52. 2026.09.29 - Andrey Mamontov, Radicals in primitive axial algebras Video

There are several concepts of the radical of an axial algebra A: 

(1) the largest ideal not containing any of the generating axes;

(2) the Frobenius form radical;  

(3) the Jacobson radical, which we define simply as the intersection of all maximal ideals of A.

In a joint work with S. Shpectorov and V. Zhelyabin we prove that 1 is a subset of 2, which is a subset of 3.

53. 2026.10.13 - Alonso Castillo-Ramírez (Tuesday, 19:00 Novosibirsk time, GMT+7),Fusion rules from the Norton inequality

The Norton inequality is one of the fundamental axioms in the theory of Majorana and axial algebras, yet its precise structural consequences have remained partially understood. In this talk, we shall see how the Norton inequality forces some fusion rules for arbitrary idempotents in a commutative real algebra equipped with a Frobenius form. If the Frobenius form is nondegenerate (as in Majorana algebras), we prove that the 0- and 1-eigenspaces of an arbitrary idempotent are subalgebras and annihilate one another, while in the degenerate case the corresponding inclusions hold modulo the radical of the form. 

54. 2026.10.27 - Roman Avdeev, An infinite series of Gorenstein local algebras failing the affine homogeneity property

Given a commutative associative finite-dimensional local algebra, its socle is the annihilator of the maximal ideal. The algebra is called Gorenstein if its socle is one-dimensional. In the talk, we will present an infinite series of Gorenstein local algebras Aₙ having the following property: the maximal ideal of every algebra Aₙ possesses a one-dimensional subspace that is different from the socle and invariant under the automorphism group of Aₙ. The latter implies that the algebras Aₙ fail the so-called affine homogeneity property. We will introduce necessary definitions, discuss an elementary proof and examine connections with additive actions on projective hypersurfaces.

The talk is based on a joint work with Yulia Zaitseva.

55. 2026.11.10 - Sergey Shpectorov, TBA

56. 2026.11.24 - Louis Rowen, TBA

57. 2026.12.08 - Vsevolod Gubarev, Roman Kozlov, Identities of 2-generated axial algebras of Monster type

If you have any questions please do not hesitate to write by e-mail: ilygor8 (AT) gmail.com

Bibliography:

1) Axial algebras (non-official lecture notes), D. Craven, S. Shpectorov, 2017.

2) Axial algebras, J. McInroy, 2018 (update 2020).