Quasi-Hopf algebras associated with sl(2) and complex curves

TL;DR

This paper constructs quasi-Hopf algebras related to sl(2) and complex curves, providing a new algebraic framework for quantum groups associated with geometric data.

Contribution

It introduces a novel construction of quasi-Hopf algebras quantizing double extensions of Drinfeld's Manin pairs linked to complex curves and sl(2).

Findings

Constructed quasi-Hopf algebras for complex curves and sl(2).

Analyzed vertex relations and computed R-matrices for these quantum groups.

Developed a factorization of the twist operator for conjugated quantum groups.

Abstract

We construct quasi-Hopf algebras quantizing double extensions of the Manin pairs of Drinfeld, associated to a curve with a meromorphic differential, and the Lie algebra sl(2). This construction makes use of an analysis of the vertex relations for the quantum groups obtained in our earlier work, PBW-type results and computation of $R$-matrices for them; its key step is a factorization of the twist operator relating ``conjugated'' versions of these quantum groups.