Research Article | Open Access
The Elliptic Dynamical Quantum Group as an -Hopf Algebroid
Using the language of -Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group, , from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra . We apply the generalized FRST construction and obtain an -bialgebroid . 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 -Hopf algebroid .
The quantum dynamical Yang-Baxter (QDYB) equation was introduced by Gervais and Neveu . It was realized by Felder  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  an interesting elliptic solution to the QDYB equation with spectral parameter was given, adapted from the solution to the Star-Triangle relation constructed in . 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  in the case.
In , the authors considered objects in Felder's category which were proposed as analogs of exterior and symmetric powers of the vector representation of . 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 ] in more detail and in  using a different approach.
An algebraic framework for studying dynamical R-matrices without spectral parameter was introduced in . There the authors defined the notion of -bialgebroids and -Hopf algebroids, a special case of the Hopf algebroids defined by Lu . See [10, Remark ] for a comparison of Hopf algebroids to related structures. In  the authors also show, using a generalized version of the FRST construction, how to associate to every solution of the nonspectral quantum dynamical Yang-Baxter equation an -bialgebroid. Under some extra condition they get an -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 -bialgebroid.
In this paper we define an -Hopf algebroid associated to the elliptic R-matrix from  with both dynamical and spectral parameters for . This generalizes the spectral elliptic dynamical quantum group from  and the nonspectral trigonometric dynamical quantum group from . As in , 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 . 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 -bialgebroids.
It would be interesting to develop harmonic analysis for the elliptic quantum group, similarly to . In this context it is valuable to have an abstract algebra to work with and not only a tensor category analogous to a category of representations. For example, the analog of the Haar measure seems most naturally defined as a certain linear functional on the algebra.
The plan of this paper is as follows. After introducing some notation in Section 2.1, we recall the definition of the elliptic R-matrix in Section 2.2. In Section 3 we review the definition of -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 obtained from the elliptic R-matrix. In particular we show that only one family of residual identities is needed.
Left and right analogs of the exterior algebra over are defined in Section 5 in a similar way as in . They are certain comodule algebras over and arise naturally from a single relation analogous to . The matrix elements of these corepresentations are generalized minors depending on a spectral parameter. Their properties are studied in Section 6. In particular we show that the left and right versions of the minors in fact coincide. In Section 6.3 we prove Laplace expansion formulas for these elliptic quantum minors.
In Section 7 we show that the -bialgebroid can be equipped with a cobraiding, in the sense of , extending the case from . 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 .
Finally, in Section 7.4 we define to be the localization of at the determinant and show that it has an antipode giving it the structure of an -Hopf algebroid.
Let , . We assume are generic in the sense that if for some , then .
Denote by the normalized Jacobi theta function: It is holomorphic on with zero set and satisfies and the addition formula where we use the notation Recall also the Jacobi triple product identity, which can be written It will sometimes be convenient to use the auxiliary function given by Relation (2.2) implies that .
The set will be denoted by .
2.2. The Elliptic R-Matrix
Let be a complex vector space, viewed as an abelian Lie algebra, its dual space, and let a diagonalizable -module. A dynamical R-matrix is by definition a meromorphic function satisfying the quantum dynamical Yang-Baxter equation with spectral parameter (QDYBE): Equation (2.8) is an equality in the algebra of meromorphic functions . Upper indices are leg-numbering notation, and indicates the action of . For example, An R-matrix is called unitary if as meromorphic functions on with values in .
In the example we study, is the Cartan subalgebra of . Thus is the abelian Lie algebra of all traceless diagonal complex matrices. Let be the -module with standard basis . Define () by We have and . Define by where are the matrix units, () is an abbrevation for , and are given by
Proposition 2.1 (see ). The map 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  which we denote by . Thus , where is the Cartan subalgebra of , is defined as in (2.13) with replaced by , Here with , and is the first Jacobi theta function: As proved in , satisfies the following version of the QDYBE: and the unitarity condition We can identify where is the trace. Since has the form (2.13), it is constant, as a function of , on the cosets modulo . So induces a map , which we also denote by , still satisfying (2.19), (2.20).
