New $R$-matrices for small quantum groups

TL;DR

This paper constructs new $R$-matrices for extended small quantum groups, broadening their applicability and generalizing known results, especially for type $A_n$ and beyond.

Contribution

It introduces a large family of $R$-matrices for extensions of small quantum groups, expanding the cases where these matrices exist and generalizing previous results for $u_q(rak{sl}_2)$.

Findings

Extended $R$-matrices for various quantum group extensions

New combinatorial theorem for roots of unity in type $A_n$

Broader applicability of $R$-matrices in quantum algebra

Abstract

In this article we construct a large family of $R$-matrices for various extensions of small quantum groups by grouplike elements. The extensions are in correspondence to lattices between root and weight lattice and admit $R$-matrices in many cases where the original small quantum group does not. The results of this work extends some results that are well-known for $u_q(\mathfrak{sl}_2)$. Especially for $A_n$ a combinatorial theorem about roots of unity was necessary, which is interesting on its own, and has been proven in a separate paper [LN14].