 
Summary: RESEARCH SUMMARY
ELI ALJADEFF
1. Introduction
My research deals with several topics which share a common ground, namely
crossed product algebras and more generally Ggraded (or Hcomodule) algebras.
These structures appear in a wide variety of topics as 1) Brauer groups 2) Group
cohomology 3) twistings and deformations 4) PItheory. I describe below several of
the main results I obtained in these topics together with my coauthors.
2. Brauer groups
Let k be a field and Br(k) the corresponding Brauer group. It consists of equiva
lence classes of kcentral simple algebras (i.e. finite dimensional over k, with center
k and with no nontrivial two sided ideals). The group structure is defined via the
tensor product over k. The Schur subgroup S(k) of the Brauer group is the subgroup
generated (and in fact consisting of) by classes represented by Schur algebras over
k. By definition, a Schur algebra over k is a kcentral simple algebra which is the
homomorphic image of a group algebra kG for some finite group G. Equivalently,
a Schur algebra over k is a kcentral simple algebra which has a finite spanning
group of unit elements. One of the main results (BrauerWitt) concerning Schur al
gebras is that such an algebra is Brauer equivalent to a cyclotomic algebra, namely
a crossed product algebra (K/k, G, ) where K/k is a cyclotomic field extension
