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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
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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

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