Automorphism-invariant Integral Forms in Griess Algebras.

Abstract

Motivated by the existence of group-invariant integral forms in various vertex operator algebras, we classify maximal automorphism-invariant integral forms in some small-dimensional Griess algebras, which are certain finite-dimensional commutative, nonassociative algebras arising in the theory of vertex operator algebras. An integral form of a rational algebra is the integer span of a basis of the algebra that is closed under the algebra product. The main method is the development of "integral form detector functions" and an investigation of their properties. Each of the small Griess algebras we analyzed - the eight Norton-Sakuma algebras and three others - have unique maximal automorphism-invariant integral forms. This provides a canonically defined lattice and subring inside these algebras.