Lattice structure of modular vertex algebras

TL;DR

This paper investigates the integral form of lattice vertex algebras, demonstrating that certain operators preserve the integral lattice structure and establishing analogs of Kostant forms within this context.

Contribution

It introduces the preservation of the integral lattice by divided powers of vertex operators and Garland operators, extending Kostant form analogs to lattice vertex algebras.

Findings

Divided powers of vertex operators preserve the integral lattice.

Garland operators also preserve the integral form.

Establishment of Kostant $ extbf{Z}$-form analogs for lattice vertex algebras.

Abstract

In this paper we study the integral form of the lattice vertex algebra $V_L$. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant $\mathbb Z$-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras.