Classical elliptic current algebras

TL;DR

This paper explores classical elliptic current algebras on complex tori, their quantizations into dynamical quasi-Hopf algebras, and their degenerations into rational and trigonometric forms, including a review of the averaging method.

Contribution

It identifies two distinct classical elliptic current algebras based on different test function algebras and discusses their quantizations and degenerations.

Findings

Two different classical elliptic current algebras are constructed.

Quantization leads to two classes of dynamical quasi-Hopf current algebras.

Degenerations produce rational and trigonometric current algebras.

Abstract

In this paper we discuss classical elliptic current algebras and show that there are two different choices of commutative test function algebras on a complex torus leading to two different elliptic current algebras. Quantization of these classical current algebras give rise to two classes of quantized dynamical quasi-Hopf current algebras studied by Enriquez-Felder-Rubtsov and Arnaudon-Buffenoir-Ragoucy-Roche-Jimbo-Konno-Odake-Shiraishi. Different degenerations of the classical elliptic algebras are considered. They yield different versions of rational and trigonometric current algebras. We also review the averaging method of Faddeev-Reshetikhin, which allows to restore elliptic algebras from the trigonometric ones.