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Transactions of the American Mathematical Society

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

 

 

Quadratic forms permitting triple composition


Author: Kevin McCrimmon
Journal: Trans. Amer. Math. Soc. 275 (1983), 107-130
MSC: Primary 17A40
DOI: https://doi.org/10.1090/S0002-9947-1983-0678338-7
MathSciNet review: 678338
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Abstract: In an algebraic investigation of isoparametric hypersurfaces, J. Dorfmeister and E. Neher encountered a nondegerate quadratic form which permitted composition with a trilinear product, $ Q\left(\{{xyz} \} \right) = Q(x)Q(y)Q(z)$. In this paper we give a complete description of such composition triples: they are all obtained as isotopes of permutations of standard triples $ \{xyz \} = (xy)z$ or $ x(yz)$ determined by a composition algebra, with the quadratic form $ Q$ the usual norm form. For any fixed $ Q$ this leads to $ 1$ isotopy class in dimensions $ 1$ and $ 2$, $ 3$ classes in the dimension $ 4$ quaternion case, and $ 6$ classes in the dimension $ 8$ octonion case.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0678338-7
Article copyright: © Copyright 1983 American Mathematical Society