The concept of Majorana algebra was introduced by A.A. Ivanov based on the work of the VOA theorists Miyamoto and Sakuma. The principal example of Majorana algebra is the Griess–Norton Monster algebra of dimension 196884, which Griess used to construct the Monster sporadic group M.
In this talk I would like to report on two projects. First, in a joint project with J.I. Hall we determine all ¼-free Majorana algebras. Every Majorana algebra comes with a nontrivial group of symmetries, which in general is a group of 6-transpositions. The condition of ¼-freeness reduces this to 3-transposition groups, which are all known by a theorem of Cuypers and Hall. Surprisingly, it turns out that every 3-transposition group leads to an algebra where only one Majorana axiom may fail, namely, positive definiteness of the associative inner product. We then determine all 3-transposition groups, for which this extra property is also satisfied. As a consequence, we show that the class of Majorana algebras is infinite.
Time permitting, I will also report on the joint project with F. Rehren. By further relaxing some axioms we obtain the category of generalized Majorana algebras, which admits universal objects, namely, universal k-generated Majorana algebras. We then extend the available proof Sakuma’s theorem to turn it into the identification of the universal 2-generated Majorana algebra.