Classification of 2-Dimensional Subalgebras and Corresponding Reductive Pairs of Lie Algebra of All Real 2 × 2 Matrices

Shtukar U

Abstract

The purpose of the article is to describe all 2-dimensional subalgebras and all corresponding reductive pairs of Lie algebra g of all 2 × 2 real matrices. This Lie algebra is 4-dimensional as a vector space, it’s not simple, and it’s not solvable. The evaluation procedure utilizes canonical bases for subspaces that were introduced. Part I of the article contains necessary basic information. In Part II, all 2-dimensional subalgebras of the given Lie algebra g are classified. All reductive pairs { , m} with 2-dimensional subalgebras h are found in Part III. The separate article contributes classification of all 3-dimensional subalgebras and its reductive pairs. Together, both articles give the total classification of all subalgebras and all reductive pairs of Lie algebra g.