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"Nilpotent Lie algebras obtained by quivers and Ricci solitons", TMU-OMU Mini-Workshop on Differential Geometry (Tokyo Metropolitan University), 2026/05/16

Abstract

Following the affirmative resolution of the generalized Alekseevsky conjecture, it becomes an important problem to determine whether a given solvable (nilpotent) Lie group admits a left-invariant Einstein (Ricci soliton) metric. In this talk, we utilize quivers to construct nilpotent Lie groups (algebras). A quiver is a directed graph that allows loops and multiple arrows between vertices. By employing the concept of paths within quivers, we introduce a method for constructing nilpotent Lie algebras from finite quivers that do not contain cycles. We prove that all of these Lie algebras correspond to simply-connected nilpotent Lie groups that admit left-invariant Ricci solitons. Additionally, we will discuss some recent progress related to this topic. This work is based on collaboration with Fumika Mizoguchi.

Slide

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