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Transactions of the American Mathematical Society

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



The automorphism group of a composition of quadratic forms

Author: C. Riehm
Journal: Trans. Amer. Math. Soc. 269 (1982), 403-414
MSC: Primary 10C05; Secondary 10C04, 20F28
MathSciNet review: 637698
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Abstract: Let $ U \times X \to X$ be a (bilinear) composition $ (u,\,x) \mapsto ux$ of two quadratic spaces $ U$ and $ X$ over a field $ F$ of characteristic $ \ne 2$ and assume there is a vector in $ U$ which induces the identity map on $ X$ via this composition. Define $ G$ to be the subgroup of $ O(U) \times O(X)$ consisting of those pairs $ (\phi ,\,\psi )$ satisfying $ \phi (u)\psi (x) = \psi (ux)$ identically and define $ {G_X}$ to be the projection of $ G$ on $ O(X)$. The group $ G$ is investigated and in particular it is shown that its connected component, as an algebraic group, is isogenous to a product of two or three classical groups and so is reductive. Necessary and sufficient conditions are given for $ {G_X}$ to be transitive on the unit sphere of $ X$ when $ U$ and $ X$ are Euclidean spaces.

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Keywords: Quadratic forms, composition of quadratic forms, algebraic groups, nilmanifolds
Article copyright: © Copyright 1982 American Mathematical Society

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