Exotic Quantum Double,Its Universal R-matrix And Their Representations

TL;DR

This paper constructs an exotic quantum double and its universal R-matrix, revealing new quasi-triangular Hopf algebra structures and their multi-parameter R-matrices, expanding the understanding of quantum Yang-Baxter solutions.

Contribution

It introduces a novel exotic quantum double with a universal R-matrix, distinct from standard quantum doubles, and explores its representation theory and associated multi-parameter R-matrices.

Findings

Constructed the exotic quantum double and its universal R-matrix.

Derived many-parameter representations related to Lie algebra A2.

Generated multi-parameter R-matrices for the quantum Yang-Baxter equation.

Abstract

The exotic quantum double and its universal R-matrix for quantum Yang-Baxter equation are constructed in terms of Drinfeld's quantum double theory.As a new quasi-triangular Hopf algebra, it is much different from those standard quantum doubles that are the q-deformations for Lie algebras or Lie superalgebras. By studying its representation theory,many-parameter representations of the exotic quantum double are obtained with an explicit example associated with Lie algebra $A_2$ .The multi-parameter R-matrices for the quantum Yang-Baxter equation can result from the universal R-matrix of this exotic quantum double and these representattions.