The l-State Boson Algebra as a Non-Standard Quantum Double and its Universal R-Matrix for Yang-Baxer Equation

TL;DR

This paper introduces a novel quantum double structure based on the l-state boson algebra, providing explicit universal R-matrices for the Yang-Baxter equation and new braid group representations.

Contribution

It constructs a non-standard quantum double with a unique Hopf algebra structure and derives explicit R-matrices, expanding the framework beyond traditional q-deformations.

Findings

Explicit universal R-matrix for the new quantum double

New braid group representations from constructed R-matrices

Non-traditional Hopf algebra structure not derived from q-deformation

Abstract

In this paper we construct a new quantum double by endowing the l-state bosonalgebra with a non-trivial Hopf algebra structure,which is not a q-deformation of the Lie algebra or superalgebra.The universal R-matrix for the Yang-Baxter equation associated with this new quantum group structure is obtained explicitly.By building the representations of this quantum double,we get some R-matrices ,which can result in new representations of the braid group.