35. 2024.02.27 - Albert Gevorgyan (Imperial College London), Monster embeddings of 3-transposition groups via standard Majorana representations video

The Monster group M is the largest sporadic simple group, which has more than 8 · 1053 elements. In addition, it is the group of automorphisms of the 196, 884-dimensional Fischer-Griess algebra VM, which is equipped with a positive definite inner product (·, ·), and a commutative, non-associative algebra ·, which satisfy to the relation (x·y, z) = (x, y·z). The algebra VM is generated by a set of axial vectors A. In 2009, A. A. Ivanov axiomatized some properties of the axes a ∈ A and introduced the notions of Majorana algebra and Majorana representation. Later, Majorana theory proved itself to be a powerful machinery to study the subgroup structure of M, and the subalgebra structure of VM.

The 3-transposition groups with a trivial center and a simple derived subgroups are categorized by B. Fischer. In addition, the Monster group M contains subgroups isomorphic to quite big 3-transposition groups, or their subgroups of index 2. Therefore, there is a motivation to study Majorana representations of 3-transposition groups. 

Firstly, we find the sizes of the maximal symmetric subgroups of the groups from the Fischer list, generated by the transpositions. Then, we use this information to find all pairs of 3-transposition groups from the Fischer list, which can be embedded into each other. Furthermore, we find groups from the Fischer list, which admit a standard Majorana representation. The main result is that a group from the Fischer list, except possibly F i24, admits a standard Majorana representation if and only if it can be embedded in the Monster group.

36. 2024.03.12 - Michael Turner (University of Birmingham), Binary Axial Algebras video slides

For an axial algebra with a C_2 graded fusion law, we can define a natural automorphism for each axis called the Miyamoto involution. Binary axial algebras have pairs of axes with the property that the pairs are invariant under the Miyamoto involutions. We will start by a quick introduction to assure that everyone is on the same page. Defining binary axial algebras formally, we can produce a binary diagram to represent how each pair acts on each other. Further, we can say a few results for general binary axial algebras before focusing the rest of the talk on the fusion law being Monster type. Firstly, we will present generalisations of some 2-generated axial algebras to n-generated algebras which are binary. Secondly, we will look at 3-generated binary axial algebras, giving examples and partial classifications as well as open problems. 

37. 2024.03.26 -  Jari Desmet (Ghent University),  A characterization of Jordan algebras using solid lines video slides

The classification of primitive axial algebras of Jordan type half is still an open problem. Recently, Gorshkov, Shpectorov and Staroletov introduced solid subalgebras to tackle this problem. In this talk, I give a method to prove that a primitive axial algebra of Jordan type half is a Jordan algebra if and only if all its 2-generated subalgebras are solid, over fields of characteristic not equal to 2. Using similar techniques, we extend the result by Gorshkov, Shpectorov and Staroletov that 2-generated subalgebras are solid whenever they contain more than 3 axes to positive characteristic. 

38. 2024.04.09 - Vsevolod Gubarev (joint with F. Mashurov and A. Panasenko), Generalized sharped cubic form and split spin factor algebra video slides

There is a well-known construction of a Jordan algebra via a sharped cubic form. We introduce a generalized sharped cubic form and prove that the split spin factor algebra is induced by this construction and satisfies the identity ((a,b,c),d,b) + ((c,b,d),a,b) + ((d,b,a),c,b) = 0.

39. 2024.04.23 - Sergey Shpectorov, The universal baric algebra of Jordan type half video slides

By the result of De Medts, Rowen and Segev, 4-generated algebras of Jordan type half have dimension at most 81. In a joint project with Yunxi Shi, we tried to see what the universal 4-generated algebra of Jordan type half looks like in the simplest case where all values of the Frobenius form on pairs of axes are equal to one. It turns out that in this case the dimension 81 is only reached in characteristic 3, while in all other characteristic there is a much tighter bound. In the final part of the talk, we will discuss the possibility of extending our methods to arbitrary values of the Frobenius form.

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).