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Sur les éléments réguliers dans les algèbres de Lie réductives. (Regular elements of reductive Lie algebras). (French) Zbl 0681.17008
Let $${\mathfrak g}$$ be a finite dimensional Lie algebra over a field k of characteristic 0 and $$f\in {\mathfrak g}^*$$. Denote by $$B_ f$$ the alternating bilinear form (x,y) $$\mapsto f([x,y])$$ on $${\mathfrak g}\times {\mathfrak g}$$ and by $${\mathfrak g}^ f$$ the kernel of $$B_ f$$. The index of $${\mathfrak g}$$ is the integer $$m_{{\mathfrak g}}=Inf\{\dim {\mathfrak g}$$; $$f\in {\mathfrak g}^*\}$$ and f is called a regular linear form on $${\mathfrak g}$$ if dim $${\mathfrak g}^ f=m_{{\mathfrak g}}$$. The set of regular forms on $${\mathfrak g}$$ is denoted by $${\mathfrak g}^*_ r$$. It is known that for $$f\in {\mathfrak g}^*$$, $${\mathfrak g}^ f$$ is a commutative Lie algebra but there exists an example of a nilpotent Lie algebra which shows that the converse is inexact.
In this paper, the author shows that if $${\mathfrak g}$$ is reductive, $$f\in {\mathfrak g}^*_ r$$ if and only if [$${\mathfrak g}^ f,{\mathfrak g}^ f]=0$$. The theorem follows from the fact that for a simple Lie algebra $${\mathfrak g}$$, a distinguished element x of $${\mathfrak g}$$ is regular if and only if the centralizer of x in $${\mathfrak g}$$ is commutative. This fact is shown for each type of simple Lie algebra separately.
Reviewer: E.Abe

##### MSC:
 17B20 Simple, semisimple, reductive (super)algebras 20G15 Linear algebraic groups over arbitrary fields