Affine Periplectic Brauer Algebras

TL;DR

This paper introduces affine periplectic Brauer algebras, establishing their connection to periplectic Lie superalgebra representations, and provides structural results including bases and tensor product actions.

Contribution

It formulates Nazarov-Wenzl type algebras for periplectic Lie superalgebras and establishes their relation to existing algebras and representations.

Findings

Connection between $rak{p}(n)$-representations and $ ext{Affine Periplectic Brauer Algebras$}$ established.

A Poincare-Birkhoff-Witt basis for $ ext{Affine Periplectic Brauer Algebras$}$ constructed.

Actions of Jucys-Murphy elements derived in tensor product representations.

Abstract

We formulate Nazarov-Wenzl type algebras ${\widehat{P}_d^-}$ for the representation theory of the Periplectic Lie superalgebras $\mathfrak{p}(n)$. We establish a Arakawa-Suzuki type theorem to provide a connection between $\mathfrak{p}(n)$-representations and $\widehat{P}_d^-$-representations. We also consider various tensor product representations for $\widehat{P}_d^-$. The periplectic Brauer algebra $A_d$ defined by Moon is an quotient of $\widehat{P}_d^-$. In particular, actions induced by Jucys-Murphy elements can be obtained under the tensor product representation of $\widehat{P}_d^-$. Also, a Poincare-Birkhoff-Witt type basis for $\widehat{P}_d^-$ is obtained.