Infinite-rank Euclidean Lattices and Loop Groups

TL;DR

This paper constructs infinite-rank Euclidean lattices linked to loop groups and affine Kac--Moody algebra representations, enabling the definition of theta-like functions for these lattices.

Contribution

It introduces a novel association between infinite-rank lattices and loop group elements, extending the concept of theta-finiteness to new infinite-dimensional settings.

Findings

Lattices are theta-finite for polynomial representatives.

Defined theta-like functions for these infinite-rank lattices.

Established connections between loop groups, Kac--Moody algebras, and lattice theory.

Abstract

In this paper, we associate a family of infinite-rank pro-Euclidean lattices to elements of a formal loop group and a highest weight representation of the underlying affine Kac--Moody algebra. In the case that the element has a polynomial representative, we can prove our lattices are theta-finite in the sense of Bost, allowing us to attach to each of our lattices a well-defined theta-like function.