TITLE: Groups, quantum groups and (fill in the blank)
QUESTION [39 upvotes]: 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?

REPLY [3 votes]: 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.