Involutions on Rank 16 Central Simple Algebras

Abstract

Let D be a central division algebra of rank 16 over a field K which represents a 2-torsion element in the Brauer group of K, A classical result of Albert asserts that D splits as a tensor product of quaternion sub-algebras. Further, D admits an involution which is trivial on K. We say that an involution σ on D splits if there exist quaternion subalgebras H₁, of D with involutions σ₁ such that D=H₁⊗H₂ and σ=σ₁⊗σ₂. An involution σ is split if and only if D has σ-invariant quaternion subalgebras. A natural question arises as to when an involution σ on D splits. Examples of rank 16 division algebras with involutions which admit no invariant quaternion subalgebras were constructed in ([2] and [8]). It was proved in ([7], Theorem B, p. 296) that if σ is of even symplectic type and char K≠2, then a splits. One can modify easily the arguments in [7] to include the case char K=2 also in the even symplectic case. In ([5], p. 196) an invariant called pfaffian discriminant was attached to any involution on a central simple algebra of even dimension with values in K*/K*^δ. It was shown in [6], using quadratic form theory and Clifford algebras that if char K≠2 and σ is of orthogonal type on a rank 16 algebra, σ splits if and only if the pfaffian discriminant of σ is trivial. The aim of this paper is to give a criterion for an involution σ on a rank 16 central simple algebra A over a field K to split, without restriction on the characteristic of K. We have included the proofs in the case char K≠2 to make the discussion self-contained.