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

Volume 2009, Article ID 545892,41pages doi:10.1155/2009/545892

Research Article

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

Jonas T. Hartwig

Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Postbus 94248, 1090 GE Amsterdam, The Netherlands

Correspondence should be addressed to Jonas T. Hartwig,jonas.hartwig@gmail.com Received 17 May 2009; Accepted 3 August 2009

Recommended by Francois Goichot

Using the language ofh-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group,FellGLn, from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebrasln. We apply the generalized FRST construction and obtain anh-bialgebroidFellMn. 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 theh-Hopf algebroidFellGLn.

Copyrightq2009 Jonas T. Hartwig. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The quantum dynamical Yang-Baxter QDYB equation was introduced by Gervais and Neveu1. It was realized by Felder2that 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. In2an interesting elliptic solution to the QDYB equation with spectral parameter was given, adapted from theA¹_n solution to the Star-Triangle relation constructed in3. 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 in4in thesl₂case.

In5, the authors considered objects in Felder’s category which were proposed as analogs of exterior and symmetric powers of the vector representation ofgl_n. To each object in the tensor category they associate an algebra of vector-valued difference operators and prove that a certain operator, constructed from the analog of the top exterior power, commutes with

all other difference operators. This is also proved in6, Appendix Bin more detail and in7 using a different approach.

An algebraic framework for studying dynamical R-matrices without spectral parame- ter was introduced in8. There the authors defined the notion ofh-bialgebroids andh-Hopf algebroids, a special case of the Hopf algebroids defined by Lu9. See10, Remark 2.1for a comparison of Hopf algebroids to related structures. In 8the authors also show, using a generalized version of the FRST construction, how to associate to every solutionRof the nonspectral quantum dynamical Yang-Baxter equation anh-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 theh-bialgebroid.

In this paper we define anh-Hopf algebroid associated to the elliptic R-matrix from2 with both dynamical and spectral parameters forgsl_n. This generalizes the spectral elliptic dynamicalGL2 quantum group from 11and the nonspectral trigonometric dynamical GLn 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 ofh-bialgebroids.

It would be interesting to develop harmonic analysis for the ellipticGLnquantum group, similarly to13. 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 inSection 2.1, we recall the definition of the elliptic R-matrix in Section 2.2. In Section 3 we review the definition ofh-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 inSection 4for the algebraFellMnobtained from the elliptic R-matrix. In particular we show that only one family of residual identities is needed.

Left and right analogs of the exterior algebra over Cⁿ are defined in Section 5 in a similar way as in 12. They are certain comodule algebras over FellMn 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. InSection 6.3we prove Laplace expansion formulas for these elliptic quantum minors.

In Section 7 we show that the h-bialgebroid FellMn can be equipped with a cobraiding, in the sense of14, extending then2 case from10. We use this and the ideas as in5,6to 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 in5.

Finally, inSection 7.4we defineFellGLnto be the localization ofFellMnat the determinant and show that it has an antipode giving it the structure of anh-Hopf algebroid.

2. Preliminaries

Letp, q ∈ R, 0 < p, q <1. We assumep, qare generic in the sense that ifp^aq^b 1 for some a, b∈Z, thenab0.

Denote byθthe normalized Jacobi theta function:

θz θ z;p

^∞

j0

1−zp^j 1− p^j1

z

. 2.1

It is holomorphic onC^× :C\ {0}with zero set{p^k:k∈Z}and satisfies θ

z⁻¹ θ

pz

−z⁻¹θz 2.2

and the addition formula θ xy,x

y, zw, z w

θ xw, x w, zy,z

y

z y

θ

xz,x z, yw, y

w

, 2.3

where we use the notation

θz1, . . . , zn θz1· · ·θzn. 2.4 Recall also the Jacobi triple product identity, which can be written

k∈Z

−z^kp^kk−1/2θz^∞

j1

1−p^j

. 2.5

It will sometimes be convenient to use the auxiliary functionEgiven by E:C−→C, Es q^sθ

q^−2s

. 2.6

Relation2.2implies thatE−s −Es.

The set{1,2, . . . , n}will be denoted by1, n.

