2021, Jan-Jun

1. 2021.01.26   - Mamontov A.S., axial algebras (Frobenius form, Miyamoto involution) video (in Russian)

2. 2021.02.02   - Mamontov A.S., axial algebras (2-generated a.a., radical of the form, Matsuo algebras) video (in Russian)

3. 2021.02.09   - Staroletov A.M., 3-generated axial algebras video (in Russian)

4. 2021.02.16   - Staroletov A.M., 3-generated axial algebras and Jordan algebras video (in Russian)

5. 2021.03.02   - Gubarev V.Yu., Jordan algebras: classification and Peirce decomposition video (in Russian)

6. 2021.03.09* - Gubarev V.Yu., Jordan algebras: properties of Peirce decomposition video (in Russian)

7. 2021.03.16   - Shpectorov S.V.,  video

(1) why we can assume bijectivity between axes and Miyamoto involutions (based on Hall, Segev, Sh); 

(2) general construction algorithm; 

(3) how to select  values of the Frobenius form to match the orders of \tau_a\tau_b;

(4) results of our calculation; 

(5) Jordan algebra for a group of reflections (based on De Medts-Rehren) 

8. 2021.03.23   - Justin McInroy (University of Bristol), The structure of axial algebras video

Axial algebras are a new class of non-associative algebra, introduced recently by Hall, Rehren and Shpectorov, which have a strong link to groups.  They generalise both a large class of Jordan algebras and also the Griess algebra. In this talk we will discuss various aspects of their structure.  What are their ideals?  How can we easily compute them?  What are the choices for a Frobenius form (a bilinear form which associates with the algebra multiplication)?  Can we decompose the algebra into a direct sum of subalgebras?  When are these subalgebras axial?  What is the `best' (finest) sum decomposition?

This is joint work with Sergey Shpectorov (Birmingham) and Sanhan Khasraw (Salahaddin University-Erbil)

9. 2021.03.30   - Discussion of open problems video

10. 2021.04.06 - Shpectorov S.V., double axis construction video

11. 2021.04.13 - Shpectorov S.V., double axis construction-II video

12. 2021.04.20 - Tkachev V. (Linköping University), Minimal cones and Hsiang algebras-I video

I will discuss a nonassociative algebra approach to certain problems of differential geometry and partial differential equations.  The basic idea comes back to the Freudenthal-Springer-Tits construction of exceptional Jordan algebras: one replaces a study of a cubic form u by the study of a certain commutative algebra A (u) recovering properties of u from the properties of the corresponding algebra, and vice versa. The algebra A(u) is not necessarily associative but it is metrized, i.e. the multiplication operator L(x) by x is self-adjoint. The correspondence u--> A(u) is natural in the sense that many well-established algebraic concepts can be intrinsically read out from the analytic structure of u. Furthermore, the algebra structure A(u) identifies many different geometric and analytic patterns of the corresponding solution u. In my talk, I will explain how this method helps to solve a  long-standing problem on classification of cubic minimal cones, the so-called Hsiang problem. The corresponding algebras have many similarities with axial algebras (any Hsiang algebra is generated by idempotents with the same spectrum and fusion laws).

13. 2021.04.27 - Clara Franchi (Università Cattolica del Sacro Cuore), On the classification of 2-generated primitive axial algebras of Monster type video

I shall talk about the general strategy for the classification of 2-generated primitive axial algebras of Monster type (\alpha, \beta). I’ll define a category for such algebras and construct its universal object. This leads to a natural trichotomy depending whether \alpha=2\beta, \alpha=4\beta and \alpha\not \in \{2\beta,\4\beta\}. I shall discuss the main questions and results in these three cases.

2021.05.04 - No seminar!

14. 2021.05.11 - Mario Mainardis (Università degli Studi di Udine), The Highwater algebra and its cover in characteristic 5 video

The Highwater algebra is essentially the unique case of an infinite-dimensional 2-generated primitive axial algebra of Monster type over an arbitrary field F of characteristic other than 2 and 3. I shall discuss its construction, its group of automorphisms and describe some of its relevant features. I’ll also show that, in case the ground field has characteristic 5 the Highwater algebra has a proper cover which is still a 2-generated primitive axial algebra of Monster type.

15. 2021.05.18 - Takahiro Yabe (The University of Tokyo), On the classification of 2-generated axial algebras of Majorana type - Seminar was stopped because of the net problems

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.

16. 2021.05.25 - Tkachev V. (Linköping University), Minimal cones and Hsiang algebras-II video

In the 2nd part of my talk I will explain how various methods of nonassociative algebra (including the Springer-Freudenthal construction, the Peirce decomposition etc) help to classify Hsiang algebras of minimal cones. In particular, the classification uses a dichotomy of Hsiang algebras (Clifford vs exceptional) which crucially depends on the Jordan algebra structure on the -1/2 Peirce subspace.

17. 2021.06.01 - Shpectorov S.V., Split spin factor algebras video

Recent classification by Yabe of symmetric 2-generated algebras of Monster type introduced several new classes of examples. Trying to make sense of these new algebras, McInroy and the speaker generalised one of them, III(al,1/2,dl), to any number of generators, obtaining a rich family, singularly similar to the class of spin factor Jordan algebras. In the talk, we will describe the new algebras and their properties, including all idempotents, the fusion law, ideals and factors.

Star denotes a seminar held in the mixed format