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Cartan subalgebras of a Malcev superalgebra. (Sous-algèbres de Cartan d’une superalgèbre de Malcev.) (French) Zbl 1080.17500
The paper under review has two different parts. The first one is devoted to results about the solvability of Mal’tsev superalgebras and the last one to results on Cartan subalgebras of these superalgebras.
A word of caution is in order, relative to the first part. Most of the new results there are consequences of Proposition 2.12, which asserts that the odd part of any non-Lie Mal’tsev superalgebra is contained in the solvable radical. Unless new hypotheses are added, this is not true as is shown by the direct sum of any non-Lie Mal’tsev algebra and a simple Lie superalgebra; the ideal generated then by the odd part is the simple Lie superalgebra, which is plainly not solvable.
The definition given in the last part of the paper of a Cartan subalgebra of a Mal’tsev superalgebra is quite restrictive and, when restricted to Lie superalgebras, does not agree with the usual definition. Some first properties of these subalgebras are shown.
17D10 Mal’tsev rings and algebras
17A70 Superalgebras