(q2#I2z))ΔΛ(vI2(z))=ΔΛ(vI1(q2#I2z)vI2(z))=ΔΛ(Sl(I1,I2;ζ)vI(z))=∑J‍Sl(I1,I2;λ)ξIJ(z)⊗vJ(z).
Equating these expressions proves (6.64) since, by Proposition 5.5, the set {vJ(z):J⊆[1,n]} is linearly independent over M𝔥*. The second part is completely analogous, using the right comodule algebra Λ′ in place of Λ.In Section7.4 we will need the following lemma, relating the left and right signums Sl(I,J;ζ) and Sr(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.
LetI,J be two disjoint subsets of [1,n]. Then
(6.68)Sl(I,J;ζ+ω(I))=Sr(J,I;ζ)-1,
where ω(I)=∑i∈Iω(i).Proof.
First we claim that, we have the following explicit formulas:(6.69)Sl(I,J;ζ)=∏i∈I,j∈JE(ζji+1),(6.70)Sr(I,J;ζ)=∏i∈I,j∈JE(ζij)-1.
Recall the definition (6.7) of vI(z). Since E is odd, relation (5.8b) implies that
(6.71)vi(q2z)vj(z)=E(ζji+1)E(ζij+1)vj(q2z)vi(z).
Also, FJ(ζ) only involves ζij with i,j∈J so it commutes with any vk(z) with k∈I (since I∩J=∅). From these facts we obtain
(6.72)vI(q2#Jz)vJ(z)=FI(ζ)-1FJ(ζ)-1FI∪J(ζ)-1∏i∈I,j∈Ji<jE(ζji+1)E(ζij+1)vI∪J(z)=∏(i,j)∈Ki<jE(ζij+1)∏i∈I,j∈Ji<jE(ζji+1)E(ζij+1)vI∪J(z)=∏i∈I,j∈JE(ζji+1)vI∪J(z),
where K=(I×J)∪(J×I). This proves (6.69). Similarly one proves (6.70). Now we have
(6.73)Sl(J,I;ζ+ω(J))-1=∏i∈I,j∈JE((ζ+ω(J))ij+1)-1=∏i∈I,j∈JE(ζij)-1=Sr(I,J;ζ).
Here we used that for any i∈I,j∈J we have ω(J)(Eii)=0, ω(J)(Ejj)=1, and hence (ω(J))ij=-1.