2.3. Useful Identities
We end this section by recording some useful identities. Recall the definitions of in (2.15). It is immediate that By induction, one generalizes (2.2) to Applying (2.24) to the definitions of , we get and, using also , for , , and . By the addition formula (2.3) with we have
We recall some definitions from . Let be a finite-dimensional complex vector space (e.g., the dual space of an abelian Lie algebra), and let be the field of meromorphic functions on .
Definition 3.1. An -algebra is a complex associative algebra with which is bigraded over , , and equipped with two algebra embeddings , called the left and right moment maps, such that where denotes the automorphism of . A morphism of -algebras is an algebra homomorphism preserving the bigrading and the moment maps.
The matrix tensor product of two -algebras , is the -bigraded vector space with , where denotes tensor product over modulo the relations: The multiplication for and and the moment maps and make into an -algebra.
Example 3.2. Let be the algebra of operators on of the form with and . It is an -algebra with bigrading , and both moment maps equal to the natural embedding.
For any -algebra , there are canonical isomorphisms defined by
Definition 3.3. An -bialgebroid is an -algebra equipped with two -algebra morphisms, the comultiplication and the counit such that and , under the identifications (3.3).
3.2. The Generalized FRST Construction
In  the authors gave a generalized FRST construction which attaches an -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 . However, when specifying the R-matrix to (2.13) with , 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  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 , such relations are consequences of the ordinary RLL relations by taking residues.
Another motivation for our procedure is that -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 ).
These considerations suggest the following procedure for constructing an -bialgebroid from a quantum dynamical R-matrix with spectral parameter.
Let be a finite-dimensional abelian Lie algebra, a finite-dimensional diagonalizable -module, and a meromorphic function. We attach to this data an -bialgebroid as follows. Let be a homogeneous basis of , where is an index set. The matrix elements of are given by They are meromorphic on . Define by . Let be the complex associative algebra with generated by and two copies of , whose elements are denoted by and , respectively, with defining relations for and for all , and . The bigrading on is given by for , and for . The moment maps are defined by , . The counit and comultiplication are defined by This makes into an -bialgebroid.
Consider the ideal in generated by the RLL relations: where , and . More precisely, to account for possible singularities of , we let be the ideal in generated by all relations of the form where , , and is a meromorphic function such that the limits exist.
We define to be . The bigrading descends to because (3.8) is homogeneous, of bidegree , by the -invariance of . One checks that and . Thus is an -bialgebroid with the induced maps.
Remark 3.4. Objects in Felder's tensor category associated to an R-matrix are certain meromorphic functions where is a finite-dimensional -module . After regularizing with respect to the spectral parameter it will give rise to a dynamical representation of the -bialgebroid in the same way as in the nonspectral case treated in . 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
Assume are defined, as meromorphic functions of for any . Define by for , . Note that the and in the left-hand side are not variables but merely indicate which moment map is to be used. For we also define by Here are the matrix units in , and acts on itself by left multiplication. The RLL relation (3.7) is equivalent to in , where for and the operator is given by where means the image in of the meromorphic function under the right moment map . This equivalence can be seen by acting on in both sides of (3.11) and collecting and equating terms of the form . The matrix elements of the R-matrix in the right-hand side can then be moved to the left using that is -invariant and using relation (3.5).
4. The Algebra
We now specialize to the case where is the Cartan subalgebra of , , and is given by (2.13)–(2.16). The case was considered in . We will show that (3.8) contains precisely one additional family of relations, as compared to (3.7), and we write down all relations explicitly.
When we apply the generalized FRST construction to this data we obtain an -bialgebroid which we denote by . The generators will be denoted by . Thus is the unital associative -algebra generated by , , , and two copies of , whose elements are denoted by and for , subject to the following relations: for all , , and , and for all . More explicitly, from (2.13) we have which substituted into (4.2) yields four families of relations: where , , and . Since has zeros precisely at , and have poles at . Thus (4.4b)–(4.4d) are to hold for with .
In (3.8), assuming , , taking , , , and using the limit formulas (2.26), we obtain the relation This identity does not follow from (4.4a)–(4.4d) in an obvious way. It will be called the residual RLL relation.
Proof. Assume that we have a relation of the form (3.8) and that a limit in one of the terms, , say, exists and is nonzero. Then one of the following cases occurs.
