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Adjoint operators in Lie algebras and the classification of simple flexible Lie-admissible algebras


Authors: Susumu Okubo and Hyo Chul Myung
Journal: Trans. Amer. Math. Soc. 264 (1981), 459-472
MSC: Primary 17B10; Secondary 17A20, 81C40
DOI: https://doi.org/10.1090/S0002-9947-1981-0603775-4
MathSciNet review: 603775
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Abstract: Let $ \mathfrak{A}$ be a finite-dimensional flexible Lie-admissible algebra over an algebraically closed field $ F$ of characteristic 0. It is shown that if $ {\mathfrak{A}^ - }$ is a simple Lie algebra which is not of type $ {A_n}(n \geqslant 2)$ then $ \mathfrak{A}$ is a Lie algebra isomorphic to $ {\mathfrak{A}^ - }$, and if $ {\mathfrak{A}^ - }$ is a simple Lie algebra of type $ {A_n}(n \geqslant 2)$ then $ \mathfrak{A}$ is either a Lie algebra or isomorphic to an algebra with multiplication $ x \ast y = \mu xy + (1 - \mu )yx - (1/(n + 1))\operatorname{Tr} (xy)I$ which is defined on the space of $ (n + 1) \times (n + 1)$ traceless matrices over $ F$, where $ xy$ is the matrix product and $ \mu \ne \frac{1} {2}$ is a fixed scalar in $ F$. This result for the complex field has been previously obtained by employing an analytic method. The present classification is applied to determine all flexible Lie-admissible algebras $ \mathfrak{A}$ such that $ {\mathfrak{A}^ - }$ is reductive and the Levi-factor of $ {\mathfrak{A}^ - }$ is simple. The central idea is the notion of adjoint operators in Lie algebras which has been studied in physical literature in conjunction with representation theory.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0603775-4
Keywords: Lie algebra, flexible Lie-admissible algebra, adjoint operator, weight, highest adjoint weight, Weyl basis, reductive Lie algebra, adjoint dimension, representation
Article copyright: © Copyright 1981 American Mathematical Society

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