## Graduation Semester and Year

2022

## Language

English

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy in Mathematics

## Department

Mathematics

## First Advisor

Dimitar Grantcharov

## 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 determine representations of the superalgebra $\mathfrak{U}_q\mathfrak{p}_n$ and derive its Drinfeld-Jimbo relations. We prove the triangular decomposition of $\mathfrak{U}_q\mathfrak{p}_n$ and introduce some preliminary results concerning the highest weight representation of $\mathfrak{U}_q\mathfrak{p}_n$. 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 $\mathbb{C}(n|n)^{\otimes \ell}$. Finally, we propose a definition for a new periplectic q-Schur superalgebra.

## Keywords

Lie superalgebras, Quantum groups, Lie algebras, Quantum supergroups, Deformations, Quantum algebras, Brauer algebras, Representation theory, Periplectic Lie superalgebras, Lie superalgebras of Type P, Quantized universal enveloping superalgebras

## Disciplines

Mathematics | Physical Sciences and Mathematics

## License

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

## Recommended Citation

Ahmed, Saber Murad, "Quantized Enveloping Superalgebra of Type P" (2022). *Mathematics Dissertations*. 240.

https://mavmatrix.uta.edu/math_dissertations/240

## Comments

Degree granted by The University of Texas at Arlington