16-dimensional Algebras with Involution (June 1990)

These notes give a simplified and more concrete approach to some of the very nice results in the paper "Pfaffians, central simple algebras, and similitudes" (Math. Z. Vol. 206 (1991), 149--165) by M.-A.Knus, R. Parimala, and R. Sridharan. We consider 16-dimensional central simple algebras A with first kind involution * over a field F, char(F) not 2. By using the eigenspaces of the Pfaffian discriminant on the 6-dimensional space of skew-symmetric elements of A (if * is of orthogonal type) (resp. symmetric elements if * is of symplectic type) we see that if disc(*) = 1, then A decomposes into Q_1 tensor_F Q_2, where each Q_i is a *-stable quaternion subalgebra of A. Moreover, if * is orthogonal, there are unique such Q_i so that * restricts to the unique symplectic involution on each Q_i.

There are a few fragmentary comments on the characteristic 2 situation. Subsequently, M.-A.Knus, R. Parimala, and R. Sridharan showed in "Involutions on rank 16 central simple algebras," J. Indian Math. Soc., Vol. 57 (1991), 143--151, that the basic result (if disc(*) = 1, then A decomposes into a product of quaternion algebras compatibly with *) still holds if char(F) = 2.

There is no plan to publish these notes.

Adrian Wadsworth / arwadsworth@ucsd.edu