Hyperbolic Involutions and Quadratic Extensions

Abstract

This is a variation on a theme of Bayer-Fluckiger, Shapiro, and Tignol related to hyperbolic involutions. More precisely, criteria for the hyperbolicity of involutions of quadratic extensions of simple algebras and involutions of the form σ ⊗ τ and σ ⊗ ρ, where σ is an involution of a central simple algebra A, τ is the nontrivial automorphism of a quadratic extension of the center of A, and ρ is an involution of a quaternion algebra are obtained.

Key Words:

• Central simple algebra
• Hermitian form
• Hyperbolic involution

2000 Mathematics Subject Classification:

• 16W10
• 11E39

ACKNOWLEDGMENTS

I am indebted to the referee for several useful suggestions which led to a significant improvement of the organization of this article. In particular, the statement of Theorem 3.1 which unified several results of an earlier version of this article, is due to the referee. I thank Behrooz Mirzaii for careful reading of an earlier version of this manuscript and for his useful suggestions. I also thank Eva Bayer-Fluckiger and Karim Becher for their constructive comments on this work. I acknowledge the financial support provided by the Research Council of Sharif University of Technology.

Notes

Communicated by A. Prestel.