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41. 2024.09.24 - Louis Rowen, Weakly primitive axial algebras (jointly with Y. Segev) slides video
In earlier work we studied the structure of primitive axial algebras of Jordan type (PAJ's), not necessarily commutative, in terms of their primitive axes. In this paper we weaken primitivity and permit several pairs of (left and right) eigenvalues satisfying a more general fusion rule, bringing in interesting new examples such as the band semigroup algebras and various noncommutative examples. Also we broaden our investigation to the case of 2-generated algebras for which only one axis satisfies the fusion rules.
As an example we describe precisely the 2-dimensional axial algebras and the 3-dimensional and 4-dimensional weakly primitive axial algebras of Jordan type (weak PAJ's), and we see, in contrast to the case for PAJ's, that there are higher dimensional weak PAJ's generated by two axes.
We also obtain a Frobenius form.
42. 2024.10.08 - Sergey Shpectorov, Radicals in Matsuo algebras and their flip subalgebras (joint project with B. Rodrigues) slides video
Matsuo algebras, introduced by Matsuo in 2007 are an important class of algebras of Jordan type. Every flip (an automorphism of order 2) \sigma of a Matsuo algebra M defines a flip subalgebra of M generated by all single and double axes fixed by \sigma. These flip subalgebras can be viewed as twisted versions of Matsuo algebras and they belong to the class of algebras of Monster type (2\eta,\eta). There is currently a project underway focussing on the classification of flips and properties of flip subalgebras.
One of the key properties of an axial algebra is whether it is (semi)simple and if not then what is its radical. It turns out that both Matsuo algebras and their flip subalgebras are generically (i.e., for all but finitely many values of the parameter \eta) (semi)simple, that is, they have trivial radical. The exceptional values of \eta, for which the radical is non-zero, are called critical. Hall and Shpectorov suggested a method of finding the critical values for an arbitrary Matsuo algebra and finding the dimension of the radical.
In the talk we will present a generalisation of this method to the class of flip subalgebra. It turns out that flip subalgebras have the same critical values as their ambient Matsuo algebras and The dimension of the radical can be found by solving a simple system of linear equations.
43. 2024.10.22 - Ching Hung Lam, Some 3-transposition groups arising from VOA theory video1 video2
We will discuss several examples of 3-transposition groups that can be realized as automorphism subgroups of vertex operator algebras and explain the ideas behind such constructions. In particular, we will discuss some generalizations of the theory of Miyamoto involutions associated with simple Virasoro VOA of central charge 1/2 to other Virasoro VOA in the unitary series.
44. 2024.11.05 - Michael Turner, Double axes and constructing axial algebras of Monster type slides video
For two orthogonal axes, we can produce a double axis by taking their sum. This area has been applied to produce subalgebras of Matsuo algebras by Galt, Joshi, Mamontov, Shpectorov, and Staroletov in 2021 as well as other papers. Similar work can be applied to axial algebras of Monster type and using double axes to help construct new algebras. We will begin with the basics around double axes and at the specific conditions we desire. Our work is only concerned with shapes of an axial algebra which have certain subalgebras of 4A, 4J, and their quotients of codimension one. Using double axes, we can produce a subalgebra which is of Jordan type and then expand to the whole algebra. Looking at each possible case, we will present classifications and open questions which depend on understanding axial algebras of Jordan type. This is joint work with Justin McInroy.
45. 2024.11.19 - Hans Cuypers, Axial algebras related to polar spaces and Jordan algebras of quasi-Clifford algebras video
46. 2024.12.17 - Bernardo Rodrigues, Axial algebras of Monster type from unitary groups slides video
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