Representations of finite-dimensional quotient algebras of the 3-string braid group

TL;DR

This paper studies finite-dimensional quotient algebras of the 3-string braid group, providing criteria for semisimplicity and explicit formulas for irreducible representations in specific cases.

Contribution

It introduces new finite-dimensional quotients of the 3-string braid group algebra and characterizes their irreducible representations.

Findings

Finite-dimensional quotient algebras for p=2,3,4,5.

Semisimplicity criteria established.

Explicit formulas for irreducible representations.

Abstract

We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give semisimplicity criteria for these algebras and present explicit formulas for all their irreducible representations.