Group gradings on finite-dimensional division algebras

Let A be an algebra over a field k, and let G be a finite group.  We say A is G-graded if there are k-subspaces A_g for all g in G such that A is the direct sum of the subspaces A_g, and A_g A_h is contained in A_gh for all elements g,h of G.  Finite group gradings play an important role in the study of finite-dimensional division algebras and, more generally, in the study of finite-dimensional central simple algebras.  For example, crossed product algebras, which provide the bridge between Brauer groups and Galois cohomology, and symbol algebras, which provide the bridge between Brauer groups and K-theory, are both naturally graded algebras.

We consider the following question: what are all possible (finite) group gradings on finite-dimensional k-central division algebras?

In this talk we give, by means of generic constructions, a complete answer in the case where the center k contains an algebraically closed field of characteristic zero.

This work is joint with Eli Aljadeff and Yakov Karasik.