Ph.D Thesis | |

Ph.D Student | Natapov Michael |
---|---|

Subject | Central Simple Algebras with a Projective Basis |

Department | Department of Mathematics |

Supervisor | PROF. Eli Aljadeff |

Let *k* be a field.
For each finite group *G* and cohomology class α in H^{2}(*G***,
***k**) one can form the twisted group algebra *k*_{}*G*. We refer to *G* as a projective
basis of *k*_{}*G*. If α
is not trivial the algebra *k*_{}*G* 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 =* *k*_{}*G*, 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

* S*_{3}⋊*C*_{6} - graded polynomial identity (of a special kind) for_{}as an example
of such application.