The theory of vertex operator algebras (VOAs) has deep roots in mathematics and physics. On the mathematical side, one of its major accomplishments is the explanation of the famous "monstrous moonshine", providing a link between number theory and the largest sporadic simple group, the so-called Fischer-Griess Monster.

The structure of the algebras arising from VOAs "of Griess type" has been axiomatized, first to Majorana algebras (2009, Ivanov), then to axial algebras (2015, Hall-Rehren-Shpectorov) and very recently to decomposition algebras (2021, De Medts-Peacock-Shpectorov-Van Couwenberghe), focusing on so-called fusion laws. This theory sheds new light on many aspects of representation theory of finite groups and algebraic groups.

However, up to now, the theory of VOAs and the theory of axial algebras and decomposition algebras have not influenced each other so much. The goal of the current research proposal is precisely to initiate such an interaction between these theories.