On quantum group SL_q(2)

TL;DR

This paper introduces a new method for constructing quantum groups with specific subgroup properties, applies it to SL(2), and discovers new quantum groups called metaplectic groups, enriching the understanding of quantum group structures.

Contribution

It develops a general construction method for quantum groups with classical subgroups and introduces new metaplectic quantum groups of SL(2)-type.

Findings

A general method for constructing quantum groups with classical subgroups.

Identification of a new series of quantum groups called metaplectic groups.

Potential applications in bimodule categories over SL_q(2) representations.

Abstract

We start with the observation that the quantum group SL_q(2), described in terms of its algebra of functions has a quantum subgroup, which is just a usual Cartan group. Based on this observation we develop a general method of constructing quantum groups with similar property. We also describe this method in the language of quantized universal enveloping algebras, which is another common method of studying quantum groups. We carry our method in detail for root systems of type SL(2); as a byproduct we find a new series of quantum groups - metaplectic groups of SL(2)-type. Representations of these groups can provide interesting examples of bimodule categories over monoidal category of representations of SL_q(2).