Birman-Wenzl-Murakami Algebra and the Topological Basis

TL;DR

This paper constructs matrix representations of the Birman-Wenzl-Murakami algebra using entangled states, introduces topological basis states, and simplifies the algebra into a lower dimension, revealing spin singlet states in special cases.

Contribution

It presents new 9x9-matrix representations of BWMA based on entangled states and introduces topological basis states to reduce the algebra's dimensionality.

Findings

Constructed 9x9-matrix representations of TLA and BWMA.

Identified topological basis states for BWMA.

Recast BWMA into a 3-dimensional form revealing spin singlet states.

Abstract

In this paper, we use entangled states to construct 9x9-matrix representations of Temperley-Lieb algebra (TLA), then a family of 9x9-matrix representations of Birman-Wenzl-Murakami algebra (BWMA) have been presented. Based on which, three topological basis states have been found. And we apply topological basis states to recast nine-dimensional BWMA into its three-dimensional counterpart. Finally, we find the topological basis states are spin singlet states in special case.