Quasitriangularity of quantum groups at roots of 1

TL;DR

This paper investigates the quasitriangularity property of quantum groups at roots of unity, revealing conditions under which these structures exhibit quasitriangular or autoquasitriangular properties, with implications for their algebraic automorphisms.

Contribution

It characterizes quasitriangularity for quantum sl_2 at roots of unity and introduces the concept of autoquasitriangularity, expanding understanding of their algebraic structure.

Findings

Quantum groups at roots of 1 can be quasitriangular or autoquasitriangular.

The braiding automorphism combines Poisson and adjoint transformations.

Most interesting cases involve specific algebraic automorphisms.

Abstract

An important property of a Hopf algebra is its quasitriangularity and it is useful various applications. This property is investigated for quantum groups $sl_2$ at roots of 1. It is shown that different forms of the quantum group $sl_2$ at roots of 1 are either quasitriangular or have similar structure which will be called autoquasitriangularity. In the most interesting cases this property means that "braiding automorphism" is a combination of some Poisson transformation and an adjoint transformation with certain element of the tensor square of the algebra.