Paper · May 18, 2024

Nilpotent Lie algebras obtained by quivers and Ricci solitons

Fumika Mizoguchi, Hiroshi Tamaru

math.DGarXiv:2405.11184
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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.

Comments: 15pages, 9figures

Ricci solitonsHomogeneous spacesDiscrete & graph Ricci

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Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups (Rohollah Bakhshandeh-Chamazkoti, 2021)5Bakhshandeh- 2021 A Hilbert–Mumford criterion for nilsolitons (Yoshinori Hashimoto, 2023)4Hashimoto 2023 Positive Hermitian curvature flow on 2-step nilpotent Lie groups (Ettore Lo Giudice, 2025)6Giudice 2025 Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two (Hamid Reza Salimi Moghaddam, 2026)8Moghaddam 2026 Nilpotent Lie algebras obtained by ordered sets and Ricci solitons (Yihao Zheng and Shenglin Zhu, 2026)1Zheng 2026 Solvable Lie algebras obtained by quivers and Einstein metrics (Fumika Mizoguchi, 2026)3Mizoguchi 2026 Sectional Curvature Pinching of Two-Step Nilmanifolds (Tomoya Tatsuno, 2026)7Tatsuno 2026 Two-step nilpotent Lie algebras obtained by quivers and geometric structures (Fumika Mizoguchi, 2026)2Mizoguchi 2026
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    Yihao Zheng and Shenglin Zhu PDF

    Topics Discrete & graph Ricci, Homogeneous spaces, Ricci solitons · Shared terms “nilpotent Lie”, “Lie algebras”, “construct nilpotent”

  2. Two-step nilpotent Lie algebras obtained by quivers and geometric structures

    Fumika Mizoguchi PDF

    Topics Discrete & graph Ricci, Homogeneous spaces, Ricci solitons · Same author Fumika Mizoguchi · Shared terms “nilpotent Lie”, “Lie algebras”, “step nilpotent”

  3. Solvable Lie algebras obtained by quivers and Einstein metrics

    Fumika Mizoguchi PDF

    Topics Discrete & graph Ricci, Homogeneous spaces, Ricci solitons · Same author Fumika Mizoguchi · Shared terms “Lie algebras”, “nilpotent Lie”, “finite quivers”

  4. A Hilbert–Mumford criterion for nilsolitons

    Yoshinori Hashimoto PDF

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilpotent Lie”, “Lie algebras”, “Lie groups”

  5. Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups

    Rohollah Bakhshandeh-Chamazkoti PDF

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilpotent Lie”, “left invariant”, “Lie groups”

  6. Positive Hermitian curvature flow on 2-step nilpotent Lie groups

    Ettore Lo Giudice PDF

    Topics Homogeneous spaces · Shared terms “nilpotent Lie”, “step nilpotent”, “Lie groups”

  7. Sectional Curvature Pinching of Two-Step Nilmanifolds

    Tomoya Tatsuno PDF

    Topics Homogeneous spaces, Ricci solitons · Shared terms “nilpotent Lie”, “step nilpotent”, “nilmanifolds”

  8. Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two

    Hamid Reza Salimi Moghaddam PDF

    Topics Homogeneous spaces, Ricci solitons · Shared terms “left invariant”, “Lie groups”, “invariant metrics”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 2002)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    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.

  2. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.