The elliptic GL(n) dynamical quantum group as an h-Hopf algebroid

TL;DR

This paper constructs a new elliptic quantum group as an h-Hopf algebroid using elliptic solutions of the quantum dynamical Yang-Baxter equation, extending the algebraic framework for quantum groups.

Contribution

It introduces a novel elliptic quantum group F_ell(GL(n)) as an h-Hopf algebroid, utilizing the generalized FRST construction and elliptic minors.

Findings

Elliptic determinant is shown to be central.

Construction of an h-Hopf algebroid from elliptic solutions.

Definition and analysis of elliptic minors and their properties.

Abstract

Using the language of h-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group, F_ell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra sl_n. We apply the generalized FRST construction and obtain an h-bialgebroid F_ell(M(n)). 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 h-Hopf algebroid F_ell(GL(n)).