Ph.D Thesis

Ph.D StudentNatapov Michael
SubjectCentral Simple Algebras with a Projective Basis
DepartmentDepartment of Mathematics
Supervisor PROF. Eli Aljadeff


Let k be a field. For each finite group G and cohomology class α in H2(G, k*) one can form the twisted group algebra kG. We refer to G as a projective basis of kG. If α is not trivial the algebra kG may be simple.  The study of central simple algebras with a projective basis is motivated by their importance in the theory of group representations and the theory of algebras graded by a group.

In this work we complete a classification of projective bases in division algebras which are finite dimensional over their centers. Namely, we prove that a finite group G is a projective basis of a division algebra if and only if G is nilpotent and every Sylow p-subgroup of G is contained in the short list of groups Λ (consisting of three families of p-groups).

The question of uniqueness of a projective basis of a k-central division algebra is considered as well.

Next, we consider central simple algebras with a nilpotent projective basis. We show that the structure of

R = kG, with G nilpotent, depends on subquotients of G which have cyclic commutator subgroups. In particular, we obtain a bound on the index of R in terms of such subquotients. In the case 2 † ord(G), or 2 | ord(G) and Ï k, we improve the structure theorem and the index bound by proving that the subquotients are on the list Λ.

In addition, we establish a connection between projective bases and fine gradings on central simple algebras. This enables us to apply the above results to the theory of graded polynomial identities for. We give an explicit construction of an

 S3C6 - graded polynomial identity (of a special kind) foras an example of such application.