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Closed G$_2$-structures on non-solvable Lie groups

Anna Fino, Alberto Raffero

TL;DR

This work investigates the existence of left-invariant closed $G_2$-structures on seven-dimensional non-solvable Lie groups, establishing a classification via Levi decomposition and unimodularity. Using stability theory, the bilinear form $b_\varphi$ and obstructions for closed $G_2$-forms, the paper identifies the precise non-solvable seven-dimensional algebras that admit such structures, proving existence in four unimodular cases and one nontrivial-Levi case, while ruling out many others. It provides explicit closed $G_2$-forms for the admissible algebras, including $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{r}$ with centerless unimodular radicals and $L_{7,3}^{-2}$, demonstrating that these are the first known examples on non-solvable Lie algebras. The results open avenues for constructing compact examples via lattices and for studying dynamics under the Laplacian $G_2$-flow in this non-solvable setting.

Abstract

We investigate the existence of left-invariant closed G$_2$-structures on seven-dimensional non-solvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a structure exists only when the semisimple part is isomorphic to $\mathfrak{sl}(2,\mathbb{R})$ and the radical is unimodular and centerless. Moreover, we classify unimodular Lie algebras with non-trivial Levi decomposition admitting closed G$_2$-structures.

Closed G$_2$-structures on non-solvable Lie groups

TL;DR

This work investigates the existence of left-invariant closed -structures on seven-dimensional non-solvable Lie groups, establishing a classification via Levi decomposition and unimodularity. Using stability theory, the bilinear form and obstructions for closed -forms, the paper identifies the precise non-solvable seven-dimensional algebras that admit such structures, proving existence in four unimodular cases and one nontrivial-Levi case, while ruling out many others. It provides explicit closed -forms for the admissible algebras, including with centerless unimodular radicals and , demonstrating that these are the first known examples on non-solvable Lie algebras. The results open avenues for constructing compact examples via lattices and for studying dynamics under the Laplacian -flow in this non-solvable setting.

Abstract

We investigate the existence of left-invariant closed G-structures on seven-dimensional non-solvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a structure exists only when the semisimple part is isomorphic to and the radical is unimodular and centerless. Moreover, we classify unimodular Lie algebras with non-trivial Levi decomposition admitting closed G-structures.

Paper Structure

This paper contains 5 sections, 14 theorems, 22 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Closed ${\rm G}_2$-structures on Lie algebras
  3. Non-solvable seven-dimensional Lie algebras
  4. Non-solvable Lie algebras with trivial Levi decomposition
  5. Unimodular Lie algebras with non-trivial Levi decomposition

Key Result

Lemma 2.2

A seven-dimensional oriented real Lie algebra $\mathfrak{g}$ does not admit any closed ${\rm G}_2$-structure if for every closed 3-form $\phi\in\Lambda^3(\mathfrak{g}^*)$ one of the following conditions hold for the map $b_\phi:\mathfrak{g}\times\mathfrak{g}\rightarrow\Lambda^7(\mathfrak{g}^*)\cong{ This result does not depend on the choice of the orientation.

Theorems & Definitions (25)

  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3: CoFe
  • Remark 3.1
  • Theorem 4.1: Chu
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3: AnBaDoOvMub
  • Theorem 4.4: Chu
  • ...and 15 more