## Open Access Theses & Dissertations

2020-01-01

#### Degree Name

Master of Science

#### Department

Mathematical Sciences

Piotr J. Wojciechowski

#### Abstract

Classification of the subalgebras of the familiar algebra of all $n\times n$ real matrices over the real numbers can get quite unwieldy as all subalgebras are of dimension ranging from $1$ to $n^2$. Classification of the subalgebras of the algebra of all $2\times 2$ real matrices over the real numbers is an interesting first start.

Since $\2$ is of dimension $4$ then its possible subalgebras are of dimension $1, 2, 3,$ or $4$. The one-dimensional subalgebra and four-dimensional subalgebra need little to no attention. The two-dimensional and three-dimensional subalgebras however turn out to be of significance.

It turns out there is only one one-dimensional subalgebra and one four-dimensional subalgebra of $\2$. The former being fairly simple and the latter being trivial. The investigation of the two-dimensional and three-dimensional subalgebras is not as brief. Therefore, the goal of this Thesis is to answer the following question:

Up to an isomorphism, how many distinct two-dimensional and three-dimensional subalgebras of $\2$ are there?

We show here that up to an isomorphism there are three distinct two-dimensional subalgebras and one distinct three-dimensional subalgebra.

en

50 pages

application/pdf

#### Rights Holder

Justin Luis Bernal

COinS