Quantized enveloping superalgebra of type $P$

TL;DR

This paper introduces a new quantized superalgebra related to the Lie superalgebra of type P, explores its properties, and connects it to a new algebraic structure called the periplectic q-Brauer algebra, proposing a related q-Schur superalgebra.

Contribution

It defines the quantized enveloping superalgebra of type P, studies its properties, and establishes its relationship with the periplectic q-Brauer algebra and a new q-Schur superalgebra.

Findings

Introduction of the superalgebra $rak{U}_q{rak{p}}_n$ and its properties

Identification of the periplectic q-Brauer algebra as a centralizer

Proposal of a new periplectic q-Schur superalgebra

Abstract

We introduce a new quantized enveloping superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ attached to the Lie superalgebra ${\mathfrak{p}}_n$ of type $P$. The superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ is a quantization of a Lie bisuperalgebra structure on ${\mathfrak{p}}_n$ and we study some of its basic properties. We also introduce the periplectic $q$-Brauer algebra and prove that it is the centralizer of the $\mathfrak{U}_q {\mathfrak{p}}_n$-module structure on ${\mathbb C}(n|n)^{\otimes l}$. We end by proposing a definition for a new periplectic $q$-Schur superalgebra.