Involutions on Rank 16 Central Simple Algebras

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Authors

  • Mathematik, ETH-Zentrum, Ramistrasse 74, CH-8092, Zurich ,CH
  • School of Mathematics, Tata Institute of Fundamental Research, Bombay-400005 ,IN
  • School of Mathematics, Tata Institute of Fundamental Research, Bombay-400005 ,IN

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.

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Published

1991-12-01

How to Cite

Knus, M. A., Parimala, R., & Sridharan, R. (1991). Involutions on Rank 16 Central Simple Algebras. The Journal of the Indian Mathematical Society, 57(1-4), 143–151. Retrieved from http://informaticsjournals.com/index.php/jims/article/view/21929