## 6.1. Definition

ForI⊆[1,n] we set
(6.1)FI(ζ)=∏i,j∈I,i<jE(ζij+1),FI(ζ)=∏i,j∈I,i<jE(ζij),
and define the left and right elliptic sign functions:
(6.2)sgnI(σ;ζ)=σ(FI(ζ))Fσ(I)(ζ)=∏i,j∈I,i<j,σ(i)>σ(j)E(ζσ(i)σ(j)+1)E(ζσ(j)σ(i)+1),sgnI(σ;ζ)=Fσ(I)(ζ)σ(FI(ζ))=∏i,j∈I,i<j,σ(i)>σ(j)E(ζσ(j)σ(i))E(ζσ(i)σ(j)),
for σ∈Sn. In fact, E(ζij)/E(ζji)=-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 subsetI⊆[1,n] by i1<i2<⋯.Proposition 6.1.
LetI⊆[1,n], d=#I, σ∈Sn, and J=σ(I). Then for z∈ℂ×,
(6.3)vσ(id)(q2(d-1)z)⋯vσ(i1)(z)=sgnI(σ;ζ)vjd(q2(d-1)z)⋯vj1(z),(6.4)wσ(i1)(z)⋯wσ(id)(q2(d-1)z)=sgnI(σ;ζ)wj1(z)⋯wjd(q2(d-1)z).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,…,id-1},J′=σ(I′). Let 1≤j1′<⋯<jd-1′≤n be the elements of J′. By the induction hypothesis, the left hand side of (6.3) equals
(6.5)vσ(id)(q2(d-1)z)sgnI′(σ,ζ)vjd-1′(q2(d-2)z)⋯vj1′(z).
Now vσ(id)(q2(d-1)z) commutes with sgnI′(σ,ζ) since the latter only involves ζij with i,j≠σ(id). Using the commutation relations (5.8b) we obtain
(6.6)sgnI′(σ,ζ)·∏j∈J′,j>σ(id)E(ζjσ(id)+1)E(ζσ(id)j+1)·vjd(q2(d-1)z)⋯vj1(z).
Replace j∈J′ such that j>σ(id) by σ(i), where i∈I,i<id,σ(i)>σ(id).Introduce the normalized monomials(6.7)vI(z)=FI(ζ)-1vir(q2(d-1)z)vir-1(q2(d-2)z)⋯vi1(z)∈Λ,(6.8)wI(z)=FI(ζ)wi1(z)wi2(q2z)⋯wid(q2(d-1)z)∈Λ′.Corollary 6.2.
LetI⊆[1,n]. For any permutation σ∈Sn,
(6.9)σ(vI(z))=vσ(I)(z),σ(wI(z))=wσ(I)(z)
for any z∈ℂ×. In particular vI(z) and wI(z) are fixed by any permutation which preserves the subset I.Proof.
LetJ=σ(I). Then
(6.10)σ(vI(z))=σ(FI(ζ)-1)vσ(id)(q2(d-1)z)⋯vσ(i1)(z)=σ(FI(ζ))-1sgnI(σ;ζ)vjd(q2(d-1)z)⋯vj1(z)=vσ(I)(z).
The proof for wI(z) is analogous.For anyI⊆[1,n], let SI denote the group of all permutations of the set I. We are now ready to define certain elements of the 𝔥-bialgebroid ℱell(M(n)) which are analogs of minors.Proposition 6.3.
ForI,J⊆[1,n] and z∈ℂ×, the left and right elliptic minors, ξ⟵IJ(z) and ξ⃗IJ(z), respectively, can be defined by
(6.11)ΔΛ(vI(z))=∑Jξ⟵IJ(z)⊗vJ(z),(6.12)ΔΛ′(wJ(z))=∑IwI(z)⊗ξ⃗IJ(z),
where the sums are taken over all subsets of [1,n].
If#I≠#J, then ξ⟵IJ(z)=0=ξ⃗IJ(z), for all z. If #I=#J=d, they are explicitly given by
(6.13)ξ⟵IJ(z)=FJ(ρ)FI(λ)∑τ∈SJ‍sgnJ(τ;ρ)sgnI(σ;λ)eσ(id)τ(jd)(q2(d-1)z)eσ(id-1)τ(jd-1)(q2(d-2)z)⋯eσ(i1)τ(j1)(z)
for any σ∈SI, and
(6.14)ξ⃗IJ(z)=FJ(ρ)FI(λ)∑σ∈SIsgnJ(τ;ρ)sgnI(σ;λ)eσ(i1)τ(j1)(z)eσ(i2)τ(j2)(q2z)⋯eσ(id)τ(jd)(q2(d-1)z)
for any τ∈SJ. Moreover,
(6.15)(σ,τ)(ξ⟵IJ(z))=ξ⟵σ(I)τ(J)(z),(σ,τ)(ξ⃗IJ(z))=ξ⃗σ(I)τ(J)(z)
for any (σ,τ)∈Sn×Sn and z∈ℂ×.Remark 6.4.
In Theorem6.10 we will prove that, in fact, ξ⟵IJ(z)=ξ⃗IJ(z).Proof.
We prove the statements concerning the left elliptic minorξ⟵IJ(z). We have
(6.16)ΔΛ(vI(z))=∑1≤k1,…,kd≤n‍FI(λ)-1eidkd(q2(d-1)z)⋯ei1k1(z)⊗vkd(q2(d-1)z)⋯vk1(z)=∑J,#J=d‍∑τ∈SJ‍FI(λ)-1eidτ(jd)(q2(d-1)z)⋯ei1τ(j1)(z)⊗vτ(jd)(q2(d-1)z)⋯vτ(j1)(z)=∑J,#J=d(∑τ∈SJ‍τ(FJ(ρ))FI(λ)eidτ(jd)(q2(d-1)z)⋯ei1τ(j1)(z))⊗vJ(z).
Thus (6.11) holds when ξ⟵IJ(z) is defined by (6.13) with σ=Id. Then the right hand side of (6.13) equals (σ,Id)(ξ⟵IJ(z)) for any σ∈SI. Thus only (6.15) remains. Using (5.20) and Corollary 6.2 we have
(6.17)ΔΛ(σ(vI(z)))=((σ,τ)⊗τ)(ΔΛ(vI(z)))=∑J(σ,τ)((ξ⟵IJ(z))⊗vτ(J)(z)).
On the other hand, again by Corollary 6.2,
(6.18)ΔΛ(σ(vI(z)))=ΔΛ(vσ(I)(z))=∑J(ξ⟵σ(I)τ(J)(z)⊗vτ(J)(z)),
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 Theorem6.10 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 ℱell(M(n)) and consider certain operators on V⊗n⊗ℱell(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 setℱ=ℱell(M(n)). Recall the operators from Section 3.3, defined for any 𝔥-bialgebroid AR obtained from the FRST construction. When specializing to ℱ we get operators R(λ,z), R(ρ,z)∈End(V⊗V⊗ℱ), where V=ℂn. For z∈ℂ×, define the following linear operators on V⊗n⊗ℱ:
(6.19)Rij(λ,z):=limw→zθ(q2w)R(λ,w)(i,j,n+1),Rij(ρ,z):=limw→zθ(q2w)R(ρ,w)(i,j,n+1).
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).Letℰn denote the algebra of all functions
(6.20)F:(ℂ×)n→End(V⊗n⊗ℱ).
The symmetric group Sn acts on ℰn by
(6.21)σ(F(z))=(σ⊗Idℱ)∘F(σ(z))∘(σ-1⊗Idℱ),
for F∈ℰn and σ∈Sn. 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
(6.22)(23)(R12(λ,z1z2))=R13(λ,z1z3).
Consider the skew group algebra ℰn*Sn, defined as the algebra with underlying space ℰn⊗ℂSn, where ℂSn is the group algebra, with the multiplication
(6.23)(F(z)⊗σ)(G(z)⊗τ)=F(z)σ(G(z))⊗στ,
for σ,τ∈Sn, F,G∈ℰn. Since σ acts on ℰn by automorphisms, ℰn*Sn is an associative algebra. The constant function z↦IdV⊗n⊗ℱ⊗(1) is the unit element. Let Bn be the monoid (set with unital associative multiplication) generated by {s1,…,sn-1} and relations
(6.24)sisi+1si=si+1sisi+1,for1≤i≤n-2,sisj=sjsi,if|i-j|>1.
Let σi=(ii+1)∈Sn. We have an epimorphism π:Bn→Sn given by π(si)=σi,  π(1)=(1). Define
(6.25)W(1)=IdV⊗n⊗ℱ⊗(1),W(si)=Ri,i+1(λ-h≥i+2,zizi+1)⊗σi.
Here and below we use h≥k to denote the expression ∑j=knhj, and operators involving shifts hi such as Rn-2,n-1(λ-hn,zi/zi+1) are defined as in Section 3.3.Proposition 6.5.
W extends to a well-defined morphism of monoids, that is, a map
(6.26)W:Bn→ℰn*Sn
satisfying W(b1b2)=W(b1)W(b2) for any b1,b2∈Bn.Proof.
We have to show the relations(6.27)W(si)W(si+1)W(si)=W(si+1)W(si)W(si+1),(6.28)W(si)W(sj)=W(sj)W(si)if|i-j|>1.
Relation (6.27) follows from the QDYBE (2.8). For example, W(si)W(si+1)W(si) equals
(6.29)Ri,i+1(λ-h≥i+2,zizi+1)Ri,i+2(λ-h≥i+3,zizi+2)Ri+1,i+2(λ-hi-h≥i+3,zi+1zi+2)⊗σiσi+1σi.
Relation (6.28) is easy to check, using the 𝔥-invariance of R.Forb∈Bn we define Wb(λ,z)∈ℰn by
(6.30)W(b)=Wb(λ,z)⊗π(b).
From this and the product rule (6.23) it  follows that
(6.31)Wb1b2(λ,z)=Wb1(λ,z)·π(b1)(Wb2(λ,z)),
for b1,b2∈Bn. By replacing λ by ρ we get similarly operators Wb(ρ,z).Recall the operatorsL(z)∈End(V⊗ℱ) from Section 3.3. Define for z∈ℂ×, i∈[1,n],
(6.32)Li(z)=L(z)(i,n+1)∈End(V⊗n⊗ℱ).Ifi,j,k are distinct, then one can check that
(6.33)Rij(λ-hk,z)Lk(w)=Lk(w)Rij(λ,z),(6.34)Rij(ρ,z)Lk(w)=Lk(w)Rij(ρ+hk,z).Due to the RLL relations (3.8) we have
(6.35)R12(λ,z1z2)L1(z1)L2(z2)=L2(z2)L1(z1)R12(ρ+h1+h2,z1z2)
for any z1,z2∈ℂ×.Definetd∈Bn, d∈[1,n], recursively by
(6.36)td={td-1sd-1sd-2⋯s1,d>11,d=1.
Let τd be the image of td in Sn:
(6.37)τd:=π(td)=(12⋯dd+1⋯ndd-1⋯1d+1⋯n)∈Sn.Proposition 6.6.
Let1≤d≤n. For any z=(z1,…,zd)∈(ℂ×)d we have
(6.38)Wtd(λ,z)L1(z1)⋯Ld(zd)=Ld(zd)⋯L1(z1)Wtd(ρ+h≤d,z).Proof.
We use induction ond. The case d=1 is trivial, while d=2 is the RLL relation (6.35). If d>2, write td=td-1ud, where ud=sd-1sd-2⋯s1. Thus, by (6.31),
(6.39)Wtd(λ,z)=Wtd-1(λ,z)·τd-1(Wud(λ,z)).
We claim that
(6.40)τd-1(Wud(λ,z))L1(z1)⋯Ld(zd)=Ld(zd)L1(z1)⋯Ld-1(zd-1)τd-1(Wud(ρ+h≤d,z)).
