Justin McInroy

The structure of axial algebras

Axial algebras are a new class of non-associative algebra, introduced recently by Hall, Rehren and Shpectorov, which generalise some properties found in vertex operator algebras and the Griess algebra. Axial algebras are generated by axes which are idempotents that decompose the algebra as a direct sum of eigenspaces. The multiplication of eigenvectors is controlled by a so-called fusion law. When this is graded, it leads naturally to a subgroup of automorphisms of the algebra called the Miyamoto group. The prototypical example is the Griess algebra which has the Monster simple sporadic group as its Miyamoto group. We will discuss some recent developments about the structure of such algebras: their ideals, sum decomposition and an alternating bilinear form.

This is joint work with Sergey Shpectorov (Birmingham).