2.2. The EllipticR-Matrix

Lethbe a complex vector space, viewed as an abelian Lie algebra,h^∗its dual space, and letV

⊕_λ∈h^∗Vλ a diagonalizable h-module. A dynamical R-matrix is by definition a meromorphic function

R:h^∗×C^× −→End_hV ⊗V 2.7

satisfying the quantum dynamical Yang-Baxter equation with spectral parameterQDYBE:

R λ,z₂ z3

₂₃

R λ−h²,z₁ z3

₁₃

R λ,z₁ z2

₁₂

R λ−h³,z₁ z2

₁₂

R λ,z₁ z3

₁₃

R λ−h¹,z₂ z3

₂₃ .

2.8

Equation2.8is an equality in the algebra of meromorphic functionsh^∗×C^× → EndV^⊗3. Upper indices are leg-numbering notation, andhindicates the action ofh. For example,

R λ−h³,z1

z_{2 12}

u⊗v⊗w R λ−α,z1

z₂

u⊗v⊗w, if w∈Vα. 2.9

An R-matrixRis called unitary if

Rλ, zR

λ, z⁻¹₂₁

Id_V⊗V 2.10

as meromorphic functions onh^∗×C^×with values in End_hV⊗V.

In the example we study, his the Cartan subalgebra ofsln. Thushis the abelian Lie algebra of all traceless diagonal complexn×nmatrices. LetV be theh-moduleCⁿwith standard basise₁, . . . , e_n. Defineωi∈h^∗i1, . . . , nby

ωih h_i, ifhdiagh1, . . . , h_n∈h. 2.11

We haveV ⊕ⁿ_i1V_ωiandV_ωiCei. Define

R:h^∗×C^×−→EndV⊗V 2.12

by

Rλ, z ⁿ

i1

E_ii⊗E_ii

i /j

α λ_ij, z

E_ii⊗E_jj

i /j

β λ_ij, z

E_ij⊗E_ji, 2.13

whereEij∈EndVare the matrix units,λijλ∈h^∗is an abbrevation forλEii−Ejj, and

α, β:C×C^×−→C 2.14

are given by

αλ, z α

λ, z;p, q

θzθ q^2λ1 θ

q²z θ

q^2λ, 2.15

βλ, z β

λ, z;p, q θ

q² θ

q^−2λz θ

q²z θ

q^−2λ. 2.16

Proposition 2.1see2. The mapRis a unitary R-matrix.

For the reader’s convenience, we give the explicit relationship between the R-matrix 2.13and Felders R-matrix as written in 5which we denote by R1. ThusR1 : h^∗₁×C → EndV ⊗V, whereh₁ is the Cartan subalgebra of gln, is defined as in 2.13 with α, β replaced byα1, β1 :C² → C,

α₁λ, x α₁

λ, x;τ, γ

θ1x;τθ1

λγ;τ θ1

x−γ;τ

θ1λ;τ, β1λ, x β1

λ, x;τ, γ

−θ₁xλ;τθ1

γ;τ θ₁

x−γ;τ

θ₁λ;τ.

2.17

Hereτ, γ ∈Cwith Imτ >0, andθ1is the first Jacobi theta function:

θ1x;τ −

j∈Z1/2

e^{πij2τ2πijx1/2}. 2.18

As proved in2,R₁satisfies the following version of the QDYBE:

R₁

λ−γh³, x₁−x₂₁₂

R₁λ, x1−x₃¹³R₁

λ−γh¹, x₂−x₃₂₃ R1λ, x2−x3²³R1λ−γh², x1−x3¹³R1λ, x1−x2¹²

2.19

and the unitarity condition

R1λ, xR²¹₁ λ,−x Id_V⊗V. 2.20 We can identifyh^∗ h^∗₁/Ctr where tr ∈ h^∗₁ is the trace. Since R1 has the form2.13, it is constant, as a function ofλ ∈ h^∗₁, on the cosets moduloCtr. SoR₁induces a maph^∗×C → EndV⊗V, which we also denote byR₁, still satisfying2.19,2.20.

Let τ, γ ∈ C with Imτ > 0 be such thatp e^πiτ,q e^πiγ. Then, as meromorphic functions ofλ, x∈h^∗×C,

R₁

γλ,−x;τ 2, γ

R

λ, z;p, q

, 2.21

whereze^2πix. Indeed, using the Jacobi triple product identity2.5we have

θ1

x;τ

2

ie^πiτ/2−xθz^∞

j1

1−p^j

, 2.22

and substituting this into2.17givesα₁γλ,−x;τ/2, γ αλ, z;p, qandβ₁γλ,−x;τ/2, γ βλ, z;p, qwhich proves2.21.

