M.Sc Thesis
M.Sc Student Shira Zerbib-Gelaki On the Projective Analogue of the Brauer-Witt Theorem Department of Mathematics Full Professor Aljadeff Eli

Abstract

A finite dimensional central simple algebra A over a field k is called projective Schur if it is spanned over k by a group of unit elements, which is finite modulo its center. In 1995 it was conjectured by Aljadeff and Sonn that every projective Schur algebra is Brauer equivalent to a radical algebra (i.e., a crossed product with a radical Galois extension of k and a 2-cocycle with radical values). This conjecture was proved in the past for fields of positive characteristic, but in characteristic zero only partial results are known. In this thesis we prove the conjecture in some cases, in characteristic zero. More precisely, to each projective Schur algebra we associate a faithful irreducible representation of the group that spans it and we prove the conjecture when this representation is monomial and satisfies certain additional conditions. We prove our results using two different approaches. The first approach is more classical in the sense that we work with the projective Schur algebra itself and prove that in our cases it is isomorphic to a matrix algebra over a radical algebra. In the second approach we use descent theory to prove that in our cases the algebras are Brauer equivalent to radical algebras. The benefit of the second approach is that it is more general and can be applied to other projective Schur algebras (i.e., not necessarily with monomial representations). For example, we use it also to prove the conjecture for those algebras with the property that the image of the group that spans it under the corresponding representation is invariant under the Galois action on the entries of the matrices.