[Paper Review] Symplectic structures on $3$-Lie algebras

Ruipu Bai, Shuangshuang Chen+1 more|arXiv (Cornell University)|Aug 19, 2014

Advanced Topics in Algebra8 references3 citations

TL;DR

This paper introduces and characterizes metric symplectic 3-Lie algebras—3-Lie algebras equipped with both a non-degenerate invariant symmetric bilinear form and a non-degenerate invariant skew-symmetric bilinear form. It proves that such structures exist infinitely for any 3-Lie algebra, establishes a characterization via invertible derivations, and constructs new examples through $T^{*}_{\theta}$-extensions and double extensions, generalizing classical Lie algebra constructions to the 3-Lie setting.

ABSTRACT

The symplectic structures on $3$-Lie algebras and metric symplectic $3$-Lie algebras are studied. For arbitrary $3$-Lie algebra $L$, infinite many metric symplectic $3$-Lie algebras are constructed. It is proved that a metric $3$-Lie algebra $(A, B)$ is a metric symplectic $3$-Lie algebra if and only if there exists an invertible derivation $D$ such that $D\in Der_B(A)$, and is also proved that every metric symplectic $3$-Lie algebra $( ilde{A}, ilde{B}, ildeω)$ is a $T^*_θ$-extension of a metric symplectic $3$-Lie algebra $(A, B, ω)$. Finally, we construct a metric symplectic double extension of a metric symplectic $3$-Lie algebra by means of a special derivation.

Motivation & Objective

• To study symplectic structures on 3-Lie algebras and their interplay with metric structures.
• To characterize metric symplectic 3-Lie algebras via invertible derivations preserving the metric.
• To generalize classical Lie algebra extension techniques—specifically $T^{*}_{\theta}$-extensions and double extensions—to the 3-Lie algebra setting.
• To construct new examples of metric symplectic 3-Lie algebras from known ones using derivations and cohomological data.
• To establish structural results analogous to those in quadratic Lie algebras, but adapted to the 3-Lie context.

Proposed method

• Constructs infinite families of metric symplectic 3-Lie algebras from any given 3-Lie algebra using a non-degenerate invariant symmetric bilinear form and an invertible derivation.
• Defines a symplectic structure $\omega(x,y) = B(Dx,y)$ on a metric 3-Lie algebra $(A,B)$, where $D$ is an invertible derivation in $Der_B(A)$, ensuring $\omega$ is non-degenerate and invariant.
• Applies the $T^{*}_{\theta}$-extension construction to build new metric symplectic 3-Lie algebras from existing ones, using a representation of the algebra on a module and a 3-cocycle $\theta$.
• Introduces a metric symplectic double extension by extending a metric symplectic 3-Lie algebra using a special derivation $\delta$ satisfying a cohomological condition (Eq. 4.16), and defines the extended symplectic form $\tilde{\omega}$ via Eq. (4.18).
• Verifies that the extended algebra $\tilde{A}$ carries both a non-degenerate invariant metric $T$ and a non-degenerate invariant symplectic form $\tilde{\omega}$, with the extended derivation $\tilde{D}$ preserving both structures.
• Uses cohomological identities and bilinear form compatibility to prove that the extended derivation $\tilde{D}$ satisfies the 3-Lie algebra derivation identity and preserves the metric and symplectic forms.

Experimental results

Research questions

• RQ1 Under what conditions does a metric 3-Lie algebra admit a compatible symplectic structure?
• RQ2 Can new metric symplectic 3-Lie algebras be systematically constructed from existing ones using algebraic extensions?
• RQ3 What role do invertible derivations play in characterizing metric symplectic 3-Lie algebras?
• RQ4 How do classical Lie algebra extension techniques like $T^{*}_{\theta}$-extensions and double extensions generalize to the 3-Lie algebra setting?
• RQ5 Is there a structural characterization of metric symplectic 3-Lie algebras analogous to that of quadratic symplectic Lie algebras?

Key findings

• For any 3-Lie algebra $L$, infinitely many metric symplectic 3-Lie algebras can be constructed, demonstrating a rich supply of such structures.
• A metric 3-Lie algebra $(A,B)$ is metric symplectic if and only if there exists an invertible derivation $D \in Der_B(A)$, providing a precise structural criterion.
• Every metric symplectic 3-Lie algebra $(\tilde{A}, \tilde{B}, \tilde{\omega})$ arises as a $T^{*}_{\theta}$-extension of a metric symplectic 3-Lie algebra $(A,B,\omega)$, showing a universal construction mechanism.
• A metric symplectic double extension is constructed using a special derivation $\delta$ satisfying Eq. (4.16), yielding a new metric symplectic 3-Lie algebra $\tilde{A}$ with extended metric $T$ and symplectic form $\tilde{\omega}$ defined by Eq. (4.18).
• The extended derivation $\tilde{D}$ preserves both the metric $T$ and the symplectic form $\tilde{\omega}$, and satisfies the 3-Lie algebra derivation identity, confirming the algebraic consistency of the construction.
• The symplectic form $\tilde{\omega}$ on the double extension is explicitly defined by $\tilde{\omega}(x,y) = \omega(x,y)$ for $x,y \in A$, and $\tilde{\omega}(e_1,e_2^*) = \tilde{\omega}(e_2,e_1^*) = -1$, with all other values zero on the dual components.