Classification theorems for central simple algebras with involution (with an appendix by R. Parimala)
Involutions \(\sigma, \sigma'\) on a central simple algebra \(A\) are said to be conjugate if there exists an inner automorphism \(\alpha\) of \(A\) such that \(\sigma' = \alpha \sigma \alpha^{-1}\). In this paper the involutions on a given central simple algebra are classified up to conjugacy. The range of allowable base fields \(F\) is the same as in the classification theorems for quadratic forms or hermitian forms [see \textit{R. Elman} and \textit{T. Y. Lam}, Comment. Math. Helv. 49, 373-381 (1974; Zbl 0297.15024), \textit{E. Bayer-Fluckiger} and \textit{R. Parimala}, Invent. Math. 122, 195-229 (1995; Zbl 0851.11024), Ann. Math., II Ser. 147, 651-693 (1998; Zbl 0909.20029)]. When \(I^3F=0\), all symplectic involutions on a central simple \(F-\)algebra are conjugate, and orthogonal involutions are conjugate if and only if their Clifford algebras are isomorphic. When the field \(F\) is formally real the conditions on \(F\) are a combination of the conditions of the following type: \(I^3F\) is torsion free, \(F\) is SAP, \(F\) and all its quadratic extensions are SAP, the virtual cohomological dimension of \(F\) is \(\leq 2\). The conditions for conjugacy of involutions are expressed in terms of signatures, Clifford algebras and discriminant algebras. The method of proof consists in reducing to Hermitian or skew-Hermitian forms and then applying the classification results of Bayer-Fluckiger and Parimala. In one case the latter are insufficient and a new result is used whose proof has been supplied by R. Parimala in an appendix.
scientific article; zbMATH DE number 1385132