New $R$-matrices for small quantum groups

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].

arXiv comment30 pages, minor improvements including referees comments

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Cited by in corpus (1)

• A theorem on roots of unity and a combinatorial principle