A link between two elliptic quantum groups

TL;DR

This paper establishes a deep connection between two elliptic quantum groups by constructing functors that relate their categories of finite-dimensional representations, revealing an equivalence in their tensor product subcategories.

Contribution

It introduces fully faithful functors linking the representation categories of Felder's elliptic quantum group and Belavin's elliptic algebra, demonstrating their categorical equivalence in certain subcategories.

Findings

Constructed a functor from Felder's elliptic quantum group to Belavin's algebra

Constructed a functor from Belavin's algebra to the same category

Proved the equivalence of tensor product subcategories of representations

Abstract

We construct a fully faithful functor from the category C_F of finite-dimensional representations of Felder's (dynamical) elliptic quantum group E_{tau,gamma}(gl(n)) to a cretain category D_B of (infinite-dimensional) representations of Belavin's quantum elliptic algebra B by difference operators, and a fully faithful functor from the category C_B of finite-dimensional representations of B to D_B. As a corollary, we show that the abelian subcategories of C_B and C_F generated by tensor products of vector representations are equivalent.