(1)At , and are both regular. If this holds for all terms, then the relation is just a multiple of one of (4.4a)–(4.4d).(2)At , has a pole while is regular. Then must vanish identically at . The only case where this is possible is when and , and . But then there is another term containing which is never identically zero for any , and hence the limit in that term does not exist.(3)At , is regular while has a pole. Since these poles are simple and occur only when , the function must have a zero of multiplicity one there. We can assume without loss of generality that has the specific form
Then, if and , (3.8) becomes the residual RLL relation (4.5).
If instead , , and we take , in (3.8), we get, using (2.26), or, rewritten, However this relation is already derivable from (4.4b) as follows. Take and in (4.4b) multiply both sides by , and then use (4.4b) on the right-hand side.
Similarly, if , , , , in (3.8), and using (2.26) we get or Similarly to the previous case, this identity follows already from (4.4c).
5. Left and Right Elliptic Exterior Algebras
5.1. Corepresentations of -Bialgebroids
We recall the definition of corepresentations of an -bialgebroid given in .
Definition 5.1. An -space is an -graded vector space over , , where each is -invariant. A morphism of -spaces is a degree-preserving -linear map.
Given an -space and an -bialgebroid , we define to be the -graded space with , where denotes modulo the relations for , , . becomes an -space with the -action . Similarly we define as an -space by , where here means modulo the relation and -action given by .
For any -space we have isomorphisms given by extended to -space morphisms.
Definition 5.2. A left corepresentation of an -bialgebroid is an -space equipped with an -space morphism such that and (under the identification (5.2).
Definition 5.3. A left -comodule algebra over an -bialgebroid is a left corepresentation and in addition a -algebra such that and such that is an algebra morphism, when is given the natural algebra structure.
Right corepresentations and comodule algebras are defined analogously.
5.2. The Comodule Algebras and .
We define in this section an elliptic analog of the exterior algebra, following , where it was carried out in the trigonometric nonspectral case. It will lead to natural definitions of elliptic minors as certain elements of . One difference between this approach and the one in  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 and , we can obtain exterior algebras for any -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 .
Let be the unital associative -algebra generated by , , and a copy of embedded as a subalgebra subject to the relations for , , , and . We require also the residual relation of (5.3c) obtained by multiplying by and letting . After simplification using (2.26), we get
becomes an -space by and requiring for each .
Proposition 5.4. is a left comodule algebra over with left coaction satisfying
Proposition 5.5. (i) The following is a complete set of relations for :
(ii) The set is a basis for over .
Proof. (i) Elimination of the -term in (5.3c) yields
Combining (5.10), (2.28), and the fact that the is zero precisely for we deduce that in ,
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 . Relations (5.8a), (5.8c) are just repetitions of (5.3a), (5.3d).
(ii) It follows from the relations that each monomial in can be uniquely written as , where and . It remains to show that the set (5.9) is linearly independent over . 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 for appropriate , we can assume that the sum is of the form for some fixed set . By the relations, a monomial can be given the "degree” , where is an indeterminate. Formally, consider , the tensor product (over ) of by the field of rational functions in . We identify with its image under and view naturally as a vector space over . By relations (5.8a)–(5.8d), there is a -algebra automorphism of satisfying , , and . Define for , and , , and extend to a -linear map by requiring for . The point is that the requirement (5.14) respects relations (5.8a)–(5.8d), making well defined. Write . Then one checks that , where . By applying repeatedly we get Inverting the Vandermonde matrix we obtain for each , that is, for each . This proves linear independence of (5.9).
Analogously one defines a right comodule algebra with generators and . The following relations will be used: has also -basis of the form (5.9). In fact and are isomorphic as algebras.
5.3. Action of the Symmetric Group
From (4.4a)–(4.4d), and (4.5) we see that acts by -algebra automorphisms on as follows: where () is given by permutation of coordinates: Also, acts on by -algebra automorphisms via Similarly we define an action on .
Lemma 5.6. For each , , and any we have
Proof. By multiplicativity, it is enough to prove these claims on the generators, which is easy.
6. Elliptic Quantum Minors
For we set and define the left and right elliptic sign functions: for . In fact, so