2021, Sep-Dec

18. 2021.09.07 - Takahiro Ybe (The University of Tokyo), On the classification of 2-generated axial algebras of Majorana type video chat

I will talk about the details of classification of 2-generated axial algebras which is called Majorana type (Monster type). A 2 generated symmetric axial algebra of Majorana type is isomorphic to one of 18 universal type or their quotients if its axial dimension is less than 6 or it is an algebra over the field of characteristic not 5. I will talk about the definition of some algebras as examples and sketch of my proof.

19. 2021.09.14 - Roman Kozlov (Sobolev Institute of Mathematics), Introduction to the vertex algebras: basic concepts video

In the first of two talks basic notions and properties will be given. Will be stated and proven basic lemmas and theorems, presented and applied technique to finding Grobner-Shirshov bases and, hence, linear bases. The presentation will be held within two different approaches to treat vertex algebras: as vector spaces with a list of axioms and defined so called "formal distributions" or as algebraic systems endowed with a structure of left-symmetric algebra and conformal algebra at the same time.

20. 2021.09.21 - Roman Kozlov (Sobolev Institute of Mathematics), Introduction to the vertex algebras: lattices and Monster video

In the second talk we introduce a concept of vertex operator algebras (VOA), particular vertex algebras with inner Virasoro structure. Then will be explicitly constructed a lattice VOA for any lattice L. At last, will be constructed and studied a particular lattice VOA, the Monster module, which contains the Griess algebra and has the monster group as the group of automorphisms.

21. 2021.09.28 - Justin McInroy (University of Bristol), Axets and shapes in axial algebras video

The shape of an axial algebra was first introduced by Ivanov and is the configuration of 2-generated subalgebras on the axes. This is analogous to a group amalgam. Similarly to the group case, we want to be able to talk about shapes without an algebra. To do this, we introduce an axet, which abstracts the properties of a closed set of axes, and shapes on an axet. In this talk, we will introduce axets and shapes and describe some of their properties. We will classify the 2-generated axets which will give us a new family which has a hitherto unseen configuration of axes. As an example, we will calculate the axet, and hence the size of a closed set of axes, for the 2-generated algebras of Jordan type 1/2. This is joint work with Sergey Shpectorov (Birmingham).

22. 2021.10.05 - Tendai M. Mudziiri Shumba (University of Johannesburg), Axial algebras for the sporadic simple group HS video

We present constructions of axial algebras for the Higman-Sims sporadic simple group via Norton algebras. Fusion laws are presented as well as the extensions of these algebras by unit. This is work that was part of a PhD thesis supervised by Bernardo Rodrigues and Sergey Shpectorov.

23. 2021.10.12 - Shpectorov S.V., How to find the full automorphism groups of Matsuo algebras video

I will discuss a possible approach to finding the full automorphism group of a general Matsuo algebra with the parameter eta not equal to 1/2. Equivalently, it means finding the complete set of axes of Jordan type in such algebras. I’ll start with reviewing the paper by Hall and the speaker on the spectra of the diagrams of 3-transposition groups focussing on the simplest case of symmetric groups. I will demonstrate how the tables in this paper can be used to solve the above problem.

24. 2021.10.19 - Gubarev V. (joint with Gorshkov I.), Quasi-definite axial algebras of Jordan type 1/2 video

We consider axial algebras of Jordan type 1/2 which satisfy additional conditions on the Frobenius form and on the property of idempotents. Under such conditions we may state when an axial algebra of Jordan type 1/2 is unital or finite-dimensional.

25. 2021.10.26 - Segev Y. (Ben-Gurion University), Axes in non-commutative algebras video

Let A be a non-associative (i.e. not necessarily associative) non-commutative algebra, with or without an identity element over a field F (usually of characteristic not 2). We start experimenting with the notion of an axis in such an algebra, in the simplest possible way. I will present results about 2-generated such algebras. There are open problems, and, indeed, this is sort of an experiment. One possible goal, is that in generalizing away from commutativity, one can come back to commutative primitive axial algebras of Jordan type (for example) and get results there. This is joint work with Louis Rowen.

26. 2021.11.02 - Yunxi Shi (University of Birmingham), Axial algebras of Monster type (2\eta, \eta) for symplectic and orthogonal groups over F_2 video

In this talk, we investigate the classes of involutions of the isometry group of a non-degenerate symplectic space over the field with two elements and we apply the double axis construction to build new axial algebras of Monster type (2\eta, \eta). We also consider the case of the orthogonal space and describe the involutions and flip subalgebras in this case.

27. 2021.11.09 - Pilar Paez Guillán (University of Santiago de Compostela), On central extensions of axial algebras video

In this talk, we will describe a method for constructing axial algebras from a given one, basing on the method of Skjelbred-Sund for classifying nilpotent Lie algebras by means of central extensions. We will discuss the usefulness of this technique and apply it to some examples, such as all axial algebras of dimension 2 over an algebraically closed field.

28. 2021.11.16 - Staroletov A.M., On 3-generated groups of Jordan type video

Inspired by the work of Gorshkov and Staroletov on 3-generated primitive axial algebras of Jordan type, we study the corresponding 3-generated Miyamoto groups. Of particular interest are the groups for algebras over quadratic fields: in these cases, the order of two Miyamoto involutions lies in a small list of values (if finite). As a consequence, we get a class of groups that extends the class of 3-transposition groups.

2021.11.23 - No seminar!

29. 2021.11.30 - Fox D. (Universidad Politécnica de Madrid), Partial associativity conditions and trace-forms video chat

There will be explained, for general not necessarily associative algebras, notions of partial and quantitative associativity, called projective and sectional nonassociativity - for which the Norton inequality is motivating - and their interaction with the invariance of certain trace forms (e.g. the Killing type trace form) will be discussed through some simple examples. As a toy model, there will be described the characterization in these terms of the algebra structure on the standard representation of the symmetric group as the unique up to isomorphism projectively associative, Killing metrized, exact commutative algebra. These notions are complementary to the axial algebra framework, and it is hoped they suggest some interesting questions.

30. 2021.12.07 - Ivanov A.A., Towards Majorana representation of U_3(5) video

This work is with contributions of Andries Brouwer, Clara Franchi, Willian Giuliano, Mario Mainardis. We are aimed to construct a Majorana representation of the group U_3(5). The shape is uniquely determined and the Majorana axis are arranged in a distance regular graph. We estimate the dimension of the representation based on embedding into the Monster and calculate the inner product to get the rank. There is still a step to be completed.