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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Some classes of flexible Lie-admissible algebras


Author: Hyo Chul Myung
Journal: Trans. Amer. Math. Soc. 167 (1972), 79-88
MSC: Primary 17A20
MathSciNet review: 0294419
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Abstract: Let $ \mathfrak{A}$ be a finite-dimensional, flexible, Lie-admissible algebra over a field of characteristic $ \ne 2$. Suppose that $ {\mathfrak{A}^ - }$ has a split abelian Cartan subalgebra $ \mathfrak{H}$ which is nil in $ \mathfrak{A}$. It is shown that if every nonzero root space of $ {\mathfrak{A}^ - }$ for $ \mathfrak{H}$ is one-dimensional and the center of $ {\mathfrak{A}^ - }$ is 0, then $ \mathfrak{A}$ is a Lie algebra isomorphic to $ {\mathfrak{A}^ - }$. This generalizes the known result obtained by Laufer and Tomber for the case that $ {\mathfrak{A}^ - }$ is simple over an algebraically closed field of characteristic 0 and $ \mathfrak{A}$ is power-associative. We also give a condition that a Levi-factor of $ {\mathfrak{A}^ - }$ be an ideal of $ \mathfrak{A}$ when the solvable radical of $ {\mathfrak{A}^ - }$ is nilpotent. These results yield some interesting applications to the case that $ {\mathfrak{A}^ - }$ is classical or reductive.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0294419-7
PII: S 0002-9947(1972)0294419-7
Keywords: Flexible algebra, power-associative algebra, Lie-admissible algebra, nilalgebra, Cartan subalgebra, root space, classical Lie algebra, Levi-factor, reductive Lie algebra
Article copyright: © Copyright 1972 American Mathematical Society