Nice pseudo-Riemannian nilsolitons

TL;DR

This paper classifies nice nilpotent Lie algebras with nilsoliton metrics across various signatures, providing a systematic approach to determine their existence through linear and polynomial systems, especially in low dimensions.

Contribution

It offers a comprehensive classification of nice nilsolitons up to dimension 9 and reduces the existence problem to linear and polynomial equations for general signatures.

Findings

Classified nice Riemannian nilsolitons up to dimension 9.

Reduced the existence problem to linear and polynomial systems.

Provided classifications for various signatures and coranks.

Abstract

We study nice nilpotent Lie algebras admitting a diagonal nilsoliton metric. We classify nice Riemannian nilsolitons up to dimension $9$. For general signature, we show that determining whether a nilpotent nice Lie algebra admits a nilsoliton metric reduces to a linear problem together with a system of as many polynomial equations as the corank of the root matrix. We classify nice nilsolitons of any signature: in dimension $\leq 7$; in dimension $8$ for corank $\leq 1$; in dimension $9$ for corank zero.