By replacingλ,xiin2.19byγλ,−xiand using2.21we obtain2.8withzie^2πixi. Similarly the unitarity2.10ofRis obtained from2.20.

2.3. Useful Identities

We end this section by recording some useful identities. Recall the definitions ofα, βin2.15.

It is immediate that

α λ, q²

β

−λ, q²

. 2.23

By induction, one generalizes2.2to θ

p^sz

−1^s

p^ss−1/2z^s₋₁

θz, fors∈Z. 2.24

Applying2.24to the definitions ofα,βwe get α

λ, p^kz

q^2kαλ, z, β λ, p^kz

q^2kλ1βλ, z, 2.25

and, using alsoθz⁻¹ −z⁻¹θz,

z→plim^−kq⁻²

q⁻¹θ q²z qθ

q⁻²zαλ, z α

λ, p^kq² ,

z→limp^−kq⁻²

q⁻¹θ q²z qθ

q⁻²zβλ, z −β

−λ, p^kq² ,

2.26

forλ∈C,z∈C^×, andk∈Z. By the addition formula2.3with x, y, z, w

z^1/2q^−λ1, z^1/2q^λ−1, z^1/2q^λ1, z^1/2q^−λ−1

, 2.27

we have

αλ, zα−λ, z−βλ, zβ−λ, z q²θ q⁻²z θ

q²z. 2.28

3. h-Bialgebroids

We recall some definitions from8. Leth^∗be a finite-dimensional complex vector spacee.g., the dual space of an abelian Lie algebra, and letM_h^∗ be the field of meromorphic functions onh^∗.

Definition 3.1. Anh-algebra is a complex associative algebraAwith 1 which is bigraded over h^∗,A⊕α,β∈h^∗Aαβ, and equipped with two algebra embeddingsμl, μr :M_h^∗ → A, called the left and right moment maps, such that

μl

f aaμl

Tαf , μr

f aaμr

Tβf

, fora∈Aαβ, f ∈M_h^∗, 3.1 whereTαdenotes the automorphismTαfζ fζαofM_h^∗. A morphism ofh-algebras is an algebra homomorphism preserving the bigrading and the moment maps.

The matrix tensor productA⊗Bof twoh-algebrasA,Bis theh^∗-bigraded vector space withA⊗B_αβ ⊕_γ∈h^∗Aαγ⊗M_h∗Bγβ, where⊗M_h∗ denotes tensor product overCmodulo the relations:

μ^A_r f

a⊗ba⊗μ^B_l f

b, ∀a∈A, b∈B, f ∈M_h^∗. 3.2 The multiplicationa⊗bc⊗d ac⊗bdfora, c ∈Aandb, d∈ Band the moment maps μ_lf μ^A_l f⊗1 andμ_rf 1⊗μ^B_rfmakeA⊗Binto anh-algebra.

Example 3.2. LetD_hbe the algebra of operators onM_h^∗ of the form

ifiTαi withfi ∈ M_h^∗ andα_i∈h^∗. It is anh-algebra with bigradingfT_−α∈D_hαα, and both moment maps equal to the natural embedding.

For anyh-algebraA, there are canonical isomorphismsA A⊗Dh D_h⊗A defined by x x⊗T_−β T_−α⊗x, forx∈A_αβ. 3.3 Definition 3.3. Anh-bialgebroid is anh-algebraAequipped with twoh-algebra morphisms, the comultiplicationΔ:A → A⊗Aand the counitε:A → D_hsuch thatΔ⊗Id◦Δ Id⊗Δ◦Δ andε⊗Id◦Δ Id Id⊗ε◦Δ, under the identifications3.3.

3.2. The Generalized FRST Construction

In8the authors gave a generalized FRST construction which attaches anh-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 to2.13withn2, 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 in4in 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 in5, such relations are consequences of the ordinary RLL relations by taking residues.

Another motivation for our procedure is that h-bialgebroids associated to gauge equivalent R-matrices should be isomorphic. In particular one should be allowed to multiply the R-matrix by any nonzero meromorphic function of the spectral variable without changing the isomorphism class of the associated algebrafor the full definition of gauge equivalent R- matrices see8.