For notational simplicity, set λ′=λ-h>d. A calculation using (6.30) shows that, compare  the proof of Proposition 6.5,
(6.41)Wud(λ,z)=Rd-1,d(λ′,zd-1zd)Rd-2,d(λ′-hd-1,zd-2zd)⋯R1,d(λ′-h[2,d-1],z1zd),
where h[a,b] means ∑a≤j≤bhj. Thus
(6.42)τd-1(Wud(λ,z))=R1,d(λ′,z1zd)R2,d(λ′-h1,z2zd)⋯Rd-1,d(λ′-h≤d-2,zd-1zd).
Using (6.33) and the RLL relation (6.35) repeatedly, we obtain (6.40). Now the proposition follows by induction on d, using that
(6.43)Wtd-1(λ,z)Ld(zd)=Ld(zd)Wtd-1(λ+hd,z)
which follows from (6.33).The operatorC(λ,z):=Wtn(λ,z) is called the Cherednik operator. For an operator F(z)∈ℰn we define its matrix elements F(z)x1,…,xna1,…,an∈ℱ by
(6.44)F(z)(ea1⊗⋯⊗ean⊗1)=∑x1,…,xn‍ex1⊗⋯⊗exn⊗F(z)x1,…,xna1,…,an.Proposition 6.7.
Let(6.45)α̃(λ,z)=limw→zθ(q2w)α(λ,w)=θ(z)θ(q2(λ+1))θ(q2λ).
Then
(6.46)C(λ,z)1,…,n1,…,n=∏i<jα̃(λij,zizj)=∏i<jqθ(zizj)·F[1,n](λ)F[1,n](λ).Proof.
The second equality follows from the definition (6.1) of FI and FI. We prove by induction on d that Wtd(λ,z)1,…,n1,…,n=∏i<j≤dα̃(λij,zi/zj). For d=2 we have td=s1 and Ws1(λ,z)1,…,n1,…,n=R12(λ-h>2,z1/z2)1,…,n1,…,n=α̃(λ12,z1/z2) as claimed. For d>2, using factorization (6.39) we have
(6.47)Wtd(λ,z)1,…,n1,…,n=∑x1,…,xn‍Wtd-1(λ,z)1,…,nx1,…,xnτd-1(Wud(λ,z))x1,…,xn1,…,n.
SinceWtd-1(λ,z) is a product of operators of the form σ(Rii+1(λ,zi/zi+1)) where 1≤i≤d-2 and σ∈Sn, σ(j)=j,j>d-1, and each of these operators preserve the subspace spanned by eτ(1)⊗⋯⊗eτ(d-1)⊗ed⊗⋯⊗en⊗a, where τ∈Sd-1 and a∈ℱ; the operator Wtd-1(λ,z) also preserves this subspace. This means that Wtd-1(λ,z)1,…,nx1,…,xn=0 unless xj=j for j≥d and {x1,…,xd-1}={1,…,d-1}. Furthermore, by (6.42),
(6.48)τd-1(Wud(λ,z))x1,…,xd-1,d,…,n1,…,n=∑y2,…,yd-1‍R̃x1d1y2(λ,z1zd)R̃x2y22y3(λ-ω(1),z2zd)⋯R̃xd-1yd-1d-1,d(λ-∑k≤d-2ω(k),zd-1zd).
Here R̃xyab(λ,z)=limw→zθ(q2w)Rxyab(λ,w). Since R̃xyab(λ,z)=0 unless {x,y}={a,b}, we deduce that, when {x1,…,xd-1}={1,…,d-1}, the terms in the sum (6.48) are zero unless xi=i for all i and yj=d for all j. Substituting into (6.47) the claim follows by induction.Lemma 6.8.
Fix2≤d≤n and i<d. Then there are elements b,c∈Bn such that td=sib and td=csi.Proof.
Sincet2=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′∈Bn such that td-1=sib′. Hence td=td-1sd-1⋯s1=sib′sd-1⋯s1. Thus we can take b=b′sd-1⋯s1. If i=d-1, write td=td-2sd-2⋯s1sd-1⋯s1. Then move each of the d-1 rightmost factors sd-1,…,s1 as far to the left as possible, using that sjsk=sksj when |j-k|>1. This gives
(6.49)td=td-2sd-2sd-1sd-3sd-2sd-4⋯s2s3s1s2s1.
Then use sjsj+1sj=sj+1sjsj+1 repeatedly, working from right to left, to obtain
(6.50)td=td-2sd-1sd-2sd-1sd-3sd-2⋯s4s2s3s1s2.
Finally, sd-1 can be moved to the left of td-2 since the latter is a product of sj's with j≤d-3.
To prove the existence ofc we note that Bn carries an involution *:Bn→Bn satisfying (a1a2)*=a2*a1* for any a1,a2∈Bn, defined by sj*=sj for j∈[1,n] and 1*=1. Thus it suffices to show that td*=td for any d. This is trivial for d=2,3. When d>3 we have, by induction on d,
(6.51)td*=(td-1sd-1⋯s1)*=s1⋯sd-1td-1=s1⋯sd-1td-2sd-2⋯s1=s1⋯sd-2td-2sd-1sd-2⋯s1=td-1*sd-1⋯s1=td(sincesd-1commuteswithtd-2).Proposition 6.9.
Letw=(z0,q2z0,…,q2(n-1)z0), where z0≠0 is arbitrary, and let σ,τ∈Sn. Then
(6.52)C(λ,w)σ(1),…,σ(n)τ(1),…,τ(n)=sgn[1,n](σ;λ)sgn[1,n](τ;λ)C(λ,w)1,…,n1,…,n.Proof.
First we claim that for allσ,τ∈Sn and each i∈[1,n],
(6.53)Wsi(λ,w)σσi(1),…,σσi(n)τ(1),…,τ(n)=σ(sgn[1,n](σi;λ))Wsi(λ,w)σ(1),…,σ(n)τ(1),…,τ(n),(6.54)Wsi(λ,w)σ(1),…,σ(n)τσi(1),…,τσi(n)=τ(sgn[1,n](σi;λ))Wsi(λ,w)σ(1),…,σ(n)τ(1),…,τ(n)=-Wsi(λ,w)σ(1),…,σ(n)τ(1),…,τ(n).
Indeed, assume that zi/zi+1=q-2 and that {a1,…,an}={b1,…,bn}=[1,n]. Then Wsi(λ,z)a1,…,anb1,…,bn≠0 if and only if {ai,ai+1}={bi,bi+1} in which case
(6.55)Wsi(λ,z)a1,…,anb1,…,bn=E(1)E(λaiai+1+1)E(λbi+1bi).
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 (σσi,τ) and (σ,τσi) where i is arbitrary.
Leti∈[1,n]. By Lemma 6.8 we have tn=sib for some b∈Bn. We have
(6.56)Wtn(λ,w)σσi(1),…,σσi(n)τ(1),…,τ(n)=(Wsi(λ,w)σi(Wb(λ,w)))σσi(1),…,σσi(n)τ(1),…,τ(n)=∑x1,…,xnWsi(λ,w)σσi(1),…,σσi(n)x1,…,xnσi(Wb(λ,w))x1,…,xnτ(1),…,τ(n).
As in the proof of Proposition 6.7, Wsi(λ,w)σσi(1),…,σσi(n)x1,…,xn is zero, if x1,…,xn is not a permutation of 1,…,n. Using (6.53) we obtain
(6.57)σ(sgn[1,n](σi;λ))∑x1,…,xn‍Wsi(λ,w)σ(1),…,σ(n)x1,…,xnσi(Wb(λ,w))x1,…,xnτ(1),…,τ(n)=σ(sgn[1,n](σi;λ))Wtn(λ,w)σ(1),…,σ(n)τ(1),…,τ(n).
Using the induction hypothesis and the relation sgn[1,n](σ;λ)σ(sgn[1,n](σi;λ))=sgn[1,n](σσi;λ) we obtain (6.52) for (σσi,τ).
For the other case, leti be arbitrary, and set j=τn(i). By Lemma 6.8 there is a c∈Bn such that tn=csj. Recall the surjective morphism π:Bn→Sn sending si to σi=(ii+1). Then σjπ(c)=π(c)σi. We have
(6.58)Wtn(λ,w)σ(1),…,σ(n)τσi(1),…,τσi(n)=(Wc(λ,w)·π(c)(Wsj(λ,w)))σ(1),…,σ(n)τσi(1),…τσi(n)=∑x1,…,xnWc(λ,w)σ(1),…,σ(n)x1,…,xnπ(c)(Wsj(λ,w))x1,…,xnτσi(1),…,τσi(n).
It is easy to check that σ(F(z))a1,…,anb1,…,bn=F(σ(z))aσ(1),…,aσ(n)bσ(1),…,bσ(n) for any F(z)∈ℰn and σ∈Sn. Define wi by (w1,…,wn)=w=(z0,q2z0,…,q2(n-1)z0). Then wi/wi+1=q-2 for each i. Set w′=(wπ(c)(1),…,wπ(c)(n)). For each i, wπ(c)(i)/wπ(c)(i+1)=wτn(i+1)/wτn(i)=q-2 also. Therefore
(6.59)Wtn(λ,w)σ(1),…,σ(n)τσi(1),…,τσi(n)=∑x1,…,xn‍Wc(λ,w)σ(1),…,σ(n)x1,…,xnWsj(λ,w′)xπ(c)(1),…,xπ(c)(n)τσiπ(c)(1),…,τσiπ(c)(n)=∑x1,…,xn‍Wc(λ,w)σ(1),…,σ(n)x1,…,xnWsj(λ,w′)xπ(c)(1),…,xπ(c)(n)τπ(c)σj(1),…,τπ(c)σj(n)=∑x1,…,xn‍Wc(λ,w)σ(1),…,σ(n)x1,…,xn(sgnσj)Wsj(λ,w′)xπ(c)(1),…,xπ(c)(n)τπ(c)(1),…,τπ(c)(n)=∑x1,…,xn‍Wc(λ,w)σ(1),…,σ(n)x1,…,xn(-1)π(c)(Wsj(λ,w))x1,…,xnτ(1),…,τ(n)=-Wtn(λ,w)σ(1),…,σ(n)τ(1),…,τ(n).
By the induction hypothesis it follows that (6.52) holds for (σ,τσi). This proves the formula (6.52).Theorem 6.10.
For any subsetsI,J⊆[1,n] and z∈ℂ×, the left and right elliptic minors coincide:
(6.60)ξ⟵IJ(z)=ξ⃗IJ(z).
We denote this common element by ξIJ(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 eij(z), i,j∈[1,d], z∈ℂ× and f(λ),f(ρ) with f∈M𝔥d*⊆M𝔥*, 𝔥d being the Cartan subalgebra of 𝔰𝔩(d), is isomorphic to ℱell(M(d)), we can also assume d=n. Identifying the matrix element 1,…,n1,…,n on both sides of (6.38) we get
(6.61)∑x1,…,xn‍C(λ,z)1,…,nx1,…,xnex1,1(z1)⋯exn,n(zn)=∑x1,…,xn‍en,xn(zn)⋯e1,x1(z1)C(ρ+h≤n,z)x1,…,xn1,…,n.
As in the proof of Proposition 6.7, C(λ,z)1,…,nx1,…,xn is zero if x1,…,xn is not a permutation of 1,…,n. Taking z=w=(z0,q2z0,…,q2(n-1)z0) and dividing both sides by ∏i<jqθ(wi/wj)=∏i<jqθ(q2(i-j)) we get
(6.62)F[1,n](λ)F[1,n](λ)∑σ∈Sn‍sgn[1,n](σ;λ)-1eσ(1)1(z0)⋯eσ(n)n(q2(n-1)z0)=F[1,n](ρ)F[1,n](ρ)∑τ∈Sn‍sgn[1,n](τ;ρ)enτ(n)(q2(n-1)z0)⋯e1σ(1)(z0).
Multiplying by F[1,n](ρ)/F[1,n](λ) and comparing with (6.13) and (6.14), we deduce that ξ⃗[1,n][1,n](z0)=ξ⟵[1,n][1,n](z0), as desired.


