The Existence of Soliton Metrics for Nilpotent Lie Groups

TL;DR

This paper characterizes soliton metrics on nilpotent Lie groups using matrix equations and Kac-Moody algebra theory, providing new examples and conditions for their existence.

Contribution

It introduces a matrix equation characterization of soliton metrics and applies Kac-Moody algebra theory to find new examples and conditions for soliton metrics on nilpotent Lie groups.

Findings

Infinitely many new nilmanifolds with soliton metrics identified.

A matrix equation Uv = [1] characterizes soliton metrics.

Sufficient conditions for sum of soliton structures to be soliton.

Abstract

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N,g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac-Moody algebras to analyze the solution spaces for such linear systems. We use these methods to find infinitely many new examples of nilmanifolds with soliton metrics. We give a sufficient condition for a sum of soliton metric nilpotent Lie algebra structures to be soliton, and we use this criterion to show that soliton metrics exist on every naturally graded filiform metric Lie algebra.