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

Let hbe a finite-dimensional abelian Lie algebra, V ⊕α∈h^∗Vα a finite-dimensional diagonalizableh-module, andR:h^∗×C^× → End_hV⊗Va meromorphic function. We attach to this data anh-bialgebroidA_Ras follows. Let{ex}_x∈Xbe a homogeneous basis ofV, where Xis an index set. The matrix elementsR^ab_xy :h^∗×C^× → CofRare given by

Rζ, zea⊗eb

x,y∈X

R^ab_xyζ, zex⊗ey. 3.4

They are meromorphic onh^∗×C^×. Defineω:X → h^∗byex ∈V_ωx. LetARbe the complex associative algebra with 1 generated by {Lxyz : x, y ∈ X, z ∈ C^×} and two copies of M_h^∗, whose elements are denoted by fλ and fρ, respectively, with defining relations fλgρ gρfλforf, g∈M_h^∗and

fλLxyz L_xyzfλωx, f ρ

L_xyz L_xyzf ρω

y

, 3.5

for allx, y ∈X,z∈C^× andf ∈M_h^∗. The bigrading onA_Ris given byL_xyz∈A_Rωx,ωy forx, y ∈X,z∈C^×andfλ, fρ∈AR₀₀forf ∈M_h^∗. The moment maps are defined by μ_lf fλ,μ_rf fρ. The counit and comultiplication are defined by

εLabz δ_abT_−ωa, ε

fλ

ε f

ρ fT0, ΔLabz

x∈X

Laxz⊗Lxbz, Δ

fλ

fλ⊗1, Δ

f ρ

1⊗f ρ

.

3.6

This makesA_Rinto anh-bialgebroid.

Consider the ideal inARgenerated by the RLL relations:

x,y∈X

R^xy_ac λ,z1

z₂

L_xbz1Lydz2

x,y∈X

R^bd_xy ρ,z1

z₂

L_cyz2Laxz1, 3.7

wherea, b, c, d∈X, andz1, z2∈C^×. More precisely, to account for possible singularities ofR, we letI_Rbe the ideal inA_Rgenerated by all relations of the form

x,y∈X

w→limz1/z2

ϕwR^xyacλ, w

L_xbz1Lydz2

x,y∈X

w→limz1/z2

ϕwR^bd_xy ρ, w

L_cyz2Laxz1, 3.8

wherea, b, c, d ∈ X,z1, z2 ∈ C^×, andϕ :C^× → Cis a meromorphic function such that the limits exist.

We define A_R to be A_R/I_R. The bigrading descends to A_R because 3.8 is homogeneous, of bidegreeωa ωc, ωb ωd, by theh-invariance ofR. One checks thatΔIR ⊆ A_R⊗I RI_R⊗A_R andεIR 0. Thus A_R is anh-bialgebroid with the induced maps.

Remark 3.4. Objects in Felder’s tensor category associated to an R-matrix R are certain meromorphic functionsL : h^∗×C^× → End_hCⁿ⊗Wwhere W is a finite-dimensionalh- module2. After regularizingLwith respect to the spectral parameter it will give rise to a dynamical representation of theh-bialgebroidA_Rin the same way as in the nonspectral case treated in8. The residual relations incorporated in3.8are 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 relation3.7can be written as a matrix relation. We show how this is done in the present setting. It will be used later inSection 6.2.

AssumeR^ab_xyζ, z are defined, as meromorphic functions of ζ ∈ h^∗ for anyz ∈ C^×. DefineRλ, z,Rρ, z∈EndV⊗V⊗ARby

Rλ, zea⊗eb⊗u

x,y∈X

ex⊗ey⊗R^ab_xyλ, zu, R

ρ, z

ea⊗e_b⊗u

x,y∈X

e_x⊗e_y⊗R^ab_xy ρ, z

u,

3.9

fora, b ∈X,u∈A_R. Note that theλandρin the left-hand side are not variables but merely indicate which moment map is to be used. Forz∈C^×we also defineLz∈EndV ⊗A_Rby

Lz

x,y∈X

E_xy⊗L_xyz. 3.10

HereE_xyare the matrix units in EndV, andA_Racts on itself by left multiplication. The RLL relation3.7is equivalent to