## 6.3. Laplace Expansions

Using the left (right)ℱell(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 Sl(I,J;ζ),Sr(I,J;ζ)∈M𝔥* by
(6.63)vI(q2#Jz)vJ(z)=Sl(I,J;ζ)vI∪J(z),wI(z)wJ(q2#Iz)=Sr(I,J;ζ)wI∪J(z).
That this is possible follows from the definitions (6.7) and (6.8) of vI(z),wI(z), and the commutation relations (5.8b)–(5.8d), (5.16). In particular Sl(I,J;ζ)=0=Sr(I,J;ζ), if I∩J≠∅.Theorem 6.11.
(i) LetI1,I2,J⊆[1,n], and set I=I1∪I2. Then
(6.64)Sl(I1,I2;λ)ξIJ(z)=∑J1∪J2=J‍Sl(J1,J2;ρ)ξI1J1(q2#I2z)ξI2J2(z).
(ii) LetJ1,J2,I⊆[1,n] and set J=J1∪J2. Then
(6.65)Sr(J1,J2;ρ)ξIJ(z)=∑I1∪I2=I‍Sr(I1,I2;λ)ξI1J1(z)ξI2J2(q2#J1z).Proof.
We have(6.66)ΔΛ(vI1(q2#I2z))ΔΛ(vI2(z))=∑J1,J2‍ξI1J1(q2#I2z)ξI2J2(z)⊗vJ1(q2#I2z)vJ2(z)=∑J1,J2‍ξI1J1(q2#I2z)ξI2J2(z)⊗Sl(J1,J2;ζ)vJ(z)=∑J(∑J1∪J2=J‍Sl(J1,J2;ρ)ξI1J1(q2#I2z)ξI2J2(z))⊗vJ(z).
On the other hand,
(6.67)ΔΛ(vI1(q2#I2z))ΔΛ(vI2(z))=ΔΛ(vI1(q2#I2z)vI2(z))=ΔΛ(Sl(I1,I2;ζ)vI(z))=∑J‍Sl(I1,I2;λ)ξIJ(z)⊗vJ(z).
Equating these expressions proves (6.64) since, by Proposition 5.5, the set {vJ(z):J⊆[1,n]} is linearly independent over M𝔥*. The second part is completely analogous, using the right comodule algebra Λ′ in place of Λ.In Section7.4 we will need the following lemma, relating the left and right signums Sl(I,J;ζ) and Sr(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.
LetI,J be two disjoint subsets of [1,n]. Then
(6.68)Sl(I,J;ζ+ω(I))=Sr(J,I;ζ)-1,
where ω(I)=∑i∈Iω(i).Proof.
First we claim that, we have the following explicit formulas:(6.69)Sl(I,J;ζ)=∏i∈I,j∈JE(ζji+1),(6.70)Sr(I,J;ζ)=∏i∈I,j∈JE(ζij)-1.
Recall the definition (6.7) of vI(z). Since E is odd, relation (5.8b) implies that
(6.71)vi(q2z)vj(z)=E(ζji+1)E(ζij+1)vj(q2z)vi(z).
Also, FJ(ζ) only involves ζij with i,j∈J so it commutes with any vk(z) with k∈I (since I∩J=∅). From these facts we obtain
(6.72)vI(q2#Jz)vJ(z)=FI(ζ)-1FJ(ζ)-1FI∪J(ζ)-1∏i∈I,j∈Ji<jE(ζji+1)E(ζij+1)vI∪J(z)=∏(i,j)∈Ki<jE(ζij+1)∏i∈I,j∈Ji<jE(ζji+1)E(ζij+1)vI∪J(z)=∏i∈I,j∈JE(ζji+1)vI∪J(z),
where K=(I×J)∪(J×I). This proves (6.69). Similarly one proves (6.70). Now we have
(6.73)Sl(J,I;ζ+ω(J))-1=∏i∈I,j∈JE((ζ+ω(J))ij+1)-1=∏i∈I,j∈JE(ζij)-1=Sr(I,J;ζ).
Here we used that for any i∈I,j∈J we have ω(J)(Eii)=0, ω(J)(Ejj)=1, and hence (ω(J))ij=-1.


