## Graduation Semester and Year

2022

## Language

English

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy in Mathematics

## Department

Mathematics

## First Advisor

Dimitar Grantcharov

## Abstract

Representation theory of Lie algebra of a finite dimensional reductive Lie algebra g is a long-standing problem. The ultimate goal is to classify all representations of g. However. the only case only case when a complete classification is obtained is the case of g = sl(2). Hence, it is natural to study certain categories of representations of g for which some finiteness conditions on the action of certain elements of g is enforced. In this thesis, we introduce a class of representations T (g, V, S) of sl(n + 1) of mixed tensor type. By varying the polynomial g, the gl(n)-module V , and the set S, we obtain important classes of weight representations over the Cartan subalgebra h of sl(n + 1), and representations that are free over h. Moreover, An isomorphism theorem and simplicity criterion for T(g,V,S) is provided.

## Keywords

Lie algebras, Representation theory

## 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

Nguyen, Khoa Hoang, "Exponential Tensor Modules" (2022). *Mathematics Dissertations*. 227.

https://mavmatrix.uta.edu/math_dissertations/227

## Comments

Degree granted by The University of Texas at Arlington