Nilpotent Lie algebras obtained by quivers and Ricci solitons

TL;DR

This paper introduces a novel method for constructing nilpotent Lie algebras from quivers, demonstrating that these algebras lead to nilpotent Lie groups with Ricci soliton metrics, expanding the class of known Ricci solitons.

Contribution

It presents a new approach using quivers to generate nilpotent Lie algebras and proves these groups admit Ricci soliton metrics, broadening the scope of known examples.

Findings

Constructed nilpotent Lie algebras from finite quivers without cycles.

Proved these Lie groups admit left-invariant Ricci solitons.

Generated a broad family of Ricci soliton nilmanifolds with high nilpotency.

Abstract

Nilpotent Lie groups with left-invariant metrics provide non-trivial examples of Ricci solitons. One typical example is given by the class of two-step nilpotent Lie algebras obtained from simple directed graphs. In this paper, however, we focus on the use of quivers to construct nilpotent Lie algebras. A quiver is a directed graph that allows loops and multiple arrows between two vertices. Utilizing the concept of paths within quivers, we introduce a method for constructing nilpotent Lie algebras from finite quivers without cycles. We prove that for all these Lie algebras, the corresponding simply-connected nilpotent Lie groups admit left-invariant Ricci solitons. The method we introduce constructs a broad family of Ricci soliton nilmanifolds with arbitrarily high degrees of nilpotency.