2022

31. 2022.09.26 - Tendai Shumba (Mathematical Center in Akademgorodok), On the automorphism groups of axial algebras video

Let A := (X, F) be an axial algebra with Miyamoto group G. We are interested in finding the full automorphism group, Aut A ≥ G, of A. If there is a grading, this is equivalent to finding all the axes in A. We discuss the techniques used and report on results obtained for some algebras of modest dimensions. In particular, we take G = S4, and F to be the Monster fusion law.

32. 2022.10.10 - Shpectorov S.V., A “solid” approach to algebras of Jordan type half video

Classification of algebras of Jordan type half remains an important open problem. We will discuss an approach to the classification via the concept of a solid subalgebra. Namely, let A be an algebra of Jordan type half. A (2-generated) subalgebra B of A will be called solid when every primitive idempotent from B is an axis of Jordan type half in A. 

In the talk we show that the 2-generated subalgebras B of A are almost always solid. In fact, if the ground field is of characteristic 0, B=<<a,b>> can be non-solid only in two specific situations, where the order of \tau_a\tau_b is 3 or 4. Hence, in characteristic 0, an algebra A without infinite solid lines should have a 4-transposition group for its Miyamoto group. 

We also discuss some examples, found by Gorshkov and Staroletov, of Matsuo algebras which are not Jordan algebras and yet contain infinite solid lines.

33. 2022.11.21 - Faarie Alharbi (Birmingham University), Automorphism groups of Matsuo algebras and aligned 3-transposition groups video

In this talk, we look for the exceptions to the general situation that the automorphism group of a Matsuo algebra (with eta not 1/2) is the same as the automorphism group of the underlying group of 3-transpositions. We will describe some examples where a Matsuo algebra has additional axes and a larger group of automorphisms. Further, we will introduce a method to investigate all cases of pairs of 3-transposition groups that are in cross characteristic and deduce that no new examples arise from this situation.

34. 2022.12.05 - Jari Desmet (Ghent University), Representations of algebraic groups with an interesting axial structure video

In an effort to understand more about the representations of algebraic groups (and in particular E_8), a paper by Tom De Medts and Michiel van Couwenberghe and a paper by Maurice Chayet and Skip Garibaldi independently constructed algebras for a simple adjoint algebraic group G, on which G acts as automorphisms.  In this talk, we give an overview of the construction by Maurice Chayet and Skip Garibaldi, and describe how it leads to idempotents with surprisingly simple fusion laws for types B_n, C_n, F_4 and G_2.