Universal axial algebras and a theorem of Sakuma

In the first half of this paper, we define axial algebras: nonassociative commutative algebras generated by axes, that is, semisimple idempotents-the prototypical example of which is Griess' algebra [2] for the Monster group. When multiplication of eigenspaces of axes is controlled by fusion rules, the structure of the axial algebra is determined to a large degree. We give a construction of the universal Frobenius axial algebra on n generators with specified fusion rules, of which all n-generated Frobenius axial algebras with the same fusion rules are quotients. In the second half, we realise this construction in the Majorana/Ising/Vir(4, 3)-case on 2 generators, and deduce a result generalising Sakuma's theorem in VOAs [13].