This lecture is a continuation of the general talk given at the Drensky conference last month, on axial algebras, which are (not necessarily commutative, not necessarily associative) algebras generated by semisimple idempotents. After a review of the definitions, we investigate the key question, being, “Under what conditions must an axial algebra be finite-dimensional?” Krupnik showed that 3 idempotents can generate arbitrarily large dimensional associative algebras (and thus infinite-dimensional algebras via an ultraproduct argument), so some restriction is needed. We consider ‘primitive’ axes, in which the left and right eigenspaces having eigenvalue 1 are one-dimensional.
Hall, Rehren, Shpectorov solves obtained a positive answer for commutative axial algebras of ‘Jordan type’ λ ≠ 1/2, although the proof relies on the classification of simple groups and the given bound of the dimension is rather high. Gorshkov and Staroletov provided a sharp bound for 3-generated commutative axial algebras of ‘Jordan type’. Our objective in this project is give a non-commutative version and indicate how to investigate 4-generated commutative axial algebras of ‘Jordan type’, in terms of the regular representation.
Our method is to build an associative algebra from the adjoint algebra of A, which has a strictly larger dimension which nevertheless also is finite-dimensional.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
