Degenerate Sklyanin Algebras

TL;DR

This paper introduces new solutions to the quantum Yang-Baxter equation through gauge transformations, relates them to Sklyanin algebras, and explores their representations and extensions to higher dimensions.

Contribution

It presents novel trigonometric and rational solutions to the QYBE derived from elliptic R-matrices, linking them to Sklyanin algebras and their representations.

Findings

New solutions related to Sklyanin algebra are obtained.

These solutions are connected via quasi-Hopf twists.

Representation by difference operators is constructed.

Abstract

New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for $sl(2,\mathbb{C})$. These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The $sl(N,\mathbb{C})$-case is discussed.