SIMPLE TWO-SIDED RATIONAL VECTOR SPACES OF RANK TWO

Author

John Walker Hart, The University of Montana

Year of Award

2009

Document Type

Dissertation

Degree Type

Doctor of Philosophy (PhD)

Degree Name

Mathematics

Department or School/College

Department of Mathematical Sciences

Committee Chair

Adam Nyman

Commitee Members

Andrew Ware, George McRae, Jennifer Halfpap, Nikolaus Vonessen

Keywords

Simple, Rank, Two-sided

Publisher

University of Montana

Abstract

The purpose of this thesis is to find sufficient conditions under which a non-commutative version of the polynomial ring in two variables exists. The non-commutative rings we construct are non-commutative symmetric algebras over a two-sided vector space. After reviewing the definition of a two-sided vector space and giving some examples, we briefly recall the theory of simple two-sided vector spaces. We then assume k is a field of characteristic zero and t is transcendental over k and we find sufficient conditions under which a simple k-central two-sided vector space V over k(t) has left and right dimension two. Given such a V, and letting ^*V and V^* denote the left and right duals we find conditions under which (V^i*,V^(i+1)*,V^(i+2)* ) has a simultaneous for all i, i an integer. This condition implies the non-commutative symmetric algebra over V can be constructed. We conclude by exhibiting a five-dimensional family of simple k-central two-sided vector spaces over k(t) of left and right dimension two who non-commutative symmetric algebras exist.

Recommended Citation

Hart, John Walker, "SIMPLE TWO-SIDED RATIONAL VECTOR SPACES OF RANK TWO" (2009). Graduate Student Theses, Dissertations, & Professional Papers. 885.
https://scholarworks.umt.edu/etd/885