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Weak super-representations of Mal’tsev superalgebras. (Super-représentations faibles de superalgèbres de Malcev.) (French) Zbl 1106.17308
The authors give some properties of a Mal’tsev superalgebra on a field \(K\) of characteristic zero (for details see M. Scheunert [On the theory of Lie super algebras. An introduction, Lectures Notes Math. 716 (1979; Zbl 0407.17001)] and A. Koulibaly [Contributions à la théorie des algèbres de Malcev. Cahier Mathematique, 33, USTL, Montpellier (1985; Zbl 0685.17016)].
It is defined the notion of weak super-representation of a Mal’tsev superalgebra on a \(\mathbb Z_2\)-graded vector space. Each Mal’tsev super-representation is a weak super-representation. Also are given the conditions in which the conversely assertion is true, too.
Lie triple superalgebras and Lie triple supersystems are defined and it is proved that a Mal’tsev superalgebra induce a Lie triple superalgebra. This result generalizes the similar result for Mal’tsev algebra of K. Yamaguti [Kumamoto J., Ser. A 6, 9–45 (1969; Zbl 0138.26203)].

MSC:
17D10 Mal’tsev rings and algebras
17A70 Superalgebras
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