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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Isomorphism of simple Lie algebras


Author: B. N. Allison
Journal: Trans. Amer. Math. Soc. 177 (1973), 173-190
MSC: Primary 17B20
MathSciNet review: 0327852
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Abstract: Let $ \mathcal{L}$ and $ \mathcal{L}'$ be finite dimensional simple Lie algebras over a field of characteristic zero. A necessary and sufficient condition is given for $ \mathcal{L}$ and $ \mathcal{L}'$ to be isomorphic. The anisotropic kernel of $ \mathcal{L}$ is also studied. In particular, a result about this kernel in the rank one reduced case is proved. This result is then used to prove a conjugacy theorem for the simple summands of the anisotropic kernel in the general reduced case. The results and methods of this paper are rational in the sense that they involve no extension of the base field.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0327852-6
Keywords: Isomorphism of Lie algebras, simple Lie algebras, anisotropic kernel, rank one Lie algebras, equivalence of representations, centrum, root systems, Weyl group, subalgebras of Lie algebras
Article copyright: © Copyright 1973 American Mathematical Society