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Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

Youssef Ayad

TL;DR

The paper classifies pseudo-Riemannian algebraic Ricci solitons on all four-dimensional Lie groups by reducing to the algebraic condition $\operatorname{Ric} = \eta \mathrm{Id} + D$ with $D$ a derivation of the Lie algebra. It treats both decomposable and indecomposable 4D Lie algebras, deriving explicit vanishing patterns and diagonal/coefficient relations for the Ricci operator that ensure compatibility with a derivation. The results provide a complete dimension-four description, including a flat, trivial soliton example and a nontrivial soliton example, and extend the Lauret framework to the pseudo-Riemannian setting relevant for Lorentzian and other indefinite geometries. The work advances understanding of self-similar solutions to the pseudo-Riemannian Ricci flow on Lie groups and yields a systematic blueprint for constructing and classifying such solitons in low dimension.

Abstract

We investigate the conditions under which pseudo-Riemannian inner products induce pseudo-Riemannian algebraic Ricci solitons on four-dimensional Lie algebras. By analyzing the algebraic Ricci soliton equation for each four-dimensional Lie algebra, we obtain a complete description of when such pseudo-Riemannian algebraic Ricci solitons arise in dimension four. We present two applications of our formalism on a chosen four-dimensional Lie algebra by exhibiting a pseudo-Riemannian algebraic Ricci soliton and a flat pseudo-Riemannian inner product, which is a trivial algebraic Ricci soliton.

Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

TL;DR

The paper classifies pseudo-Riemannian algebraic Ricci solitons on all four-dimensional Lie groups by reducing to the algebraic condition with a derivation of the Lie algebra. It treats both decomposable and indecomposable 4D Lie algebras, deriving explicit vanishing patterns and diagonal/coefficient relations for the Ricci operator that ensure compatibility with a derivation. The results provide a complete dimension-four description, including a flat, trivial soliton example and a nontrivial soliton example, and extend the Lauret framework to the pseudo-Riemannian setting relevant for Lorentzian and other indefinite geometries. The work advances understanding of self-similar solutions to the pseudo-Riemannian Ricci flow on Lie groups and yields a systematic blueprint for constructing and classifying such solitons in low dimension.

Abstract

We investigate the conditions under which pseudo-Riemannian inner products induce pseudo-Riemannian algebraic Ricci solitons on four-dimensional Lie algebras. By analyzing the algebraic Ricci soliton equation for each four-dimensional Lie algebra, we obtain a complete description of when such pseudo-Riemannian algebraic Ricci solitons arise in dimension four. We present two applications of our formalism on a chosen four-dimensional Lie algebra by exhibiting a pseudo-Riemannian algebraic Ricci soliton and a flat pseudo-Riemannian inner product, which is a trivial algebraic Ricci soliton.
Paper Structure (30 sections, 23 theorems, 253 equations)

This paper contains 30 sections, 23 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Ricci operator of a pseudo-Riemannian Lie algebra
  4. Procedure to compute the Ricci operator of a pseudo-Riemannian Lie algebra
  5. Pseudo-Riemannian algebraic Ricci solitons on decomposable four-dimensional Lie algebras
  6. The Lie algebra $4\mathfrak{g}_1$
  7. The Lie algebra $\mathfrak{g}_{2.1} \oplus 2\mathfrak{g}_1$
  8. The Lie algebra $2\mathfrak{g}_{2.1}$
  9. The Lie algebra $\mathfrak{g}_{3.1} \oplus \mathfrak{g}_1$
  10. The Lie algebra $\mathfrak{g}_{3.2} \oplus \mathfrak{g}_1$
  11. The Lie algebra $\mathfrak{g}_{3.3} \oplus \mathfrak{g}_1$
  12. The Lie algebra $\mathfrak{g}_{3.4}^0 \oplus \mathfrak{g}_1$
  13. The Lie algebra $\mathfrak{g}_{3.4}^{\alpha} \oplus \mathfrak{g}_1$
  14. The Lie algebra $\mathfrak{g}_{3.5}^0 \oplus \mathfrak{g}_1$
  15. The Lie algebra $\mathfrak{g}_{3.5}^{\alpha} \oplus \mathfrak{g}_1$
  16. ...and 15 more sections

Key Result

Theorem 3.1

A pseudo-Riemannian inner product on $\mathfrak{g}_{2.1} \oplus 2\mathfrak{g}_1$ is an algebraic Ricci soliton if its Ricci operator satisfies In this situation, the Ricci operator can be written as $\operatorname{Ric} = \eta I_4 + D$ where $\eta = R_{22}$ and is a derivation of $\mathfrak{g}_{2.1} \oplus 2\mathfrak{g}_1$ with respect to the basis $\mathcal{B}$.

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 51 more