Highest Weight Modules Over The Quantum Periplectic Superalgebra of Type $P$

TL;DR

This paper explores the structure of highest weight modules over the quantum superalgebra of type P, introducing new representations, establishing a triangular decomposition, and analyzing module categories.

Contribution

It introduces a Drinfeld-Jimbo representation for the quantum superalgebra of type P and studies the category of tensor modules.

Findings

Triangular decomposition of ${f U}_q {f p}_n$ established

Relation between modules over ${f U}_q {f p}_n$ and ${f p}_n$ clarified

Category of tensor modules shown to be non-semisimple

Abstract

In this paper, we begin the study of highest weight representations of the quantized enveloping superalgebra ${\mathfrak U}_q {\mathfrak p}_n$ of type $P$. We introduce a Drinfeld-Jimbo representation and establish a triangular-decomposition of ${\mathfrak U}_q {\mathfrak p}_n$. We explain how to relate modules over ${\mathfrak U}_q {\mathfrak p}_n$ to modules over ${\mathfrak p}_n$, the Lie superalgebra of type $P$, and we prove that the category of tensor modules over ${\mathfrak U}_q {\mathfrak p}_n$ is not semisimple.