R λ,z₁ z₂

L¹z1L²z2 L²z2L¹z1R ρh¹h²,z₁ z₂

3.11

in EndV ⊗V⊗AR, whereLⁱz Lz^i,3 ∈EndV⊗V ⊗ARfori1,2 and the operator Rρh¹h², z₁/z₂∈EndV⊗V⊗A_Ris given by

ea⊗eb⊗u−→

x,y∈X

ex⊗ey⊗R^ab_xy ρωa ωb,z₁ z2

u, 3.12

whereR^ab_xyρωa ωb, z1/z₂means the image inA_Rof the meromorphic functionh^∗ ζ→R^ab_xyζωaωb, z1/z2under the right moment mapμr. This equivalence can be seen by acting one_b⊗e_d⊗1 in both sides of3.11and collecting and equating terms of the form e_a⊗e_c⊗u. The matrix elements of the R-matrix in the right-hand side can then be moved to the left using thatRish-invariant and using relation3.5.

4. The Algebra F

Mn

We now specialize to the case wherehis the Cartan subalgebra ofsln,V Cⁿ, andRis given by2.13–2.16. The case n 2 was considered in11. We will show that3.8contains precisely one additional family of relations, as compared to3.7, and we write down all relations explicitly.

When we apply the generalized FRST construction to this data we obtain an h- bialgebroid which we denote byFellMn. The generatorsL_ijzwill be denoted bye_ijz.

ThusFellMnis the unital associativeC-algebra generated byeijz,i, j ∈1, n,z ∈C^×, and two copies ofM_h^∗, whose elements are denoted byfλandfρforf ∈M_h^∗, subject to the following relations:

fλeijz e_ijzfλωi, f ρ

e_ijz e_ijzf ρω

j

, 4.1

for allf∈M_h^∗,i, j∈1, n, andz∈C^×, and

n x,y1

R^xy_ac λ,z₁ z2

e_xbz1eydz2

n x,y1

R^bd_xy ρ,z₁ z2

e_cyz2eaxz1, 4.2

for alla, b, c, d∈1, n. More explicitly, from2.13we have

R^ab_xyζ, z

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

1, abxy,

α ζ_xy, z

, a /b, xa, yb, β

ζxy, z

, a /b, xb, ya,

0, otherwise,

4.3

which substituted into4.2yields four families of relations:

e_abz1eabz2 e_abz2eabz1, 4.4a

eabz1eadz2 α ρbd,z1

z₂

eadz2eabz1 β ρdb,z1

z₂

eabz2eadz1, 4.4b α λ_ac,z₁

z2

e_abz1ecbz2 β λ_ac,z₁ z2

e_cbz1eabz2 e_cbz2eabz1, 4.4c α λ_ac,z1

z₂

e_abz1ecdz2 β λ_ac,z1

z₂

e_cbz1eadz2 α ρ_bd,z₁

z₂

e_cdz2eabz1 β ρ_db,z₁ z₂

e_cbz2eadz1,

4.4d

wherea, b, c, d∈1, n,a /c, andb /d. Sinceθhas zeros precisely atp^k, k ∈Z,αandβhave poles atzq⁻²p^k, k∈Z. Thus4.4b–4.4dare to hold forz1, z2 ∈C^×withz1/z2/∈ {p^kq⁻² : k∈Z}.

In3.8, assuminga /c,b /d, takingz₁ z,z₂ p^kq²z,ϕw q⁻¹θq²w/qθq⁻²w, and using the limit formulas2.26, we obtain the relation

α

λ_ac, q²

e_abzecd

p^kq²z

−q^2kλcae_cbzead

p^kq²z

α ρ_bd, q²

e_cd p^kq²z

e_abz−q^2kρbdβ ρ_bd, q²

e_cb p^kq²z

e_adz. 4.5 This identity does not follow from4.4a–4.4din 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, and4.5consitute the defining relations of the algebraFellMn.

Proof. Assume that we have a relation of the form3.8and that a limit in one of the terms, limw→zϕwR^ab_xyλ, w, say, exists and is nonzero. Then one of the following cases occurs.

1Atw z,ϕwandR^ab_xyλ, ware both regular. If this holds for all terms, then the relation is just a multiple of one of4.4a–4.4d.

2Atw z,ϕwhas a pole whileR^ab_xyλ, wis regular. ThenR^ab_xyλ, wmust vanish identically at w z. The only case where this is possible is when x /y and R^ab_xyλ, w αλxy, w, and z p^k. But then there is another term containingβ which is never identically zero for anyz, and hence the limit in that term does not exist.