## 7. The Cobraiding and the Elliptic Determinant

### 7.1. Cobraidings for𝔥-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.
Acobraiding on an 𝔥-bialgebroid A is a ℂ-bilinear map 〈·,·〉:A×A→D𝔥 such that, for any a,b,c∈A and f∈M𝔥*,
(7.1a)〈Aαβ,Aγδ〉⊆(D𝔥)α+γ,β+δ,(7.1b)〈μr(f)a,b〉=〈a,μl(f)b〉=fT0∘〈a,b〉,(7.1c)〈aμl(f),b〉=〈a,bμr(f)〉=〈a,b〉∘fT0,(7.1d)〈ab,c〉=∑i‍〈a,ci′〉Tβi〈b,ci′′〉,Δ(c)=∑ici′⊗ci′′,ci′′∈Aβiγ,(7.1e)〈a,bc〉=∑i‍〈ai′′,b〉Tβi〈ai′,c〉,Δ(a)=∑iai′⊗ai′′,ai′′∈Aβiγ,(7.1f)〈a,1〉=〈1,a〉=ε(a),(7.1g)∑ij‍μl(〈ai′,bj′〉1)ai′′bj′′=∑ijμr(〈ai′′,bj′′〉1)bj′ai′.
The following definition was given in unpublished notes by Rosengren [16]. The terminology is motivated by Proposition 7.6 concerning FRST algebras AR, but it makes sense for arbitrary 𝔥-bialgebroids.Definition 7.2.
A cobraiding〈·,·〉 on an 𝔥-bialgebroid A is called unitary if
(7.2)ε(ab)=∑(a),(b)〈a′,b′〉Tω12(a)+ω12(b)〈a′′,b′′〉
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′∈Aαγ for some α (or equivalently, if a′′∈Aγβ for some β).


### 7.2. Cobraidings for the FRST AlgebrasAR

Now letR:𝔥*×ℂ×→End𝔥(V⊗V) be a meromorphic function, and let AR be the 𝔥-bialgebroid associated to R as in Section 3.2.Proposition 7.3.
Assume thatφ:ℂ×→ℂ is a holomorphic function, not vanishing identically, such that, for each x,y,a,b∈X, z∈ℂ×, the limit limw→z(φ(w)Rxyab(ζ,w)) exists and defines a meromorphic function in M𝔥*. Then the following statements are equivalent:
(i)
there exists a cobraiding〈·,·〉:AR×AR→D𝔥 satisfying
(7.3)〈Lij(z1),Lkl(z2)〉=limw→z1/z2(φ(w)Rikjl(ζ,w))T-ω(i)-ω(k),(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 onAcop×A. See [14].
(b) Without the factorφ(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 onAR to account for spectral singularities in the R-matrix as follows.Calla∈ARspectrally homogenous if there exist k∈ℤ≥0 and z1,…,zk∈ℂ× such that
(7.4)a∈∑σ∈Sk‍∑il,jl∈X‍M𝔥*⊗M𝔥*Li1j1(zσ(1))⋯Likjk(zσ(k)).
The multiset {zi}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.Letφ:ℂ×→ℂ be holomorphic. For spectrally homogenous elements a,b∈AR, define the regularizing factorφ̂(a,b) by
(7.5)φ̂(a,b)=∏1≤i≤k,1≤j≤lφ(ziwj),
where {zi}i=sdeg(a), {wj}j=sdeg(b).Definition 7.5.
Letφ:ℂ×→ℂ be holomorphic. A cobraiding 〈·,·〉 on AR is unitary with respect toφ if
(7.6)φ̂(a,b)φ̂(b,a)ε(ab)=∑(a),(b)〈a′,b′〉Tω12(a)+ω12(b)〈a′′,b′′〉
for all spectrally homogenous a,b∈AR.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 thatR:𝔥*×ℂ×→Endℂ(V⊗V) satisfies the QDYBE and is unitary: R(ζ,z)R(ζ,z-1)(21)=IdV⊗V. Suppose that φ:ℂ×→ℂ is nonzero holomorphic such that limw→z(φ(w)Rxyab(ζ,w)) exists and is a holomorphic function in M𝔥*. Then the cobraiding 〈·,·〉 on AR 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,φ̂(a,b)〈a,b〉R=〈a,b〉 where 〈·,·〉R is the cobraiding, defined only for generic spectral values, determined by 〈Lij(z),Lkl(w)〉R=Rikjl(ζ,z/w)T-ω(i)-ω(k). Indeed, this claim follows by induction from the identities (7.1d) and  (7.1e) using that φ̂(a1,b)φ̂(a2,c)=φ̂(a3,bc) and φ̂(c,a1)φ̂(b,a2)=φ̂(cb,a3) for spectrally homogenous ai,b,c∈AR, the ai having the same spectral degree.
Since the R-matrixR is unitary, the statement of the lemma now follows from the identity
(7.7)ε(ab)=∑(a),(b)〈a′,b′〉RTω12(a)+ω12(b)〈a′′,b′′〉R
holding for generic spectral values which was proved by Rosengren [16].


