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

Comments

Degree granted by The University of Texas at Arlington

Included in

Mathematics Commons

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