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The Representations of Lie Algebras of Prime Characteristic

1954; Cambridge University Press; Volume: 2; Issue: 1 Linguagem: Inglês

10.1017/s2040618500032974

2051-2104

Hans Zassenhaus,

Advanced Algebra and Geometry

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]).