#### 7.2.1. The Case ofℱell(M(n))

Specializing further to the algebra of interest,ℱell(M(n)), we obtain the following corollary.Corollary 7.7.
The𝔥-bialgebroid ℱell(M(n)) carries a cobraiding 〈·,·〉 satisfying
(7.8)〈eij(z),ekl(w)〉=R̃ikjl(ζ,zw)T-ω(i)-ω(k)∀z,w∈ℂ×,i,j∈[1,n],
where
(7.9)R̃ikjl(ζ,z)=limw→z(θ(q2w)Rikjl(ζ,w)).
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), R̃ 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 ℱell(M(n)) and how to resolve technical issues connected with the spectral singularities of the elliptic R-matrix.LetA=ℱell(M(n)). When I=J=[1,n] we set
(7.10)det(z)=ξIJ(z)
for z∈ℂ×, where ξIJ(z) is the elliptic minor given in Theorem 6.10. Thus one possible expression for det(z) is
(7.11)det(z)=∑σ∈SnF[1,n](ρ)σ(F[1,n](λ))eσ(1)1(z)eσ(2)2(q2z)⋯eσ(n)n(q2(n-1)z).Theorem 7.8.
(a)det(z) is a grouplike element of A for each z∈ℂ×, that is,
(7.12)Δ(det(z))=det(z)⊗det(z),ε(det(z))=1.
(b)
det(z) is a central element in ℱell(M(n)):
(7.13)[eij(z),det(w)]=[f(λ),det(w)]=[f(ρ),det(w)]=0
for all f∈M𝔥*, i,j∈[1,n] and all z,w∈ℂ×.Proof.
LetΛn(z)=M𝔥*vI(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,
(7.14)Δ(vI(z))=det(z)⊗vI(z).
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 thatdet(z)∈A00 and thus it commutes with f(ρ) and f(λ) for any f∈M𝔥*. To prove that it commutes with the generators eij(z) we need several lemmas which we now state and prove.
Lemma 7.9.
Fori,j∈[1,n], I,J⊆[1,n], #I=#J=2, we have
(7.15)〈ξIJ(w),eij(z)〉=0,ifwz∈pℤ,(7.16)〈eij(z),ξIJ(w)〉=0,ifq2wz∈pℤ.Proof.
LetI={i1,i2}, i1<i2, J={j1,j2}, j1<j2. Using the left expansion formula (6.13) and (7.1b), (7.1c) we have
(7.17)〈ξIJ(w),eij(z)〉=〈E(ρj1j2+1)E(λi1i2+1)ei2j2(q2w)ei1j1(w),eij(z)〉+[j1⟷j2]=E(ζj1j2+1+1)〈ei2j2(q2w)ei1j1(w),eij(z)〉1E(ζi1i2+1)+[j1⟷j2].
Thus we need to prove that for w/z∈pℤ, the first term is antisymmetric in j1,j2. By (7.1d),
(7.18)E(ζj1j2+1)〈ei2j2(q2w)ei1j1(w),eij(z)〉=E(ζj1j2+1)∑x〈ei2j2(q2w),eix(z)〉Tω(x)〈ei1j1(w),exj(z)〉=E(ζj1j2+1)∑xR̃i2ij2x(ζ,q2wz)R̃i1xj1j(ζ-ω(j2),wz)T-ω(j1)-ω(j2)-ω(j).
Take now w=pkz where k∈ℤ. One checks that
(7.19)R̃xyab(ζ,pk)=θ(q2pk)q2k(ζba+1-δab)δayδbx,
where δxy 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
(7.20)R̃jj1j1j(ζ-ω(j2),pk)=q2kζj2j1·R̃jj2j2j(ζ-ω(j1),pk),(7.21)E(ζj1j2+1)R̃i2ij2j1(ζ,q2pk)=-q2kζj1j2·E(ζj2j1+1)R̃i2ij1j2(ζ,q2pk).
Relation (7.20) can be proved directly from (7.19) while for (7.21) one can use that
(7.22)R̃xyab(ζ,pkz)R̃xyba(ζ,pkz)=q2kζbaR̃xyab(ζ,z)R̃xyba(ζ,z)
together with the relation
(7.23)E(ζj1j2+1)α̃(ζj2j1,q2)=-E(ζj2j1+1)β̃(ζj2j1,q2)
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 ξIJ(w) instead.Since the cobraiding depends holomorphically on the spectral variables, and all zeros of θ are simple and of the form pk, we conclude that the following limits exist for all z,w∈ℂ×, i,j,I,J, #I=#J=2:
(7.24)〈ξIJ(w),eij(z)〉′:=lim(z1,w1)→(z,w)〈ξIJ(w1),eij(z1)〉θ(z1/w1),〈eij(z),ξIJ(w)〉′:=lim(z1,w1)→(z,w)〈eij(z1),ξIJ(w1)〉θ(q2w1/z1).
Taking a=eij(z1),b=ξIJ(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
(7.25)ψ(z,w)ε(eij(z)ξIJ(w))=∑x,X〈ξIX(w),eix(z)〉′Tω(x)+ω(x1)+ω(x2)〈exj(z),ξXJ(w)〉′,
and interchanging a and b,
(7.26)ψ(z,w)ε(ξIJ(w)eij(z))=∑x,X〈eix(z),ξIX(w)〉′Tω(x)+ω(x1)+ω(x2)〈ξXJ(w),exj(z)〉′,
for all z,w∈ℂ×, where ψ:(ℂ×)2→ℂ is given by
(7.27)ψ(z,w)=θ(q2zw)θ(q4wz).
We are now ready to prove the key identity.Lemma 7.10.
For anyi,j∈[1,n], I,J⊆[1,n], #I=#J=2 and any z,w∈ℂ×, q2w/z∉pℤ we have
(7.28)ψ(z,w)∑x,Xμl(〈ξIX(w),eix(z)〉′1)ξXJ(w)exj(z)=ψ(z,w)∑x,Xμr(〈ξXJ(w),exj(z)〉′1)eix(z)ξIX(w).Proof.
Using the counit axiom followed by (7.25) we have
(7.29)ψ(z,w)eij(z)ξIJ(w)=ψ(z,w)∑x,X‍μl(ε(eix(z)ξIX(w))1)exj(z)ξXJ(w)=∑x,y,X,Y‍μl(〈ξIY(w),eiy(z)〉′1)μl(〈eyx(z),ξYX(w)〉′1)exj(z)ξXJ(w).
Applying the identity obtained by dividing by θ(q2w/z) in both sides of the cobraiding identity (7.1g) with a=eyj(z),b=ξYJ(w) in the right hand side of (7.29) gives
(7.30)ψ(z,w)eij(z)ξIJ(w)=∑x,y,X,Y‍μl(〈ξIY(w),eiy(z)〉′1)μr(〈exj(z),ξXJ(w)〉′1)ξYX(w)eyx(z).
Now multiply both sides by μr(〈ξJK(w),ejk(z)〉′1), and sum over j,J. After applying (7.26) in the right hand side we get
(7.31)ψ(z,w)∑j,J‍μr(〈ξJK(w),ejk(z)〉′1)eij(z)ξIJ(w)=ψ(z,w)∑x,y,X,Y‍μl(〈ξIY(w),eiy(z)〉′1)μr(ε(ξXK(w)exk(z))1)ξYX(w)eyx(z).
By the counit axiom the last expression equals
(7.32)ψ(z,w)∑y,Yμl(〈ξIY(w),eiy(z)〉′1)ξYK(w)eyk(z).Lemma 7.11.
(a) The limit(7.33)〈det(w),eij(z)〉′:=lim(z1,w1)→(z,w)〈det(w1),eij(z1)〉θ(w1/z1)θ(q2w1/z1)⋯θ(q2(n-2)w1/z1)
exists for any z,w∈ℂ×.
(b)
We have(7.34)μl(〈det(w),e11(z)〉′1)det(w)e11(z)=μr(〈det(w),e11(z)〉′1)e11(z)det(w)
for any z,w∈ℂ×.Proof.
(a) We must show that〈det(w),eij(z)〉 vanishes for q2kw/z∈pℤ, where k∈{0,1,…,n-2}. Applying the Laplace expansion (6.65) twice we get
(7.35)det(w)=∑I1∪I2∪I3=[1,n]‍Sr(I1,I2,I3;λ)ξI1J1(w)ξI2J2(q2#J1w)ξI3J3(q2(#J1+2)w),
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 〈ξXY(q2kw),exy(z)〉, where #X=#Y=2, which indeed vanishes for q2kw/z∈pℤ by Lemma 7.9.
(b) By (7.1a), 〈det(w),exy(z)〉=0, if x≠y. Thus (7.34) can be written
(7.36)∑xμl(〈det(w),e1x(z)〉′1)det(w)ex1(z)=∑xμr(〈det(w),ex1(z)〉′1)e1x(z)det(w).
If q2kw/z∉pℤ 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 ∏k=0n-2θ(qkw/z).
So assumeq2kw/z∈pℤ 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
(7.37)∑xμl(〈det(w),e1x(z)〉′1)det(w)ex1(z)=1∏m=0,m≠kn-2θ(q2mw/z)∑(a),(b)‍μl(〈a1′,b′〉1)a1′′μl(〈a2′,b′′〉′1)a2′′μl(〈a3′,b(3)〉1)a3′′b(4).
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(7.38)〈det(w),e11(z)〉′1=qn-1θ(q2nwz).Proof.
Expandingdet(w) using the left expansion formula (6.13) with
(7.39)σ=(12⋯nnn-1⋯1),
the longest element in Sn, and applying (7.1d) repeatedly we have (putting I=[1,n])
(7.40)〈det(w),e11(z)〉=∑τ∈Sn〈τ(FI(ρ))σ(FI(λ))e1τ(n)(q2(n-1)w)⋯enτ(1)(w),e11(z)〉=∑τ∈Snx1,…,xn-1τ(FI(ζ))〈e1τ(n)(q2(n-1)w),e1x1(z)〉Tω(x1)⋯Tω(xn-1)〈enτ(1)(w),exn-11(z)〉σ(FI(ζ)-1).
One proves inductively that in all nonzero terms we haveτ(j)=n+1-j and xn-j=1 for all 1≤j≤n-1 by looking from right to left: 〈enτ(1)(w),exn-11(z)〉=R̃nxn-1τ(1)1(ζ,w/z)T-ω(n)-ω(xn-1) which, if 1≠n, is nonzero only for τ(1)=n and xn-1=1 by (4.3). Then looking at the second pairing from the right we see that τ(2)=n-1 and xn-2=1 if it is nonzero, and so on. Thus only the term τ=σ and x1=⋯=xn-1=1 survives and it equals
(7.41)σ(FI(ζ))R̃1111(ζ,q2(n-1)wz)T-ω(1)R̃2121(ζ,q2(n-2)wz)T-ω(2)⋯T-ω(n-1)R̃n1n1(ζ,wz)T-ω(n)-ω(1)σ(FI(ζ))-1.
Using that σ(FI(ζ))=∏i<jE(ζji+1) and that R̃j1j1(ζ-ω(1),z)=α̃(ζj1+1,z)=qθ(z)(E(ζj1+2)/E(ζj1+1)) we get
(7.42)qn-1θ(q2nwz)·θ(q2(n-2)wz)θ(q2(n-3)wz)⋯θ(wz)·∏i<jE(ζji+1)∏1<jE(ζj1+2)E(ζj1+1)T-ω(1)∏i<jE(ζji+1)-1.
The factors involving the dynamical variable ζ cancel and the claim follows.
By Lemmas7.11(b) and 7.12 we conclude that det(w) commutes with e11(z) if q2nw/z∉pℤ. By applying an automorphism from the Sn×Sn-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 eij(z) as long as q2nw/z∉pℤ.
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:ℱell(M(n))→ℱell(M(n)) such that T(ab)=T(b)T(a) for all a,b∈ℱell(M(n)), given by
(7.43)T(eij(z))=eij(z-1),T(f(λ))=f(-λ),T(f(ρ))=f(-ρ),
for all f∈M𝔥*, 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),eij(z)]=0 if q2nw/z∉pℤ. Assume q2nw/z∈pℤ. Then
(7.44)T([det(w),eij(z)])=[eij(z-1),det(q-2(n-1)w-1)]=0
since q-2(n-1)w-1/z-1=q2(q2nw/z)-1∉pℤ. 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.
An𝔥-Hopf algebroid is an 𝔥-bialgebroid A equipped with a ℂ-linear map S:A→A, called the antipode, such that
(7.45)S(μr(f)a)=S(a)μl(f),S(aμl(f))=μr(f)S(a),a∈A,f∈M𝔥*,(7.46)m∘(Id⊗S)∘Δ(a)=μl(ε(a)1),a∈A,m∘(S⊗Id)∘Δ(a)=μr(Tα(ε(a)1)),a∈Aαβ,
where m denotes the multiplication, and ε(a)1 is the result of applying the difference operator ε(a) to the constant function 1∈M𝔥*.Letℱell(M(n))[det(z)-1:z∈ℂ×] be the polynomial algebra in uncountably many variables det(z)-1, z∈ℂ×, with coefficients in ℱell(M(n)). We define ℱell(GL(n)) to be
(7.47)ℱell(M(n))[det(z)-1:z∈ℂ×]J,
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 ℱell(M(n)) to ℱell(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 ℱell(GL(n)). We extend the comultiplication and counit by requiring that det(z)-1 is grouplike for each z∈ℂ×, that is,
(7.48)Δ(det(z)-1)=det(z)-1⊗det(z)-1,ε(det(z)-1)=1.
Here 1 denotes the identity operator in D𝔥. One verifies that J is a coideal and that ε(J)=0, which induces operations Δ,ε on ℱell(GL(n)). In this way ℱell(GL(n)) becomes an 𝔥-bialgebroid. This algebra is nontrivial since ε(J)=0 implies that J is a proper ideal.Fori∈[1,n] we set î={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 ℱell(GL(n)) also by f(λ) and f(ρ) respectively.Theorem 7.14.
ℱell(GL(n)) is an 𝔥-Hopf algebroid with antipode S given by
(7.49)S(f(λ))=f(ρ),S(f(ρ))=f(λ),(7.50)S(eij(z))=Sr(ĵ,{j};λ)Sr(î,{i};ρ)det(q-2(n-1)z)-1ξĵî(q-2(n-1)z),(7.51)S(det(z)-1)=det(z),
for all f∈M𝔥*, i,j∈[1,n] and z∈ℂ×.Proof.
We proceed in steps.
Step 1.
DefineS on the generators of ℱell(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∈M𝔥*, this is easily checked. Let a=eij(z). Using the right Laplace expansion (6.65) with J1=î, J2={j}, I=[1,n] and z replaced by q-2(n-1)z we obtain
(7.52)∑x=1n‍S(eix(z))exj(z)=δij.
Similarly, using the left Laplace expansion (6.64) with I1={i}, I2=ĵ, J=[1,n], and z replaced by q-2(n-1)z, together with the identity (6.68), we get
(7.53)∑x=1n‍eix(z)S(exj(z))=δij,
using also the crucial fact that, by Theorem 7.8, eij(z) commutes in ℱell(M(n)) with det(q-2(n-1)z) and hence in ℱell(GL(n)) with det(q-2(n-1)z)-1. This proves that the antipode axiom (7.46) is satisfied for a=eij(z).
Step 2.
We show thatS extends to a ℂ-linear map S:ℱell(M(n))→ℱell(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 ℱell(M(n)). Since S(eij(z))∈ℱell(GL(n))ω(ĵ),ω(î) and ω(i)+ω(î)=0, we have
(7.54)S(eij(z))S(f(λ))=S(eij(z))f(ρ)=f(ρ-ω(î))S(eij(z))=f(ρ+ω(i))S(eij(z))=S(f(λ+ω(i)))S(eij(z)).
Similarly, S(eij(z))S(f(ρ))=S(f(ρ+ω(j)))S(eij(z)) so relations (4.1) are preserved. Next, consider the RLL relation
(7.55)∑x,y=1n‍Racxy(λ,z1z2)exb(z1)eyd(z2)=∑x,y=1n‍Rxybd(ρ,z1z2)ecy(z2)eax(z1).
Multiply (7.55) from the left by S(eic(z2)) and from the right by S(edk(z2)), sum over c,d, and use (7.52), (7.53) to obtain
(7.56)∑x,c‍Racxk(λ-ω(ĉ),z1z2)S(eic(z2))exb(z1)=∑x,d‍Rxibd(ρ-ω(î),z1z2)eax(z1)S(edk(z2)).
Then multiply from the left by S(eja(z1)) and from the right by S(ebl(z1)), sum over a,b, and use (7.52), (7.53) again to get
(7.57)∑a,cRaclk(λ-ω(â)-ω(ĉ),z1z2)S(eja(z1))S(eic(z2))=∑b,d‍Rjibd(ρ-ω(ĵ)-ω(î),z1z2)S(edk(z2))S(ebl(z1)).
Since S(eij(z))∈ℱell(GL(n))ĵ,î and Rjibd(ρ-ω(ĵ)-ω(î),z1/z2)=Rjibd(ρ-ω(b̂)-ω(d̂),z1/z2) by the 𝔥-invariance of R, (7.57) can be rewritten
(7.58)∑a,c‍S(eja(z1))S(eic(z2))Raclk(λ,z1z2)=∑b,d‍S(edk(z2))S(ebl(z1))Rjibd(ρ,z1z2).
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 ℱell(M(n)) and S(ab)=S(b)S(a) for all a,b∈ℱell(M(n)), it follows that (7.46) holds for any a∈ℱell(M(n)). By taking in particular a=det(z) we get
(7.59)det(z)S(det(z))=1,S(det(z))det(z)=1,
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:ℱell(GL(n))→ℱell(GL(n)) satisfying the antipode axiom (7.46) on ℱell(M(n)) and on det(z)-1. Hence (7.46) holds for any a∈ℱell(GL(n)).


