Ph.D Thesis
Ph.D Student Elimelekh Geva On the Structure of Projective Schur Algebras Department of Mathematics Full Professor Aljadeff Eli

Abstract

Let k be a field. A projective Schur algebra over k is a k-central simple algebra A which is spanned over k by a subgroup of units in A which is finite modulo the center. The projective Schur group of a field k is the subgroup of the Brauer group of k generated by (and in-fact consisting of) classes which may be represented by projective Schur algebras. We prove that the projective Schur group is generated in the Brauer group by cyclic algebras.

This assertion was proved in the case where the field k is of positive-characteristic by E.Aljadeff and J.Sonn in 2001. In this work we establish the case where the characteristic is zero. We also give a bound for the number of cyclic algebras needed to present (up to Brauer equivalence) a projective Schur algebra which is spanned over k by a super-solvable group. The question of whether or not the whole Brauer group is generated by cyclic algebras is a major open problem.