LisMath Seminar 

15/07/2019, 16:45 — 17:15 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
Jocelyn Lochon, LisMath, Faculdade de Ciências, Universidade de Lisboa

A Supercharacter Theory for approximately finite algebra groups

By an algebra group over a field $\mathbb{K}$ it is meant a group of the form $G = 1+A$, where $A$ is a nil algebra over $\mathbb{K}$ and product rule given as $(1+a)(1+b) = 1+a+b+ab$; the group $G = 1 + A$ is said to be an approximately finite algebra group if there is a family $\{G_n\}_{n \in \mathbb{N}}$ of finite algebra subgroups for which $G$ is the direct limit $\lim_{\rightarrow} G_n$.

Assuming mild conditions on a topological group, there is a well defined notion of characters that extend the usual Character Theory of finite, or more generally compact groups; in this setting indecomposable characters play the role of irreducible characters as they fully determine the Character Theory and serve as minimal group invariants. However, the set of indecomposable characters may be too large or even too complicated to characterize, for this reason it is of interest to consider a smaller family of characters that mimics the behaviour of indecomposable ones.

In this talk we generalize the definition of a Supercharacter Theory for finite groups into the topological group scenario, and using essentially ergodic theoretical tools we define and characterize a Supercharacter Theory for an arbitrary approximately finite algebra group.