3Atw z,ϕwis regular whileR^ab_xyλ, whas a pole. Since these poles are simple and occur only whenz∈q⁻²p^Z, the functionϕmust have a zero of multiplicity one there. We can assume without loss of generality thatϕhas the specific form

ϕw q⁻¹θ q²w qθ

q⁻²w. 4.6

Then, ifa /candb /d,3.8becomes the residual RLL relation4.5.

If insteadca,b /d, and we takez1z,z2p^kq²zin3.8, we get, using2.26, 0α

ρbd, p^kq² ead

p^kq²z

eabz−β

ρbd, p^kq² eab

p^kq²z

eadz, 4.7

or, rewritten,

e_ad p^kq²z

e_abz q^2kρbdE

ρbd−1 E

ρbd1e_ab p^kq²z

e_adz. 4.8

However this relation is already derivable from4.4bas follows. Takez₁p^kq²zandz₂z in 4.4b multiply both sides byq^2kρbdEρbd−1/Eρbd1, and then use 4.4bon the right-hand side.

Similarly, ifa /c,d b,z₁ z,z₂ p^kq²z,ϕw q⁻¹θq²w/qθq⁻²win3.8, and using2.26we get

α

λ_ac, p^kq²

e_abzecb

p^kq²z

−β

λ_ca, p^kq²

e_cbzeab

p^kq²z

0, 4.9

or

eabzecb

p^kq²z

q^2kλcaecbzeab

p^kq²z

. 4.10

Similarly to the previous case, this identity follows already from4.4c.

5. Left and Right Elliptic Exterior Algebras

We recall the definition of corepresentations of anh-bialgebroid given in13.

Definition 5.1. Anh-spaceV is anh^∗-graded vector space overM_h^∗,V ⊕_α∈h^∗Vα, where each V_αisM_h^∗-invariant. A morphism ofh-spaces is a degree-preservingM_h^∗-linear map.

Given anh-spaceVand anh-bialgebroidA, we defineA⊗V to be theh^∗-graded space withA⊗V_α⊕_β∈h^∗Aαβ⊗Mh∗V_β, where⊗Mh∗ denotes⊗_Cmodulo the relations

μ_r f

a⊗va⊗fv, 5.1

forf ∈M_h^∗,a∈A,v∈V.A⊗Vbecomes anh-space with theM_h^∗-actionfa⊗v μ_lfa⊗v.

Similarly we defineV⊗A as anh-space byV⊗A _β⊕αVα⊗M_h∗Aαβ, where⊗M_h∗ here means

⊗_Cmodulo the relationv⊗μ_lfafv⊗aandM_h^∗-action given byfv⊗a v⊗μ_rfa.

For anyh-spaceV we have isomorphismsD_h⊗V V V⊗D hgiven by

T_−α⊗v v v⊗Tα, forv∈Vα, 5.2 extended toh-space morphisms.

Definition 5.2. A left corepresentationVof anh-bialgebroidAis anh-space equipped with an h-space morphismΔV :V → A⊗Vsuch thatΔV⊗1◦ΔV 1⊗Δ◦ΔVandε⊗1◦ΔV Id_V under the identification5.2.

Definition 5.3. A lefth-comodule algebraV over anh-bialgebroidAis a left corepresentation ΔV : V → A⊗V and in addition aC-algebra such thatV_αV_β ⊆ V_αβand such thatΔV is an algebra morphism, whenA⊗Vis given the natural algebra structure.

Right corepresentations and comodule algebras are defined analogously.

5.2. The Comodule AlgebrasΛandΛ.

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 ofFellMn. One difference between this approach and the one in5is 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 relations5.3a,5.3b, and5.5 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 ofαand 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 ofR.

LetΛbe the unital associativeC-algebra generated byv_iz, 1≤ i≤ n,z ∈C^×and a copy ofM_h^∗embedded as a subalgebra subject to the relations

fζviz v_izfζωi, 5.3a

v_izviw 0, 5.3b

α ζ_kj, z

w

v_kzvjw β ζ_kj, z

w

v_jzvkw 0, 5.3c

fori, j, k ∈ 1, n,j /k,z, w ∈ C^×,z/w /∈ {p^kq⁻² :k ∈ Z}andf ∈ M_h^∗. We require also the residual relation of5.3cobtained by multiplying byϕz/w q⁻¹θq²z/w/qθq⁻²z/w and lettingz/w → p^−kq⁻². After simplification using2.26, we get

v_kzvj

p^kq²z

q^2kζjkv_jzvk

p^kq²z

. 5.3d

Λbecomes anh-space by

μ_Λ f

vfζv 5.4

and requiringv_iz∈Λ_ωifor eachi, z.

