M.Sc Thesis | |

M.Sc Student | Zerbib-Gelaki Shira |
---|---|

Subject | On the Projective Analogue of the Brauer-Witt Theorem |

Department | Department of Mathematics |

Supervisor | PROF. Eli Aljadeff |

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.