Almost-reductive and almost-algebraic Leibniz algebra. (English) Zbl 07903866
Summary: This paper examines whether the concept of an almost-algebraic Lie algebra developed by Auslander and Brezin in [J. Algebra, 8(1968), 295313] can be introduced for Leibniz algebras. Two possible analogues are considered: almost-reductive and almost-algebraic Leibniz algebras. For Lie algebras these two concepts are the same, but that is not the case for Leibniz algebras, the class of almost-algebraic Leibniz algebras strictly containing that of the almost-reductive ones. Various properties of these two classes of algebras are obtained, together with some relationships between \(\phi\)-free, elementary, \(E\) algebras and \(A\)-algebras.
MSC:
| 17B05 | Structure theory for Lie algebras and superalgebras |
| 17B20 | Simple, semisimple, reductive (super)algebras |
| 17B30 | Solvable, nilpotent (super)algebras |
| 17B50 | Modular Lie (super)algebras |
Keywords:
Leibniz algebra; symmetric Leibniz algebra; Frattini ideal; \(\phi\)-free; elementary; \(E\)-algebra; \(A\)-algebra; almost-algebraic; almost-reductiveReferences:
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| [16] | David A. Towers Department of Mathematics and Statistics Lancaster University Lancaster LA1 4YF, England e-mail: d.towers@lancaster.ac.uk |
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