Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T08:19:59.189Z Has data issue: false hasContentIssue false
  • English
  • Français

A decomposition theorem for homogeneous algebras

Part of: Other nonassociative rings and algebras General nonassociative rings

Published online by Cambridge University Press:  09 April 2009

L. G. Sweet  and
J. A. Macdougall
L. G. Sweet
Affiliation:
Department of Mathematics, and Computer Science, University of Prince Edward Island, Charlottetown PEI C1A 4P3, Canada e-mail: [email protected]
J. A. Macdougall
Affiliation:
Department of Mathematics, University of Newcastle, Callaghan NSW 2308, Australia e-mail: mmjam @cc.newcastle.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An algebra A is homogeneous if the automorphism group of A acts transitively on the one dimensional subspaces of A. Suppose A is a homogeneous algebra over an infinite field k. Let La denote left multiplication by any nonzero element a ∈ A. Several results are proved concerning the structure of A in terms of La. In particular, it is shown that A decomposes as the direct sum A = ker La Im La. These results are then successfully applied to the problem of classifying the infinite homogeneous algebras of small dimension.

Keywords

Non-associative algebrasautomorphism group

MSC classification

Secondary: 17D99: None of the above, but in this section 17A36: Automorphisms, derivations, other operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 1 , February 2002 , pp. 47 - 56
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Djoković, D. ž., ‘Real homogeneous algebras’, Proc. Amer. Math. Soc. 41 (1973), 457–462.CrossRefGoogle Scholar
[2]Djoković, D. ž. and Sweet, L. G., ‘Infinite homogeneous algebras are anti-commutative’, Proc. Amer Math. Soc., 127 (1999) 3169–3174.CrossRefGoogle Scholar
[3]Gross, F., ‘Finite automorphic algebras over G F(2)’, Proc. Amer. Math. Soc. 31 (1971), 10–14.Google Scholar
[4]Ivanov, D. N., ‘On homogeneous algebras over G F(2)’, Vestnik Moskov. Univ. Matematika 37 (1982), 69–72.Google Scholar
[5]Kostrikin, A. I., ‘On homogeneous algebras’, Izvestiya Akad. Nauk USSR 29 (1965), 471–484.Google Scholar
[6]MacDougall, J. A. and Sweet, L. G., ‘Three dimensional homogeneous algebras’, Pacific J. Math. 74 (1978), 153–162.CrossRefGoogle Scholar
[7]Shult, E. E., ‘On the triviality of finite automorphic algebras’, Illinois J. Math. 13 (1969), 654–659.Google Scholar
[8]Sweet, L. G. and MacDougall, J. A., ‘Four dimensional homogeneous algebras’, Pacific J. Math. 129 (1987), 375–383.CrossRefGoogle Scholar
[9]Sweet, L. G., ‘On homogeneous algebras’, Pacific J. Math. 59 (1975), 385–594.Google Scholar
[10]Sweet, L. G., ‘On the triviality of homogeneous algebras over an algebraically closed field’, Proc. Amer. Math. Soc. 48 (1975), 321–324.CrossRefGoogle Scholar