Classification of the Subalgebras of the Algebra of All 2 by 2 Matrices
Classification of the subalgebras of the familiar algebra of all n × n real matrices over the real numbers can get quite unwieldy as all subalgebras are of dimension ranging from 1 to n2. Classification of the subalgebras of the algebra of all 2 × 2 real matrices over the real numbers is an interesting first start. Since M2(R) 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 M2(R). 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 M2(R) are there? We show here that up to an isomorphism there are three distinct two-dimensional subalgebras and one distinct three-dimensional subalgebra.
Bernal, Justin Luis, "Classification of the Subalgebras of the Algebra of All 2 by 2 Matrices" (2020). ETD Collection for University of Texas, El Paso. AAI27964538.