Classification of Maximal Subalgebras and Corresponding Reductive Pairs of Lie Algebra of All 2 ÃÂ 2 Real Matrices
ISSN: 1736-4337

# Journal of Generalized Lie Theory and ApplicationsOpen Access

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## Classification of Maximal Subalgebras and Corresponding Reductive Pairs of Lie Algebra of All 2 ÃÂ 2 Real Matrices

Shtukar U*

Math/Physics Department, North Carolina Central University, Durham, USA

*Corresponding Author:
Shtukar U
Associate Professor, Math/Physics Department
North Carolina Central University
1801 Fayetteville Street, Durham, NC 27707, USA
Tel: 919-597-0375
E-mail: [email protected]

Received date: February 19, 2017; Accepted date: April 10, 2017; Published date: April 17, 2017

Citation: Shtukar U (2017) Classification of Maximal Subalgebras and Corresponding Reductive Pairs of Lie Algebra of All 2 × 2 Real Matrices. J Generalized Lie Theory Appl 11: 263. doi: 10.4172/1736-4337.1000263

Copyright: © 2017 Shtukar U. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

### Abstract

The purpose of the article is to describe all 3-dimensional subalgebras and all corresponding reductive pairs of Lie algebra 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 the canonical bases for subspaces that were introduced. In Part I of this article, all 3-dimensional subalgebras of the given Lie algebra g are classified. All reductive pairs {h, m} with 3-dimensional subalgebras h are found in Part II. Surprisingly, there is only one reductive pair {h, m} with special 3-dimensional subalgebra h and 1-dimensional complement m. Finally, all reductive pairs {h, m} with 1-dimensional subalgebras h of algebra g are classified in Part III of the article.

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