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arXiv · 0808.1743

Central Simple Algebras with Involution: A Geometric Approach

Nikolaus Vonessen
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Abstract

Let $k$ be an algebraically closed base field of characteristic zero. The category equivalence between central simple algebras and irreducible, generically free $PGL_n$-varieties is extended to the context of central simple algebras with involution. The associated variety of a central simple algebra with involution comes with an action of the semidirect product $P_{n,τ}:=PGL_n\rtimes<τ>$, where $τ$ is the automorphism of $\PGLn$ given by $τ(h)=(h^{-1})^transpose$. Basic properties of an involution are described in terms of the action of $P_{n,τ}$ on the associated variety, and in particular in terms of the stabilizer in general position for this action.

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Nikolaus Vonessen. 2008-08-12. Central Simple Algebras with Involution: A Geometric Approach. https://doi.org/10.1007/s00229-008-0242-3

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