## 7.1. Cobraidings for𝔥-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.
Acobraiding on an 𝔥-bialgebroid A is a ℂ-bilinear map 〈·,·〉:A×A→D𝔥 such that, for any a,b,c∈A and f∈M𝔥*,
(7.1a)〈Aαβ,Aγδ〉⊆(D𝔥)α+γ,β+δ,(7.1b)〈μr(f)a,b〉=〈a,μl(f)b〉=fT0∘〈a,b〉,(7.1c)〈aμl(f),b〉=〈a,bμr(f)〉=〈a,b〉∘fT0,(7.1d)〈ab,c〉=∑i‍〈a,ci′〉Tβi〈b,ci′′〉,Δ(c)=∑ici′⊗ci′′,ci′′∈Aβiγ,(7.1e)〈a,bc〉=∑i‍〈ai′′,b〉Tβi〈ai′,c〉,Δ(a)=∑iai′⊗ai′′,ai′′∈Aβiγ,(7.1f)〈a,1〉=〈1,a〉=ε(a),(7.1g)∑ij‍μl(〈ai′,bj′〉1)ai′′bj′′=∑ijμr(〈ai′′,bj′′〉1)bj′ai′.
The following definition was given in unpublished notes by Rosengren [16]. The terminology is motivated by Proposition 7.6 concerning FRST algebras AR, but it makes sense for arbitrary 𝔥-bialgebroids.Definition 7.2.
A cobraiding〈·,·〉 on an 𝔥-bialgebroid A is called unitary if
(7.2)ε(ab)=∑(a),(b)〈a′,b′〉Tω12(a)+ω12(b)〈a′′,b′′〉
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′∈Aαγ for some α (or equivalently, if a′′∈Aγβ for some β).


## 7.2. Cobraidings for the FRST AlgebrasAR

Now letR:𝔥*×ℂ×→End𝔥(V⊗V) be a meromorphic function, and let AR be the 𝔥-bialgebroid associated to R as in Section 3.2.Proposition 7.3.
Assume thatφ:ℂ×→ℂ is a holomorphic function, not vanishing identically, such that, for each x,y,a,b∈X, z∈ℂ×, the limit limw→z(φ(w)Rxyab(ζ,w)) exists and defines a meromorphic function in M𝔥*. Then the following statements are equivalent:
(i)
there exists a cobraiding〈·,·〉:AR×AR→D𝔥 satisfying
(7.3)〈Lij(z1),Lkl(z2)〉=limw→z1/z2(φ(w)Rikjl(ζ,w))T-ω(i)-ω(k),(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 onAcop×A. See [14].
(b) Without the factorφ(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 onAR to account for spectral singularities in the R-matrix as follows.Calla∈ARspectrally homogenous if there exist k∈ℤ≥0 and z1,…,zk∈ℂ× such that
(7.4)a∈∑σ∈Sk‍∑il,jl∈X‍M𝔥*⊗M𝔥*Li1j1(zσ(1))⋯Likjk(zσ(k)).
The multiset {zi}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.Letφ:ℂ×→ℂ be holomorphic. For spectrally homogenous elements a,b∈AR, define the regularizing factorφ̂(a,b) by
(7.5)φ̂(a,b)=∏1≤i≤k,1≤j≤lφ(ziwj),
where {zi}i=sdeg(a), {wj}j=sdeg(b).Definition 7.5.
Letφ:ℂ×→ℂ be holomorphic. A cobraiding 〈·,·〉 on AR is unitary with respect toφ if
(7.6)φ̂(a,b)φ̂(b,a)ε(ab)=∑(a),(b)〈a′,b′〉Tω12(a)+ω12(b)〈a′′,b′′〉
for all spectrally homogenous a,b∈AR.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 thatR:𝔥*×ℂ×→Endℂ(V⊗V) satisfies the QDYBE and is unitary: R(ζ,z)R(ζ,z-1)(21)=IdV⊗V. Suppose that φ:ℂ×→ℂ is nonzero holomorphic such that limw→z(φ(w)Rxyab(ζ,w)) exists and is a holomorphic function in M𝔥*. Then the cobraiding 〈·,·〉 on AR 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,φ̂(a,b)〈a,b〉R=〈a,b〉 where 〈·,·〉R is the cobraiding, defined only for generic spectral values, determined by 〈Lij(z),Lkl(w)〉R=Rikjl(ζ,z/w)T-ω(i)-ω(k). Indeed, this claim follows by induction from the identities (7.1d) and  (7.1e) using that φ̂(a1,b)φ̂(a2,c)=φ̂(a3,bc) and φ̂(c,a1)φ̂(b,a2)=φ̂(cb,a3) for spectrally homogenous ai,b,c∈AR, the ai having the same spectral degree.
Since the R-matrixR is unitary, the statement of the lemma now follows from the identity
(7.7)ε(ab)=∑(a),(b)〈a′,b′〉RTω12(a)+ω12(b)〈a′′,b′′〉R
holding for generic spectral values which was proved by Rosengren [16].


