Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-07T03:48:29.862Z Has data issue: false hasContentIssue false
  • English
  • Français

The Representations of Lie Algebras of Prime Characteristic

Published online by Cambridge University Press:  18 May 2009

Hans Zassenhaus
Hans Zassenhaus
Affiliation:
McGill University, Montreal, P.Q., Canada
Rights & Permissions [Opens in a new window]

Extract

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.

There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are

(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.

(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf. [4]).

(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of prime characteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf. [4]).

(4) There are faithful fully reducible representations of every Lie-algebra of prime characteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf. [6], [2], [4]).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 2 , Issue 1 , April 1954 , pp. 1 - 36
Copyright
Copyright © Glasgow Mathematical Journal Trust 1954

References

REFERENCES

(1)Birkhoff, G., “Representability of Lie Algebras”, Annals of Math., (2), 38 (1937), pp. 526–532.CrossRefGoogle Scholar
(2)Chang, Ho-Jui, “Ueber Wittsche Lie-Ringe”, Abh. Math. Sem. Univ. Hamburg, 14 (1941), pp. 151–196.CrossRefGoogle Scholar
(3)Jacobson, N., “Abstract derivation and Lie Algebras”, Trans. Am. Math. Soc., 42 (1937), pp. 206–224.CrossRefGoogle Scholar
(4)Jacobson, N., “ A note on Lie algebras of characteristic p”, Am. J. Math., 74 (1952), pp. 357–359.CrossRefGoogle Scholar
(5)Witt, E., “Treue Darstellung Lie'scher Ringe”, Crelles Journal f. Reine u. Angew. Math., 177 (1937), pp. 152–160.CrossRefGoogle Scholar
(6)Zassenhaus, H., “Ueber Lie'sche Ringe mit Primzahl Characteristik”, Abh. Math. Sem. Univ. Hamburg, 13 (1939), pp. 1–100.CrossRefGoogle Scholar
(7)Zassenhaus, H., “Ein Verfahren jeder endlichen p-Gruppe einen Lie-Ring der Characteristik p zuzuordnen”, Abh. Math. Sem. Univ. Hamburg, 13 (1939), pp. 200–207.CrossRefGoogle Scholar
(8)Zassenhaus, H., “Darstellungstheorie nilpotenter Lie-Ringe bei Characteristik p>0” Crelles Journal f. Reine u. Angew. Math., 185 (1940).Google Scholar
(9)Zassenhaus, H., The Theory of Groups, Chelsea, 1949.Google Scholar