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Note on Three-Dimensional Lie Groups

Published online by Cambridge University Press:  18 May 2009

E. M. Patterson
E. M. Patterson
Affiliation:
United College, University of St. Andrews.
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The object of this note is to construct a set of real three-dimensional Lie groups such that every real three-dimensional Lie group is locally isomorphic with some group in the set. The construction is effected by first finding canonical forms for the constants of structure of real three-dimensional Lie algebras; these canonical forms give rise to certain bilinear forms, and the Lie groups are obtained as linear groups isomorphic with groups of automorphisms which leave these bilinear forms invariant.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 2 , Issue 3 , July 1955 , pp. 112 - 115
Copyright
Copyright © Glasgow Mathematical Journal Trust 1955

References

† See, for example, C. Chevalley, “Theory of Lie groups” (princeton University press), Chapter IV.

‡ The suffixes i, j, k, p, q take the values 1, 2, 3 and the summation convention for repeated indices is used.

§ Chevalley, loc. cit.

∥ |P| denotes the determinant of P, and P' denotes the transpose of P.