On the Representation Theory of an Algebra of Braids and Ties

TL;DR

This paper studies the algebra of braids and ties, constructing a faithful tensor space representation, providing a basis, and classifying its irreducible representations, thus advancing understanding of its structure.

Contribution

It introduces a faithful tensor space representation for the algebra, enabling basis construction and irreducible representation classification.

Findings

Tensor space representation is faithful

A basis for the algebra is established

Irreducible representations are classified

Abstract

We consider the algebra ${\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\cal E}_n(u)$ and to classify its irreducible representations.