Applications of vertex algebra covering procedures to Chevalley groups and modular moonshine

Abstract

A vertex operator algebra of lattice type ADE has a standard integral form which extends a Chevalley basis for its degree 1 Lie algebra. This integral form may be used to define a vertex algebra over a commutative ring $R$ and to get a Chevalley group over $R$ of the same type, acting as automorphisms of this vertex algebra. We define vertex algebras of types BCFG over a commutative ring and certain reduced VAs, then get analogous results about automorphism groups. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for Lie algebras is applied to the vertex algebra situation. We use integral form and covering procedures for vertex algebras to complete the modular moonshine program of Borcherds and Ryba for proving an embedding of the sporadic group $F_3$ in $E_8(3)$.

29 pages

Cited by in corpus (3)

• On integral forms for vertex algebras associated with affine Lie algebras and lattices
• Integral forms for tensor powers of the Virasoro vertex operator algebra $L(\frac{1}{2},0)$ and their modules
• Intertwining operators among modules for affine Lie algebra and lattice vertex operator algebras which respect integral forms