Central Simple Algebras with Involution: A Geometric Approach

Details Central Simple Algebras with Involution: A Geometric Approach Central Simple Algebras with Involution: A Geometric Approach

2008-08-12

arxiv.org/abs/0808.1743v1

Manuscripta Mathematica 128 (2009), no. 4, 453-467

Mathematics

Rings and Algebras

16 pages, latex

Scientific paper

10.1007/s00229-008-0242-3

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 $P_{n,\tau}:=PGL_n\rtimes<\tau>$, where $\tau$ is the automorphism of $\PGLn$ given by $\tau(h)=(h^{-1})^transpose$. Basic properties of an involution are described in terms of the action of $P_{n,\tau}$ on the associated variety, and in particular in terms of the stabilizer in general position for this action.

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