The automorphism group of a composition of quadratic forms
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- Trans. Amer. Math. Soc. 269 (1982), 403-414 Request permission
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.References
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Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 269 (1982), 403-414
- MSC: Primary 10C05; Secondary 10C04, 20F28
- DOI: https://doi.org/10.1090/S0002-9947-1982-0637698-2
- MathSciNet review: 637698