The Department of Mathematics and Statistics invites students, faculty and staff to attend a talk given by Dr. Basil Nanayakkara on Thursday, August 22nd, 2024 from 2:00 PM to 3:00 PM. The talk is entitled Quaternion algebras and the Frobenius theorem – Part 2.

For location room number, please email Neil Marshall (nmarshall@brocku.ca).

Abstract:
(This is a continuation of the seminar on July 25th. However, you will be able to understand this, even if you didn’t attend Part 1, as we will be discussing mainly the Frobenius theorem).

Given a field k and nonzero elements a, b in k, we’ll construct a quaternion algebra over k associated with a and b. We’ll show that it is a central simple algebra, and hence determines an element in the Brauer group of k. Moreover, we’ll see that this quaternion algebra is a division algebra iff the norm function admits a nontrivial zero.

A related result is the Frobenius theorem, which states that the finite dimensional division algebras over R are R, C and H only, thereby proving that the Brauer group of R is the cyclic group of order 2.

The geometric counterpart of a quaternion algebra is a one-dimensional Brauer-Severi variety. If time permits, we’ll discuss this notion as well.