[Paper Review] Construction of Lie algebras with special G2-structures

Víctor Manero|arXiv (Cornell University)|Jul 27, 2015

Advanced Topics in Algebra6 references3 citations

TL;DR

This paper presents a systematic method to construct 7-dimensional Lie algebras with closed and coclosed G₂-structures by extending 6-dimensional solvable Lie algebras equipped with symplectic half-flat or half-flat SU(3)-structures. The key contribution is a complete classification of all such 7-dimensional Lie algebras arising from 6-dimensional solvable Lie algebras with symplectic half-flat SU(3)-structures, explicitly listing their structure equations and verifying the closure of the G₂-form.

ABSTRACT

We give a method to obtain new 7-dimensional Lie algebras endowed with closed and coclosed G2-structures starting from 6-dimensional Lie algebras with symplectic half- at SU(3)-structures and half- at SU(3)- structures, respectively. Finally, we describe all the 7-dimensional Lie algebras with a closed G2-structure that are obtained with this method from the 6-dimensional solvable Lie algebras admitting a symplectic half- at SU(3)- structure.

Motivation & Objective

• To develop a general construction method for 7-dimensional Lie algebras with special G₂-structures using 6-dimensional Lie algebras with SU(3)-structures.
• To classify all 7D Lie algebras with closed G₂-structures that arise from 6D solvable Lie algebras admitting symplectic half-flat SU(3)-structures.
• To provide explicit structure equations for all such 7D Lie algebras and verify the closure of the G₂-form.
• To establish a correspondence between symplectic half-flat SU(3)-structures on 6D Lie algebras and closed G₂-structures on 7D Lie algebras via a specific extension procedure.
• To extend known classifications of special structures on nilpotent and solvable Lie algebras to the 7D G₂ setting.

Proposed method

• The method constructs a 7D Lie algebra as a semidirect product of a 6D Lie algebra with a 1D Lie algebra, using a derivation D on the 6D algebra.
• The construction relies on lifting a symplectic half-flat SU(3)-structure (dω = dψ₊ = 0) on a 6D Lie algebra to a closed G₂-structure (dφ = 0) on the 7D extension.
• The fundamental G₂ 3-form φ is defined as φ = ω ∧ e⁷ + ψ₊, where ω and ψ₊ are the Kähler and real part of the complex volume form of the 6D SU(3)-structure.
• The derivation D is chosen such that the resulting 7D Lie algebra satisfies dφ = 0, which imposes conditions on the structure constants of the 6D algebra and the matrix entries of D.
• The method is applied to all 6D solvable Lie algebras admitting symplectic half-flat SU(3)-structures, leading to a complete list of resulting 7D Lie algebras.
• The closure of the G₂-structure is verified by computing dφ and showing it vanishes using the structure equations and the symplectic half-flat condition.

Experimental results

Research questions

• RQ1 Which 7-dimensional Lie algebras admit closed G₂-structures that arise from 6-dimensional solvable Lie algebras with symplectic half-flat SU(3)-structures?
• RQ2 What is the explicit form of the structure equations for all such 7D Lie algebras constructed via the symplectic half-flat extension method?
• RQ3 How do the derivations D on the 6D Lie algebras affect the resulting 7D G₂-structure and its closure?
• RQ4 What is the complete classification of 7D Lie algebras with closed G₂-structures obtained from 6D solvable Lie algebras with symplectic half-flat SU(3)-structures?
• RQ5 Can the construction method be systematically applied to all known 6D solvable Lie algebras with symplectic half-flat SU(3)-structures to yield all possible 7D closed G₂-structures?

Key findings

• The paper constructs 7D Lie algebras with closed G₂-structures from 6D solvable Lie algebras with symplectic half-flat SU(3)-structures via a semidirect product with a 1D Lie algebra.
• The resulting 7D Lie algebras are explicitly listed in Table 1 with their structure equations, including parameters a_{i,j} that define the derivation D.
• For the Lie algebra N_{6,13}^{0,-2,0,-2}, the only derivation D that yields a closed G₂-structure is the zero matrix, resulting in a direct sum decomposition N_{6,13}^{0,-2,0,-2} ⊕ ℝe₇.
• The G₂-structure φ = ω ∧ e⁷ + ψ₊ is closed (dφ = 0) if and only if the 6D SU(3)-structure is symplectic half-flat (dω = dψ₊ = 0), confirming the construction's consistency.
• The method yields 11 distinct 7D Lie algebras with closed G₂-structures, including algebras such as (g_{5,1} ⊕ ℝ) ⋊_D ℝe₇ and A_{6,70}^{α,α/2} ⋊_D ℝe₇, each with specific structure equations.
• The construction confirms that the closed G₂-structure is preserved under the extension, and the resulting 7D Lie algebras are all non-compact and solvable.