Classification Theorems for Central Simple Algebras with Involution with an Appendix by R. Parimala
نویسنده
چکیده
The involutions in this paper are algebra anti-automorphisms of period two. Involutions on endomorphism algebras of finite-dimensional vector spaces are adjoint to symmetric or skew-symmetric bilinear forms, or to hermitian forms. Analogues of the classical invariants of quadratic forms (discriminant, Clifford algebra, signature) have been defined for arbitrary central simple algebras with involution. In this paper it is shown that over certain fields these invariants are sufficient to classify involutions up to conjugation. For algebras of low degree a classification is obtained over an arbitrary field.
منابع مشابه
Discriminant of Symplectic Involutions
We define an invariant of torsors under adjoint linear algebraic groups of type Cn—equivalently, central simple algebras of degree 2n with symplectic involution—for n divisible by 4 that takes values in H(k, μ2). The invariant is distinct from the few known examples of cohomological invariants of torsors under adjoint groups. We also prove that the invariant detects whether a central simple alg...
متن کاملPfister Involutions
The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven that, up to degree 16, over any extension over which the algebra splits, the involution is adjoint to...
متن کاملOn Totally Decomposable Algebras with Involution in Characteristic Two
A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, in terms of its Frobenius subalgebras, is given. It is also proved that a bilinear Pfister form, recently introduced by A. Dolphin, can classify totally decomposable central simple algebras of orthogon...
متن کاملA Weak Hasse Principle for Central Simple Algebras with an Involution
The notions of totally indefinite and weakly isotropic algebras with involution are introduced and a proof is given of the fact that a field satisfies the Effective Diagonalization Property (ED) if and only if it satisfies the following weak Hasse principle: every totally indefinite central simple algebra with involution of the first kind over the given field is weakly isotropic. This generaliz...
متن کاملGeneric Algebras with Involution of Degree 8m
The centers of the generic central simple algebras with involution are interesting objects in the theory of central simple algebras. These fields also arise as invariant fields for linear actions of projective orthogonal or symplectic groups. In this paper, we prove that when the characteristic is not 2, these fields are retract rational, in the case the degree is 8m and m is odd. We achieve th...
متن کامل