Axial algebras

Non-associative algebras play an important role in many areas of mathematics. The most prominent examples are Lie algebras, introduced in the 1930s to study infinitesimal transformations. Other classes of non-associative algebras also proved vary fruitful in other areas; Jordan algebas, for instance, played a crucial role in Zel'manov's solution to the restricted Burnside problem in group theory. The goal of the proposed project is to explore new connections between certain classes of non-associative algebras and group theory (and related areas). We will focus on certain vertex operator algebras (VOAs) and axial algebras. Perhaps one of the most spectacular connections between non-associative algebras and finite groups is the description of the Monster, the largest sporadic finite simple group, as the automorphism group of the Griess algebra. This algebra arises naturally inside a VOA. Recently, some attempts have been made (e.g. by Hall, Rehren and Shpectorov) to axiomatize those algebras, which has led to the notion of axial algebras. In my master thesis, I have introduced modules for axial algebras and I have discovered a connection between the representation theory of Matsuo algebras and the representation theory of 3-transposition groups. Our goal is to extend the contemporary theory of axial algebras; this includes a deeper understanding of the finite groups that arise, and a further development of a suitable representation theory.