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In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and elliptic hypergeometric functions.

In combinatorics these notions are related to enumeration, q-enumeration and "elliptic enumeration" (see this article of Schlosser).

Now, I always related the passing to q-analogs by analogy to the "way" one passes from groups to quantum groups. And indeed, q-analogs and quantum groups are not entirely unrelated concepts. But this makes me ask the question in the title, whether someone has considered quantum groups at the "elliptic level", and if so what are they?

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Great question! Here's another spin: If one views groups through their associated symmetric monoidal categories of representations, and quantum groups by their braided monoidal categories of representations, what sort of category is the category of representations for the hypothetical 'elliptic' groups? – David Roberts Mar 10 '11 at 6:50
In reply to David Roberts's question, I would suggest looking at the category M of modules for an algebra object A in a braided tensor category C. Such a category M will not be itself a tensor category, but instead a "module category" over the braided tensor category C. This is consistent with the fact that configurations of points on an elliptic curve don't naturally form an operad, the way points on C form an E_2 operad, but rather the operad for C acts on configurations of points of a curve (it makes sense to glue in little copies of C; one cannot glue in a non-contractible space). – David Jordan Mar 15 '11 at 16:25
up vote 26 down vote accepted

That's a wonderful question, but I think there's a fundamental confusion here about two possible roles of the rational/trigonometric/elliptic trichotomy -- the one asked in the question and the one that leads to elliptic quantum groups -- which are in some sense "Fourier dual". (Everything I understand about this I learned talking to Tom Nevins.) For example the R/T/E trichotomy in R-matrices corresponds in quantum group world to the trichotomy Yangians/quantum affine algebras/elliptic quantum groups, not to group/quantum group/elliptic quantum group, or in nonquantum world to the trichotomy "Lie algebra/Lie group/elliptic group", with no quantums around. Not that there's an independent notion of an "elliptic group", but you can define a lot about it in terms of moduli of bundles on an elliptic curve. What the question asks is to fill in the trichotomy group/quantum group/"elliptically quantum" group..

Perhaps the easiest setting to see these two roles is in the study of many-body systems, eg the Calogero-Moser systems (or equivalently of meromorphic solutions of the KP and Toda hierarchies). These are completely integrable hamiltonian systems describing the motion of particles in the line. Initially it looks like they come in three flavors - rational, trigonometric and elliptic - labeled by whether the dependence of the potential on positions is rational, periodic or doubly periodic. This trichotomy is nicely explained by the trichotomy in R-matrices or in one-dimensional algebraic groups over $C$ or in irreducible genus one curves (Weierstrass cubics). The phase space can be described in terms of a cotangent bundle to a configuration space of points (on $C$, $C^\times$ or an elliptic curve), and the corresponding quantum systems can be described in terms of differential operators on the corresponding configuration spaces. In representation theory this trichotomy appears in studying three versions of the loop algebra - current algebras from $C$, $C^\times$ or $E$ (the latter needs to be interpreted more sophisticatedly).One can (as I mentioned) invent something you call an "elliptic group" by studying G-bundles on an elliptic curve, in such a way that if your elliptic curve acquires a node you get the usual group, and if it acquires a cusp you get the Lie algebra.. but again this is not the question.

I claim this R/T/E trichotomy is naturally identified with the one in the above answers, but "linearly independent" (in fact Fourier dual) to the one in the question. In integrable systems world this is expressed as follows: there's a deformation of the Calogero-Moser particle systems (called Ruijsenaars or Ruijsenaars-Schneider or Macdonald) in which we change the rational dependence on MOMENTUM to trigonometric dependence on momentum. This corresponds to changing from the linear Poisson structure on the cotangent bundle to configuration space to a quadratic one on the multiplicative cotangent bundle to config space (replace all $C$'s by $C^\times$'s). When we quantize this quadratic Poisson bracket we get DIFFERENCE rather than differential operators, and a relation to quantum groups rather than groups as in the CM case.

Integrable systems people (eg Nekrasov, Gorsky and collaborators) like to express this by a 3 x 3 square : we can havce R/T/E dependence on position as above, or R/T/E dependence on momenta.. except no one has a good definition (AFAIK) of what elliptic dependence on momenta MEANS - except via Fourier transform, exchanging position and momenta - so you can define elliptic momenta/rational positions eg by switching the order..

