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Some properties of the Schur multiplier and covers of Lie algebras. (English) Zbl 1132.17003

Summary: We give the structure of all covers of Lie algebras that their Schur multipliers are finite dimensional, which generalizes the work of P. Batten and E. Stitzinger [Commun. Algebra 24, No. 14, 4301–4317 (1996; Zbl 0893.17004)]. Also, similar to a result of K. Yamazaki [J. Fac. Sci., Univ. Tokyo, Sect. I 10, 147–195 (1964; Zbl 0125.01601)] in the group case, it is shown that each stem extension of a finite dimensional Lie algebra is a homomorphic image of a stem cover for it. Moreover, we introduce an ideal in every Lie algebra, which is the smallest ideal contained in the center whose factor algebra is capable, and give some different forms of this ideal. Finally, we study the connection between this ideal and the concept of the Schur multiplier.

MSC:

17B30 Solvable, nilpotent (super)algebras
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17B99 Lie algebras and Lie superalgebras
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References:

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