Algebras of Jordan type η generalize in the axial context the class of Jordan algebras generated by primitive idempotents. In addition to these examples, arising for η = 1/2, the class of algebras of Jordan type includes the Matsuo algebras, constructed in terms of 3-transposition groups for all values of η. The classification of algebras of Jordan type for η ≠ 1/2 was completed by Hall, Rerhen and Shpectorov in 2015, with a correction by Hall, Segev and Shpectorov in 2018. The case of η = 1/2 remains open. Among the known results about algebras of Jordan type half are the classification, in the above mentioned paper from 2015, of 2-generated algebras, the classification of 3-generated algebras by Gorshkov and Staroletov in 2020, and the recent (from 2023) result by De Medts, Rowen and Segev bounding the dimension of 4-generated algebras by 81. In the talk we will discuss another recent (in preparation, 2023) result on the subject, by Gorshkov, Staroletov and Shpectorov. A 2-generated subalgebra B of an algebra A of Jordan type half is called solid if every primitive idempotent from B is an axis in the entire A. Surprisingly, it turns out that, at least in characteristic zero, almost all 2-generated subalgebras are solid. More, precisely, a non-solid 2-generated subalgebra is necessarily of type 3C(1/2). Consequently, if a finite-dimensional algebra of Jordan type half has a finite automorphism group then it is either a Matsuo algebra or a factor of Matsuo algebra. The above result hints of a possibility of a geometric theory of algebras of Jordan type half.
This video is part of the Non-Associative Day in Online, run by the European Non-Associative Algebra Seminar series.