In any case the row "rational momenta" relates to (loop) groups, "trig momenta" relates to quantum groups. So the question is what is the elliptic version? There are various hints from string theory (see works of Nekrasov eg--- the rational row relates to SUSY 4dimensional gauge theory via the Seiberg-Witten solution, the trigonometric row to 5dimensional gauge theory, and the elliptic row should come from the mysterious "5-brane theory" or "6-dimensional (0,2) CFT")...

But maybe the most concrete answer I can give is motivated by Nakajima's work (see eg his ICM). You can realize representations of (loop) groups (of simple groups of type ADE) in equivariant cohomology of quiver varieties. If you replace equivariant cohomology by equivariant K-theory, you find the representation theory of quantum (affine) algebras. So this gives a natural place to look for the elliptic analogue --- try to understand the equivariant elliptic cohomology of quiver varieties! such ideas were put forth by Grojnowski, Ginzburg-Kapranov-Vasserot, and others but a good theory of equivariant elliptic cohomology was only developed fairly recently by Lurie, so one can contemplate such questions anew.

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Elliptic solutions of the classical Yang-Baxter equation (CYBE) exists only in the $\mathfrak sl_n$ case. In the general case, there are elliptic solutions of the dynamical CYBE, which is an algebraico-differential equation generalizing the CYBE. You can use such an elliptic dynamical $r$-matrix to define a flat connexion (the KZB connexion) on the configuration space of $n$ points on the torus which leads to monodromy representations of the torus braid group.

This monodromy morphism can be expressed using the good old KZ associator, and the monodromy operators allows one to construct an exemple of an elliptic structure over a braided monoidal category. Elliptic structures plays the same role for the torus braid group as braided monoidal categories does for the usual braid group.

Edit: the basic reference concerning the anwser of Bruce and Elliptic quantum groups : . Rmk: the dynamical CYBE is called modified CYBE in this paper.

Edit in light of David Ben-Zvi's answer: I was not quite speaking about this trichotomy, but rather mentionning the existence of elliptic categories leading to representations of the elliptic braid group, following the suggestion of David Roberts that one should view quantum groups by their braided monoidal category of modules. But the fact that $r$-matrices and the (dynamical) Yang Baxter equation comes into play is not really a coincidence.

Roughlys speaking, in the KZ world, rational $r$-matrices leads to monodromy representations of the braid group $B_n$, which can be used to construct braided monoidal categories. If you take the trivial $r$-matrice $r=0$ these representations factor through representations of $S_n$, so in some sense you recover the symmetric categories of module over a Lie group. For non trivial rational $r$-matrices, you get braided monoidal categories, and quantum groups by the Kohno--Drinfeld theorem. For trigonometric one, you get representations of the cylinder braid group, the associated categorical notion being that of braided module categorie (hence comodules over quantum groups). Finally, for elliptic solutions of the (dynamical) Yang-Baxter equation, you get elliptic categories. The associated algebraic notion seems to be strongly related with quantum analogs of the algebra of differential operators on a Lie group :

I hop it makes my answer more clear :)

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Yes, but I can't find the references! There is a trichotomy for solutions to the Yang-Baxter equation. The three categories are rational, trigonometric, elliptic. Elliptic solutions lead to elliptic quantum groups. Hopefully you will get a better answer.

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Browsing through the answers to this old question, I am surprised that nobody mentions the Sklyanin algebra, which is surely the first and most fundamental example of an elliptic quantum group. It was introduced by Sklyanin in 1982, before Drinfeld explained quantum groups to the mathematics community. The Sklyanin algebra is constructed from the R-matrix of the eight-vertex model in much the same way as the standard SL(2) quantum group is constructed from the six-vertex model. In Baxter's solution of the eight-vertex model, a key step is that the R-matrix is equivalent (via a vertex-face transformation) to a dynamical R-matrix. Starting with this dynamical R-matrix leads to a Felder-type elliptic quantum group as mentioned in Adrien's answer.

You mention elliptic hypergeometric functions. They are linked to both Sklyanin- and Felder-type elliptic quantum groups in a completely similar way as classical hypergeometric functions are linked to Lie groups and basic hypergeometric functions to standard (non-elliptic) quantum groups.

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Hitoshi Konno, along with some earlier work with Jimbo et al has worked out the elliptic quantum groups, especially sl2, very explicitly and all the calculations are wonderful!

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