Structure of primitive axial algebras

A “fusion law” is a collection of multiplication rules among eigenspaces of an idempotent. This terminology is relatively new and is closely related to primitive axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras are closely related to 3-transposition groups and vertex operator algebras. In earlier work we studied primitive axial algebras, not necessarily commutative, and showed that they all have Jordan type. In this paper, we show that all finitely generated primitive axial algebras are direct sums of specifically described flexible finite-dimensional noncommutative algebras, and commutative axial algebras generated by primitive axes of the same type. In particular, all primitive axial algebras are flexible. They also have Frobenius forms. We give a precise description of all the primitive axes of axial algebras generated by two primitive axes.