Hostname: page-component-84b7d79bbc-lrf7s Total loading time: 0 Render date: 2024-07-30T04:57:05.537Z Has data issue: false hasContentIssue false

XXII.—Some Nilpotent Algebras with Centres of Given Dimension

Published online by Cambridge University Press:  14 February 2012

E. W. Wallace
Affiliation:
Department of Mathematics, University of Leeds.

Synopsis

Algebras which are nilpotent and anti-commutative are studied. Canonical forms are found for all such algebras of dimension n whose centres have dimension n−r (r < 3), and characters are given which enable any two non-isomorphic algebras to be distinguished.

A metrisable Lie algebra is a Lie algebra for which there is a non-singular, symmetric, adjoint-invariant bilinear form a(λ, μ), and such an algebra is reduced if its centre is contained in its derived algebra. The importance of the reduced algebras follows from the fact that every metrisable Lie algebra is the direct sum of a reduced metrisable Lie algebra and an abelian Lie algebra. Tsou (Thesis 1955) introduced metrisable Lie algebras, and obtained canonical forms for all real reduced metrisable Lie algebras whose derived algebras have dimension 3. We conclude this paper by providing an alternative derivation, two of the algebras being nilpotent.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1961

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES TO LITERATURE

Lee, H. C., 1947. “Sur les groupes de Lie réels à trois paramètres” J. Math. Pures Appl., 26, 251267.Google Scholar
Patterson, E. M., 1954. “Note on Nilpotent and Solvable algebras”, Proc. Camb. Phil. Soc., 51, 3740.CrossRefGoogle Scholar
Tsou, S-T., 1955. “On metrisable Lie groups and Lie algebras”, Thesis, University of Liverpool, 1955.Google Scholar
Tsou, S-T., and Walker, A. G., 1957. “Metrisable Lie groups and algebras”, Proc. Roy. Soc. Edin., 64A, 290304.Google Scholar