Laplacian spectrum for the nilpotent Kac-Moody Lie algebras

TL;DR

This paper proves that the maximal nilpotent subalgebra of a Kac-Moody Lie algebra admits a unique Euclidean metric making the Laplace operator scalar on each degree component, uniquely determining the algebra and metric.

Contribution

It establishes the existence and uniqueness of a Euclidean metric on the nilpotent subalgebra that simplifies the Laplace operator, linking algebraic structure and metric uniquely.

Findings

Unique Euclidean metric exists for the subalgebra

Laplace operator is scalar on each degree component

Both algebra structure and metric are uniquely determined

Abstract

We prove that the maximal nilpotent subalgebra of a Kac-Moody Lie algebra has an (essentially unique) Euclidean metric with respect to which the Laplace operator in the chain complex is scalar on each component of a given degree. Moreover, both the Lie algebra structure and the metric are uniquely determined by this property.