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Lie Algebras with Nilpotent Centralizers

Published online by Cambridge University Press:  20 November 2018

G. M. Benkart  and
I. M. Isaacs
G. M. Benkart
Affiliation:
The University of Wisconsin, Madison, Wisconsin
I. M. Isaacs
Affiliation:
The University of Wisconsin, Madison, Wisconsin
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We consider finite dimensional Lie algebras over an algebraically closed field F of arbitrary characteristic. Such an algebra L will be called a centralizer nilpotent Lie algebra (abbreviated c.n.) provided that the centralizer C(x) is a nilpotent subalgebra of L for all nonzero x ∈ L.

For each algebraically closed F, there is a unique simple Lie algebra of dimension 3 over F which we shall denote S(F). This algebra has a basis e−1, e0, e1 such that [e−1e0] = e−1, [e−1e1] = e0 and [e0e1] = e1. (If char(F) ≠ 2, then S(F) ≅ sl2(F).) It is trivial to check that S(F) is a c.n. algebra for all F.

There are two other types of simple Lie algebras we consider. If char (F) = 3, construct the octonion (Cayley) algebra over F.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 5 , 01 October 1979 , pp. 929 - 941
Copyright
Copyright © Canadian Mathematical Society 1979

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