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