Group Graded Azumaya Algebras and Generic Constructions

Mini-course by

Professor Eli Aljadeff

Technion University

Israel

February 17 -22, 2023

Professor Eli Aljadeff's last lecture of the mini course (see more pictures below, including the group photo)

Brief description of the mini course

The main theme of this mini-course is gradings by finite groups on finite-dimensional algebras. Similar to the classical situation of ungraded algebras, we will be interested in finite-dimensional graded simple algebras and finite-dimensional graded division algebras. An important role is played by a generalization of central simple algebras, called Azumaya algebras.

Our main tool will be polynomial identities and, in particular, graded polynomial identities. This tool will allow us to construct generic graded Azumaya algebras.

In the first lecture of the mini-course, as a part of the motivation to discuss group graded algebras, I will recall some classical topics such as division algebras, Brauer groups, crossed products and Galois cohomology. Then I will introduce G-graded polynomial identities, where G is a finite group, and discuss generic constructions. In particular, for an arbitrary finite-dimensional G-graded simple algebra A over an algebraically closed field F of characteristic 0, I will construct a generic G-graded Azumaya algebra from which one can obtain by specialization all forms of A in the sense of Galois descent.

As a key application, I will discuss the following problem. It is not difficult to see that for any finite group G, finite-dimensional G-graded division algebras are G-graded simple and they remain G-graded simple upon any extension of the field of scalars.

We will be interested in the opposite direction. Unlike the situation in the ungraded case, where the algebras of n × n matrices always admit forms which are division algebras, this is not generally true in the setting of G-graded algebras.

Suppose that A is a finite-dimensional G-graded simple algebra over an algebraically closed field F. One of the goals of these lectures is to provide necessary and sufficient conditions on the graded structure of A under which A admits forms that are G-graded division algebras. In particular we show that A must be a G-graded simple algebra for which the corresponding generic G-graded simple algebra is a G-graded division algebra.

The lectures have been delivered during three time periods, as shown below. They took place at the St. John's campus of Memorial University and broadcast via Webex. All the times are in Newfoundland Time (NST=UTC-3:30).

Friday, February 17th: 11 am - 12:15 pm, A-1045

Monday, February 20th: 11 am - 12:15 pm, A-1046

Wednesday, February 22nd: 11 am - 12:15 pm, A-1046

The lectures were available online via Webex.