Proposition 5.4. Λ is a left comodule algebra over FellMn with left coaction ΔΛ : Λ → FellMn⊗Λsatisfying

Δ_Λviz

j

e_ijz⊗v_jz, 5.5

Δ_Λ fζ

fλ⊗1. 5.6

Proof. We have

Δ_ΛvizΔ_Λviw

jk

e_ijzeikw⊗v_jzvkw

j /k

α

μjk, z w

eikweijz β μkj, z

w

eijweikz

⊗vjzvkw

j /k

eijweikz⊗ α

ζkj, z w

vkzvjw β ζkj, z

w

vjzvkw 0.

5.7 Similarly one proves that5.3c,5.3dare preserved.

Relation5.3cis not symmetric under interchange ofjandk. We now derive a more explicit, independent, set of relations forΛ. We will use the functionE, defined in2.6.

Proposition 5.5. iThe following is a complete set of relations forΛ:

fζviz v_izfζωi, 5.8a

v_k p^sq²z

v_jz −q^2sζkjE ζ_kj−1 E

ζkj1v_j p^sq²z

v_kz, ∀s∈Z, k /j, 5.8b v_kzvj

p^sq²z

q^2sζjkv_jzvk

p^sq²z

, 5.8c

vkzvjw 0 if z w/∈

p^sq^±2|s∈Z

or ifkj. 5.8d

iiThe set

v_idzd· · ·v_i1z1: 1≤i₁<· · ·< i_d≤n, z_i1

z_i ∈p^Zq^±2

5.9

is a basis forΛoverM_h^∗.

Proof. iElimination of thevjzvkw-term in5.3cyields

α ζ_jk, z

w

α ζ_kj, z

w −β

ζ_kj, z w

β

ζ_jk, z w

v_kzvjw 0. 5.10

Combining5.10,2.28, and the fact that theθzis zero precisely forz∈ {p^k |k ∈Z}we deduce that inΛ,

vkzvjw/0⇒ z

w p^sq² for somes∈Z. 5.11

Using2.25we obtain from5.11,5.3b, and5.3cthat relations5.8b,5.8dhold in the left elliptic exterior algebraΛ. Relations5.8a,5.8care just repetitions of5.3a,5.3d.

iiIt follows from the relations that each monomial inΛcan be uniquely written as fζvidzd· · ·v_i1z1, where 1≤i₁ <· · ·< i_d ≤nandf∈M_h^∗. It remains to show that the set 5.9is linearly independent overM_h^∗. Assume that a linear combination of basis elements is zero and that the sum has minimal number of terms. By multiplying from the right or left by v_jwfor appropriatej,wwe can assume that the sum is of the form

f1ζvid

z¹_d

· · ·vi1

z¹₁

· · ·frζvid

z^r_d

· · ·vi1

z^r₁

0, 5.12

for some fixed set{i1, . . . , i_d}. By the relations, a monomial v_idzd· · ·v_i1z1 can be given the ”degree”_d

i1zitⁱ⁻¹ ∈Ct, wheretis an indeterminate. Formally, considerCt⊗Λ, the tensor productoverCofΛ by the field of rational functions int. We identifyΛ with its image underΛ v → 1⊗v ∈Ct⊗Λand viewCt⊗Λnaturally as a vector space over Ct. By relations5.8a–5.8d, there is aC-algebra automorphismT ofCt⊗Λsatisfying Tvjz tv_jz,Tfζ fζ, andTp⊗1 p⊗1. Define

Dviz zv_iz, D fζ

0, D p⊗1

0, 5.13

forf ∈M_h^∗,p∈Ctandi∈1, n,z∈C^×, and extendDto aC-linear mapD :Ct⊗Λ → Ct⊗Λby requiring

Dab DaTb aDb, 5.14

fora, b ∈ Ct⊗Λ. The point is that the requirement5.14respects relations5.8a–5.8d, making D well defined. Write u_j f_jζvidz^j_d· · ·v_i1z^j₁. Then one checks that Duj

pjtuj, wherepjt _d

i1z^j_itⁱ⁻¹. By applyingDrepeatedly we get u₁

z¹

· · ·u_rz^r 0, p₁tu1

z¹

· · ·p_rturz^r 0, ...

p₁t^r−1u₁ z¹

· · ·p_rt^r−1u_rz^r 0.

