Automorphism group schemes of lattice vertex algebras

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.