Automorphism group schemes of lattice vertex algebras

TL;DR

This paper describes the automorphism groups of lattice vertex algebras constructed from positive definite even lattices, revealing their structure as extensions of reductive groups by lattice automorphisms.

Contribution

It provides a detailed description of the automorphism group schemes of lattice vertex algebras, including their structure as extensions of reductive groups by lattice automorphisms.

Findings

Automorphism groups are affine group schemes.

Each automorphism group is an extension of a split reductive group of ADE type.

The automorphism groups include outer automorphisms of the lattice.

Abstract

Given a positive definite even lattice and a commutative ring, there is a standard construction of a lattice vertex algebra over the commutative ring, and it admits a natural grading by non-negative integers. We describe the groups of automorphisms of these graded vertex algebras as affine group schemes, showing in particular that each is an extension of an explicitly described split reductive group of ADE type by the outer automorphism group of the lattice.