### 7.2.1. The Case ofℱell(M(n))

Specializing further to the algebra of interest,ℱell(M(n)), we obtain the following corollary.Corollary 7.7.
The𝔥-bialgebroid ℱell(M(n)) carries a cobraiding 〈·,·〉 satisfying
(7.8)〈eij(z),ekl(w)〉=R̃ikjl(ζ,zw)T-ω(i)-ω(k)∀z,w∈ℂ×,i,j∈[1,n],
where
(7.9)R̃ikjl(ζ,z)=limw→z(θ(q2w)Rikjl(ζ,w)).
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), R̃ is regular in z and apply Propositions 7.3 and 7.6.


## 7.2.1. The Case ofℱell(M(n))

Specializing further to the algebra of interest,ℱell(M(n)), we obtain the following corollary.Corollary 7.7.
The𝔥-bialgebroid ℱell(M(n)) carries a cobraiding 〈·,·〉 satisfying
(7.8)〈eij(z),ekl(w)〉=R̃ikjl(ζ,zw)T-ω(i)-ω(k)∀z,w∈ℂ×,i,j∈[1,n],
where
(7.9)R̃ikjl(ζ,z)=limw→z(θ(q2w)Rikjl(ζ,w)).
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), R̃ 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 ℱell(M(n)) and how to resolve technical issues connected with the spectral singularities of the elliptic R-matrix.LetA=ℱell(M(n)). When I=J=[1,n] we set
(7.10)det(z)=ξIJ(z)
for z∈ℂ×, where ξIJ(z) is the elliptic minor given in Theorem 6.10. Thus one possible expression for det(z) is
(7.11)det(z)=∑σ∈SnF[1,n](ρ)σ(F[1,n](λ))eσ(1)1(z)eσ(2)2(q2z)⋯eσ(n)n(q2(n-1)z).Theorem 7.8.
(a)det(z) is a grouplike element of A for each z∈ℂ×, that is,
(7.12)Δ(det(z))=det(z)⊗det(z),ε(det(z))=1.
(b)
det(z) is a central element in ℱell(M(n)):
(7.13)[eij(z),det(w)]=[f(λ),det(w)]=[f(ρ),det(w)]=0
for all f∈M𝔥*, i,j∈[1,n] and all z,w∈ℂ×.Proof.
LetΛn(z)=M𝔥*vI(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,
(7.14)Δ(vI(z))=det(z)⊗vI(z).
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 thatdet(z)∈A00 and thus it commutes with f(ρ) and f(λ) for any f∈M𝔥*. To prove that it commutes with the generators eij(z) we need several lemmas which we now state and prove.
Lemma 7.9.
Fori,j∈[1,n], I,J⊆[1,n], #I=#J=2, we have
(7.15)〈ξIJ(w),eij(z)〉=0,ifwz∈pℤ,(7.16)〈eij(z),ξIJ(w)〉=0,ifq2wz∈pℤ.Proof.
LetI={i1,i2}, i1<i2, J={j1,j2}, j1<j2. Using the left expansion formula (6.13) and (7.1b), (7.1c) we have
(7.17)〈ξIJ(w),eij(z)〉=〈E(ρj1j2+1)E(λi1i2+1)ei2j2(q2w)ei1j1(w),eij(z)〉+[j1⟷j2]=E(ζj1j2+1+1)〈ei2j2(q2w)ei1j1(w),eij(z)〉1E(ζi1i2+1)+[j1⟷j2].
Thus we need to prove that for w/z∈pℤ, the first term is antisymmetric in j1,j2. By (7.1d),
(7.18)E(ζj1j2+1)〈ei2j2(q2w)ei1j1(w),eij(z)〉=E(ζj1j2+1)∑x〈ei2j2(q2w),eix(z)〉Tω(x)〈ei1j1(w),exj(z)〉=E(ζj1j2+1)∑xR̃i2ij2x(ζ,q2wz)R̃i1xj1j(ζ-ω(j2),wz)T-ω(j1)-ω(j2)-ω(j).
Take now w=pkz where k∈ℤ. One checks that
(7.19)R̃xyab(ζ,pk)=θ(q2pk)q2k(ζba+1-δab)δayδbx,
where δxy 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
(7.20)R̃jj1j1j(ζ-ω(j2),pk)=q2kζj2j1·R̃jj2j2j(ζ-ω(j1),pk),(7.21)E(ζj1j2+1)R̃i2ij2j1(ζ,q2pk)=-q2kζj1j2·E(ζj2j1+1)R̃i2ij1j2(ζ,q2pk).
Relation (7.20) can be proved directly from (7.19) while for (7.21) one can use that
(7.22)R̃xyab(ζ,pkz)R̃xyba(ζ,pkz)=q2kζbaR̃xyab(ζ,z)R̃xyba(ζ,z)
together with the relation
(7.23)E(ζj1j2+1)α̃(ζj2j1,q2)=-E(ζj2j1+1)β̃(ζj2j1,q2)
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 ξIJ(w) instead.Since the cobraiding depends holomorphically on the spectral variables, and all zeros of θ are simple and of the form pk, we conclude that the following limits exist for all z,w∈ℂ×, i,j,I,J, #I=#J=2:
(7.24)〈ξIJ(w),eij(z)〉′:=lim(z1,w1)→(z,w)〈ξIJ(w1),eij(z1)〉θ(z1/w1),〈eij(z),ξIJ(w)〉′:=lim(z1,w1)→(z,w)〈eij(z1),ξIJ(w1)〉θ(q2w1/z1).
Taking a=eij(z1),b=ξIJ(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
(7.25)ψ(z,w)ε(eij(z)ξIJ(w))=∑x,X〈ξIX(w),eix(z)〉′Tω(x)+ω(x1)+ω(x2)〈exj(z),ξXJ(w)〉′,
and interchanging a and b,
(7.26)ψ(z,w)ε(ξIJ(w)eij(z))=∑x,X〈eix(z),ξIX(w)〉′Tω(x)+ω(x1)+ω(x2)〈ξXJ(w),exj(z)〉′,
for all z,w∈ℂ×, where ψ:(ℂ×)2→ℂ is given by
(7.27)ψ(z,w)=θ(q2zw)θ(q4wz).
We are now ready to prove the key identity.Lemma 7.10.
For anyi,j∈[1,n], I,J⊆[1,n], #I=#J=2 and any z,w∈ℂ×, q2w/z∉pℤ we have
(7.28)ψ(z,w)∑x,Xμl(〈ξIX(w),eix(z)〉′1)ξXJ(w)exj(z)=ψ(z,w)∑x,Xμr(〈ξXJ(w),exj(z)〉′1)eix(z)ξIX(w).Proof.
Using the counit axiom followed by (7.25) we have
(7.29)ψ(z,w)eij(z)ξIJ(w)=ψ(z,w)∑x,X‍μl(ε(eix(z)ξIX(w))1)exj(z)ξXJ(w)=∑x,y,X,Y‍μl(〈ξIY(w),eiy(z)〉′1)μl(〈eyx(z),ξYX(w)〉′1)exj(z)ξXJ(w).
Applying the identity obtained by dividing by θ(q2w/z) in both sides of the cobraiding identity (7.1g) with a=eyj(z),b=ξYJ(w) in the right hand side of (7.29) gives
(7.30)ψ(z,w)eij(z)ξIJ(w)=∑x,y,X,Y‍μl(〈ξIY(w),eiy(z)〉′1)μr(〈exj(z),ξXJ(w)〉′1)ξYX(w)eyx(z).
Now multiply both sides by μr(〈ξJK(w),ejk(z)〉′1), and sum over j,J. After applying (7.26) in the right hand side we get
(7.31)ψ(z,w)∑j,J‍μr(〈ξJK(w),ejk(z)〉′1)eij(z)ξIJ(w)=ψ(z,w)∑x,y,X,Y‍μl(〈ξIY(w),eiy(z)〉′1)μr(ε(ξXK(w)exk(z))1)ξYX(w)eyx(z).
By the counit axiom the last expression equals
(7.32)ψ(z,w)∑y,Yμl(〈ξIY(w),eiy(z)〉′1)ξYK(w)eyk(z).Lemma 7.11.
(a) The limit(7.33)〈det(w),eij(z)〉′:=lim(z1,w1)→(z,w)〈det(w1),eij(z1)〉θ(w1/z1)θ(q2w1/z1)⋯θ(q2(n-2)w1/z1)
exists for any z,w∈ℂ×.
(b)
We have(7.34)μl(〈det(w),e11(z)〉′1)det(w)e11(z)=μr(〈det(w),e11(z)〉′1)e11(z)det(w)
for any z,w∈ℂ×.Proof.
(a) We must show that〈det(w),eij(z)〉 vanishes for q2kw/z∈pℤ, where k∈{0,1,…,n-2}. Applying the Laplace expansion (6.65) twice we get
(7.35)det(w)=∑I1∪I2∪I3=[1,n]‍Sr(I1,I2,I3;λ)ξI1J1(w)ξI2J2(q2#J1w)ξI3J3(q2(#J1+2)w),
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 〈ξXY(q2kw),exy(z)〉, where #X=#Y=2, which indeed vanishes for q2kw/z∈pℤ by Lemma 7.9.
(b) By (7.1a), 〈det(w),exy(z)〉=0, if x≠y. Thus (7.34) can be written
(7.36)∑xμl(〈det(w),e1x(z)〉′1)det(w)ex1(z)=∑xμr(〈det(w),ex1(z)〉′1)e1x(z)det(w).
If q2kw/z∉pℤ 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 ∏k=0n-2θ(qkw/z).
So assumeq2kw/z∈pℤ 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
(7.37)∑xμl(〈det(w),e1x(z)〉′1)det(w)ex1(z)=1∏m=0,m≠kn-2θ(q2mw/z)∑(a),(b)‍μl(〈a1′,b′〉1)a1′′μl(〈a2′,b′′〉′1)a2′′μl(〈a3′,b(3)〉1)a3′′b(4).
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(7.38)〈det(w),e11(z)〉′1=qn-1θ(q2nwz).Proof.
Expandingdet(w) using the left expansion formula (6.13) with
(7.39)σ=(12⋯nnn-1⋯1),
the longest element in Sn, and applying (7.1d) repeatedly we have (putting I=[1,n])
(7.40)〈det(w),e11(z)〉=∑τ∈Sn〈τ(FI(ρ))σ(FI(λ))e1τ(n)(q2(n-1)w)⋯enτ(1)(w),e11(z)〉=∑τ∈Snx1,…,xn-1τ(FI(ζ))〈e1τ(n)(q2(n-1)w),e1x1(z)〉Tω(x1)⋯Tω(xn-1)〈enτ(1)(w),exn-11(z)〉σ(FI(ζ)-1).
One proves inductively that in all nonzero terms we haveτ(j)=n+1-j and xn-j=1 for all 1≤j≤n-1 by looking from right to left: 〈enτ(1)(w),exn-11(z)〉=R̃nxn-1τ(1)1(ζ,w/z)T-ω(n)-ω(xn-1) which, if 1≠n, is nonzero only for τ(1)=n and xn-1=1 by (4.3). Then looking at the second pairing from the right we see that τ(2)=n-1 and xn-2=1 if it is nonzero, and so on. Thus only the term τ=σ and x1=⋯=xn-1=1 survives and it equals
(7.41)σ(FI(ζ))R̃1111(ζ,q2(n-1)wz)T-ω(1)R̃2121(ζ,q2(n-2)wz)T-ω(2)⋯T-ω(n-1)R̃n1n1(ζ,wz)T-ω(n)-ω(1)σ(FI(ζ))-1.
Using that σ(FI(ζ))=∏i<jE(ζji+1) and that R̃j1j1(ζ-ω(1),z)=α̃(ζj1+1,z)=qθ(z)(E(ζj1+2)/E(ζj1+1)) we get
(7.42)qn-1θ(q2nwz)·θ(q2(n-2)wz)θ(q2(n-3)wz)⋯θ(wz)·∏i<jE(ζji+1)∏1<jE(ζj1+2)E(ζj1+1)T-ω(1)∏i<jE(ζji+1)-1.
The factors involving the dynamical variable ζ cancel and the claim follows.
By Lemmas7.11(b) and 7.12 we conclude that det(w) commutes with e11(z) if q2nw/z∉pℤ. By applying an automorphism from the Sn×Sn-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 eij(z) as long as q2nw/z∉pℤ.
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:ℱell(M(n))→ℱell(M(n)) such that T(ab)=T(b)T(a) for all a,b∈ℱell(M(n)), given by
(7.43)T(eij(z))=eij(z-1),T(f(λ))=f(-λ),T(f(ρ))=f(-ρ),
for all f∈M𝔥*, 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),eij(z)]=0 if q2nw/z∉pℤ. Assume q2nw/z∈pℤ. Then
(7.44)T([det(w),eij(z)])=[eij(z-1),det(q-2(n-1)w-1)]=0
since q-2(n-1)w-1/z-1=q2(q2nw/z)-1∉pℤ. 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.
An𝔥-Hopf algebroid is an 𝔥-bialgebroid A equipped with a ℂ-linear map S:A→A, called the antipode, such that
(7.45)S(μr(f)a)=S(a)μl(f),S(aμl(f))=μr(f)S(a),a∈A,f∈M𝔥*,(7.46)m∘(Id⊗S)∘Δ(a)=μl(ε(a)1),a∈A,m∘(S⊗Id)∘Δ(a)=μr(Tα(ε(a)1)),a∈Aαβ,
where m denotes the multiplication, and ε(a)1 is the result of applying the difference operator ε(a) to the constant function 1∈M𝔥*.Letℱell(M(n))[det(z)-1:z∈ℂ×] be the polynomial algebra in uncountably many variables det(z)-1, z∈ℂ×, with coefficients in ℱell(M(n)). We define ℱell(GL(n)) to be
(7.47)ℱell(M(n))[det(z)-1:z∈ℂ×]J,
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 ℱell(M(n)) to ℱell(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 ℱell(GL(n)). We extend the comultiplication and counit by requiring that det(z)-1 is grouplike for each z∈ℂ×, that is,
(7.48)Δ(det(z)-1)=det(z)-1⊗det(z)-1,ε(det(z)-1)=1.
Here 1 denotes the identity operator in D𝔥. One verifies that J is a coideal and that ε(J)=0, which induces operations Δ,ε on ℱell(GL(n)). In this way ℱell(GL(n)) becomes an 𝔥-bialgebroid. This algebra is nontrivial since ε(J)=0 implies that J is a proper ideal.Fori∈[1,n] we set î={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 ℱell(GL(n)) also by f(λ) and f(ρ) respectively.Theorem 7.14.
ℱell(GL(n)) is an 𝔥-Hopf algebroid with antipode S given by
(7.49)S(f(λ))=f(ρ),S(f(ρ))=f(λ),(7.50)S(eij(z))=Sr(ĵ,{j};λ)Sr(î,{i};ρ)det(q-2(n-1)z)-1ξĵî(q-2(n-1)z),(7.51)S(det(z)-1)=det(z),
for all f∈M𝔥*, i,j∈[1,n] and z∈ℂ×.Proof.
We proceed in steps.
Step 1.
DefineS on the generators of ℱell(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∈M𝔥*, this is easily checked. Let a=eij(z). Using the right Laplace expansion (6.65) with J1=î, J2={j}, I=[1,n] and z replaced by q-2(n-1)z we obtain
(7.52)∑x=1n‍S(eix(z))exj(z)=δij.
Similarly, using the left Laplace expansion (6.64) with I1={i}, I2=ĵ, J=[1,n], and z replaced by q-2(n-1)z, together with the identity (6.68), we get
(7.53)∑x=1n‍eix(z)S(exj(z))=δij,
using also the crucial fact that, by Theorem 7.8, eij(z) commutes in ℱell(M(n)) with det(q-2(n-1)z) and hence in ℱell(GL(n)) with det(q-2(n-1)z)-1. This proves that the antipode axiom (7.46) is satisfied for a=eij(z).
Step 2.
We show thatS extends to a ℂ-linear map S:ℱell(M(n))→ℱell(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 ℱell(M(n)). Since S(eij(z))∈ℱell(GL(n))ω(ĵ),ω(î) and ω(i)+ω(î)=0, we have
(7.54)S(eij(z))S(f(λ))=S(eij(z))f(ρ)=f(ρ-ω(î))S(eij(z))=f(ρ+ω(i))S(eij(z))=S(f(λ+ω(i)))S(eij(z)).
Similarly, S(eij(z))S(f(ρ))=S(f(ρ+ω(j)))S(eij(z)) so relations (4.1) are preserved. Next, consider the RLL relation
(7.55)∑x,y=1n‍Racxy(λ,z1z2)exb(z1)eyd(z2)=∑x,y=1n‍Rxybd(ρ,z1z2)ecy(z2)eax(z1).
Multiply (7.55) from the left by S(eic(z2)) and from the right by S(edk(z2)), sum over c,d, and use (7.52), (7.53) to obtain
(7.56)∑x,c‍Racxk(λ-ω(ĉ),z1z2)S(eic(z2))exb(z1)=∑x,d‍Rxibd(ρ-ω(î),z1z2)eax(z1)S(edk(z2)).
Then multiply from the left by S(eja(z1)) and from the right by S(ebl(z1)), sum over a,b, and use (7.52), (7.53) again to get
(7.57)∑a,cRaclk(λ-ω(â)-ω(ĉ),z1z2)S(eja(z1))S(eic(z2))=∑b,d‍Rjibd(ρ-ω(ĵ)-ω(î),z1z2)S(edk(z2))S(ebl(z1)).
Since S(eij(z))∈ℱell(GL(n))ĵ,î and Rjibd(ρ-ω(ĵ)-ω(î),z1/z2)=Rjibd(ρ-ω(b̂)-ω(d̂),z1/z2) by the 𝔥-invariance of R, (7.57) can be rewritten
(7.58)∑a,c‍S(eja(z1))S(eic(z2))Raclk(λ,z1z2)=∑b,d‍S(edk(z2))S(ebl(z1))Rjibd(ρ,z1z2).
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 ℱell(M(n)) and S(ab)=S(b)S(a) for all a,b∈ℱell(M(n)), it follows that (7.46) holds for any a∈ℱell(M(n)). By taking in particular a=det(z) we get
(7.59)det(z)S(det(z))=1,S(det(z))det(z)=1,
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:ℱell(GL(n))→ℱell(GL(n)) satisfying the antipode axiom (7.46) on ℱell(M(n)) and on det(z)-1. Hence (7.46) holds for any a∈ℱell(GL(n)).


## 8. Concluding Remarks

To define the antipode we only needed thateij(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 aM𝔥*⊕𝔥*-line bundle over the elliptic curve ℂ×/{z~pz}. In this direction we found that the relation eij(pz)=qλi-ρjeij(z) respects the RLL relation (here 𝔥 should be the Cartan subalgebra of 𝔤𝔩n). This relation should then most likely be added to the algebra.



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*Source: 545892-2009-09-07.xml*