The class of non-associative axial algebras was introduced in 2015 as a broad generalization of Majorana algebras of Ivanov that were modelled after the properties of the Griess algebra, the algebra whose automorphism group is the Monster sporadic simple group. Sakuma’s theorem classifies 2-generated Majorana algebras, which in axial terms correspond to algebras of Monster type (1/4,1/32). The quest to classify all 2-generated algebras of arbitrary Monster type (α,β) was started by Rehren who proved an upper bound on the dimension and generalised the Norton-Sakuma algebras to arbitrary (α,β). Recently, new results emerged from the work of Franchi, Mainardis and the speaker, and independently, of Yabe, who classified symmetric 2-generated algebras of Monster type. Several new classes of algebras have been found.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
