Paper · Jul 8, 2026
Solvable Lie algebras obtained by quivers and Einstein metrics
As on arXiv
Abstract
In geometry, it is important to study whether a given Lie group admits a special left-invariant geometric structure. In our previous work, we constructed nilpotent Lie algebras from finite quivers without cycles by utilizing paths within these quivers. Also, we proved that the simply-connected Lie groups corresponding to these nilpotent Lie algebras always admit left-invariant Ricci solitons. In this paper, we extend our approach by constructing solvable Lie algebras from finite quivers without cycles, adding vertices as paths of length zero. We demonstrate that the simply-connected Lie groups corresponding to these solvable Lie algebras always admit left-invariant Ricci solitons. Moreover, we prove that when the quivers are oriented multi-trees, these Ricci soliton Lie groups are rigid, that is the direct product manifold of a flat manifold and an Einstein manifold.
Comments: 10pages, 4figures
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Nilpotent Lie algebras obtained by quivers and Ricci solitons
Topics Discrete & graph Ricci, Homogeneous spaces, Ricci solitons · Same author Fumika Mizoguchi · Shared terms “Lie algebras”, “nilpotent Lie”, “finite quivers”
-
Two-step nilpotent Lie algebras obtained by quivers and geometric structures
Topics Discrete & graph Ricci, Homogeneous spaces, Ricci solitons · Same author Fumika Mizoguchi · Shared terms “Lie algebras”, “finite quivers”, “nilpotent Lie”
-
Nilpotent Lie algebras obtained by ordered sets and Ricci solitons
Topics Discrete & graph Ricci, Homogeneous spaces, Ricci solitons · Shared terms “Lie algebras”, “nilpotent Lie”, “admit left”
-
Infinitesimal maximal symmetry and Ricci soliton solvmanifolds
Topics Einstein metrics, Homogeneous spaces, Ricci solitons · Shared terms “solvable Lie”, “invariant Ricci”, “left invariant”
-
A Hilbert–Mumford criterion for nilsolitons
Topics Homogeneous spaces, Ricci solitons · Shared terms “Lie algebras”, “nilpotent Lie”, “admit left”
-
Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds
Topics Einstein metrics, Homogeneous spaces, Ricci solitons · Shared terms “Lie algebras”, “invariant Ricci”, “nilpotent Lie”
-
Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two
Topics Homogeneous spaces, Ricci solitons · Shared terms “left invariant”, “Lie groups”
-
Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups
Topics Homogeneous spaces, Ricci solitons · Shared terms “nilpotent Lie”, “Lie groups”, “left invariant”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6D.11Computing Geometric Flows
Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.
- 11A.13Homogeneous Flows
A computable laboratory: the flow on Thurston’s geometries.
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.