5.15

Inverting the Vandermonde matrixpjtⁱ⁻¹_ijwe obtainujz^j 0 for eachj, that is,fjζ 0 for eachj. This proves linear independence of5.9.

Analogously one defines a right comodule algebraΛwith generatorswⁱzandfζ∈ M_h∗. The following relations will be used:

w^kzw^j p^sq²z

−q^2sζkjw^jzw^k p^sq²z

, ∀s∈Z, k /j, w^kz1w^jz2 0, if z₂

z1/∈

p^sq^±2|s∈Z

, or if kj.

5.16

Λhas alsoM_h^∗-basis of the form5.9. In factΛandΛare 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 FellMnas follows:

σ, τ fλ

fλ◦L_σ, σ, τ f

μ f

μ◦L_τ , σ, τ

e_ijz

e_σiτjz, 5.17

whereLσ :h → hσ∈Snis given by permutation of coordinates:

L_σ

diagh1, . . . , h_n

diag

h_σ1, . . . , h_σn

. 5.18

Also,S_nacts onΛbyC-algebra automorphisms via σ

fζ

fζ◦L_σ, σviz v_σiz. 5.19 Similarly we define anS_naction onΛ.

Lemma 5.6. For eachv∈Λ,w∈Λ, and anyσ, τ∈Snwe have

Δ_Λσv σ, τ⊗τΔ_Λv, 5.20 Δ_Λτw σ⊗σ, τΔΛw. 5.21 Proof. By multiplicativity, it is enough to prove these claims on the generators, which is easy.

6. Elliptic Quantum Minors

ForI⊆1, nwe set

F_Iζ

i,j∈I,i<j

E ζ_ij1

, F^Iζ

i,j∈I,i<j

E ζ_ij

, 6.1

and define the left and right elliptic sign functions:

sgn_Iσ;ζ σFIζ

F_σIζ

i,j∈I,i<j,σi>σj

E

ζ_σiσj1 E

ζ_σjσi1, sgn^Iσ;ζ F^σIζ

σ

F^Iζ

i,j∈I,i<j,σi>σj

E

ζ_σjσi E

ζ_σiσj,

6.2

forσ∈Sn. In fact,Eζij/Eζji −1 so sgn^1,nσ;ζis just the usual sign sgnσ. 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, nbyi1< i2<· · ·. Proposition 6.1. LetI ⊆1, n,d#I,σ∈S_n, andJσI. Then forz∈C^×,

v_σid

q^2d−1z

· · ·v_σi1z sgn_Iσ;ζvjd

q^2d−1z

· · ·vj1z, 6.3 w^σi1z· · ·w^σid

q^2d−1z

sgn^Iσ;ζw^j1z· · ·w^jd

q^2d−1z

. 6.4

Proof. We prove6.3. The proof of6.4is analogous. We proceed by induction on #I d, the cased1 being clear. Ifd >1, setI {i1, . . . , i_d−1}, JσI. Let 1≤j₁ <· · ·< j_d−1 ≤nbe the elements ofJ. By the induction hypothesis, the left hand side of6.3equals

v_σid

q^2d−1z

sgn_Iσ, ζvj_d−1

q^2d−2z

· · ·v_j

1z. 6.5

Now v_σidq^2d−1z commutes with sgn_Iσ, ζ since the latter only involves ζij with i, j /σid. Using the commutation relations5.8bwe obtain

sgn_Iσ, ζ·

j∈J,j>σid

E

ζ_jσid1 E

ζ_σidj1 ·vjd

q^2d−1z

· · ·vj1z. 6.6

Replacej ∈Jsuch thatj > σidbyσi, wherei∈I, i < i_d, σi> σid. Introduce the normalized monomials

v_Iz F_Iζ⁻¹v_ir

q^2d−1z v_ir−1

q^2d−2z

· · ·v_i1z∈Λ, 6.7 w^Iz F^Iζwⁱ¹zwⁱ²

q²z

· · ·w^id

q^2d−1z

∈Λ. 6.8

Corollary 6.2. LetI ⊆1, n. For any permutationσ∈Sn, σvIz v_σIz, σ

w^Iz

w^σIz 6.9