The Classification of Low Dimensional Nilpotent Lie Algebras

Author

Kimberli C. Tripp, Utah State University

Date of Award

2002

Degree Type

Report

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

Committee Chair(s)

Mark Fels

Committee

Mark Fels

Committee

Jerry Ridenhour

Committee

Ian Anderson

Abstract

Nilpotent Lie algebras are the fundamental building blocks for generic (not semi-simple) Lie algebras. In particular, the classification of nilpotent algebras is the first step in classifying and identifying solvable Lie Algebras. The problem of classifying nilpotent Lie algebras was first studied by Umlauf [9] in 1891. More recently, classifications have been given up to dimension six using different techniques by Morosov (1958) [7], Skjelbred and Sund (1977) [8], and up to dimension five by Dixmier (1958) [2]. Using Morosov's method of classification by maximal abelian ideals, Winternitz reproduced the Morosov classification obtaining different canonical forms for the algebras. The Winternitz tables are included in Appendix A.

In chapter 1, we present some of the basic theory behind Lie Algebras and define some of the fundamental invariants, which are useful in determining whether or not two Lie a lgebras are isomorphic. Chapter 2 addresses more of the technical theory of nilpotent Lie algebras. The definitions associated with and methods of classification by maximal abelian ideal and centra l extensions are briefly discussed here. Chapter 3 gives examples of classification by basic invariants and an example of a central extension, as well as a rough idea of how maximal abelian ideals can be used to form a classification. In chapter 4, we give tables of the invariants which allow us to identify a nilpotent algebra of dimension five and six in the classification by Morosov /Winternitz. In this chapter we also give a table of how to find the appropriate maximal abelian ideal for the Winternitz classification for nilpotent algebras in dimension five and six.

Recommended Citation

Tripp, Kimberli C., "The Classification of Low Dimensional Nilpotent Lie Algebras" (2002). All Graduate Plan B and other Reports, Spring 1920 to Spring 2023. 1196.
https://digitalcommons.usu.edu/gradreports/1196