Transactions of the American Mathematical Society

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



Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras

Author: B. N. Allison
Journal: Trans. Amer. Math. Soc. 306 (1988), 667-695
MSC: Primary 17A75; Secondary 11E04, 17B25, 17B70
MathSciNet review: 933312
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Abstract: In this paper, we study algebras with involution that are isomorphic after base field extension to the tensor product of two composition algebras. To any such algebra $ (\mathcal{A},\, - )$, we associate a quadratic form $ Q$ called the Albert form of $ (\mathcal{A},\, - )$. The Albert form is used to give necessary and sufficient conditions for two such algebras to be isotopic. Using a Lie algebra construction of Kantor, we are then able to give a description of the isomorphism classes of Lie algebras of index $ F_{4,1}^{21}$, $ {}^2E_{6,1}^{29}$, $ E_{7,1}^{48}$ and $ E_{8,1}^{91}$. That description is used to obtain a classification of the indicated Lie algebras over $ {\mathbf{R}}(({T_1}, \ldots ,{T_n})),\;n \leqslant 3$.

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Article copyright: © Copyright 1988 American Mathematical Society