A generalization of the brauer algebra

TL;DR

This paper explores two new algebraic structures extending the Brauer algebra, analyzing their semisimplicity properties and providing a broader understanding of their algebraic behavior.

Contribution

It introduces the algebras A_n(x) and L_n(x) as generalizations of the Brauer algebra and studies their semisimplicity conditions.

Findings

A_n(x) is semisimple if x is not an integer.

L_n(x) is semisimple if x is nonzero.

The paper extends previous work on partition and Brauer algebras.

Abstract

We study two variations of the Brauer algebra $B_n(x)$. The first is the algebra $A_n(x)$, which generalizes the Brauer algebra by considering loops. The second is the algebra $L_n(x)$, the $A_n(x)$-subalgebra generated by diagrams without horizontal arcs. $A_n(x)$ and $L_n(x)$ have for $x \neq 0$ an hereditary-chain indexed by all integers. Following the ideas of Martin in the context of the partition algebra, and Doran et al. for the Brauer algebra, we study semisimplicity of $A_n(x)$ using restriction and induction in $A_n(x)$ and $L_n(x)$. Our main result is that $A_n(x)$ is semisimple if $x \notin Z$ and that $L_n(x)$ is semisimple if $x \neq 0$.