Quantum groups obtained from solutions to the parametrized Yang-Baxter equation

TL;DR

This paper constructs new coquasitriangular Hopf algebras from solutions to the parametrized Yang-Baxter equation, revealing dualities and irreducibility properties, and introduces objects not derived from known quantum groups.

Contribution

It develops a parametrized FRT construction to create novel Hopf algebras and explores their dualities and representation theory, including irreducibility criteria.

Findings

Constructed $oldsymbol{ ilde{SL}_q(2)}$ as a quantization of affine $SL(2)$

Established duality between $oldsymbol{ ilde{SL}_q(2)}$ and $oldsymbol{U_q( ilde{rak{sl}_2})}$

Identified a new solution to the parametrized Yang-Baxter equation not from existing quantum groups

Abstract

In this article we use a parametrized version of the FRT construction to construct two new coquasitriangular Hopf algebras. The first one, $\widehat{SL_q(2)}$, is a quantization of the coordinate ring on affine $SL(2)$. We show that there is a duality relation between this object and the more well-known $U_q(\widehat{\mathfrak{sl}_2})$. We then build certain irreducible comodules of this Hopf algebra and prove an irreducibility criterion for their tensor product in the spirit of Chari and Pressley. The second object is built from a solution of the parametrized Yang-Baxter equation with parameter group $GL(2, \mathbb{C}) \times GL (1,\mathbb{C})$. This solution doesn't come from any known quantum group, though it is related to both solutions coming from $U_{\pm i}(\widehat{\mathfrak{sl}_2})$ and $U_q(\widehat{\mathfrak{gl}(1|1)})$. We then study certain irreducible comodules of this newly built object.