[Paper Review] Lie algebra of an n-Lie algebra

Basile Guy Richard Bossoto, Eugène Okassa+1 more|arXiv (Cornell University)|Oct 9, 2013

Advanced Topics in Algebra1 references3 citations

TL;DR

This paper constructs a Lie algebra from an $n$-Lie algebra using the adjoint representation on the $(n-1)$-exterior power of the space, establishing a canonical Lie algebra structure via derivations. It defines cohomology for $n$-Lie algebras using this Lie algebra action, proving that the 0th cohomology group corresponds to the invariants of the $n$-Lie algebra, thus extending classical Lie algebra cohomology to the $n$-ary setting.

ABSTRACT

We construct the Lie algebra of an n-Lie algebra and we also define the notion of cohomology of an n-Lie algebra.

Motivation & Objective

• To construct a Lie algebra structure from an $n$-Lie algebra using the adjoint representation on the $(n-1)$-exterior power of the underlying vector space.
• To define a cohomology theory for $n$-Lie algebras based on the derived Lie algebra of derivations.
• To establish connections between invariants of the $n$-Lie algebra and the 0th cohomology group.
• To generalize classical Lie algebra concepts—ideals, solvability, nilpotency, Cartan subalgebras—to the $n$-Lie algebra setting via the constructed Lie algebra.

Proposed method

• Define the adjoint map $ad_{\mathcal{G}}: \Lambda_K^{n-1}(\mathcal{G}) \to \text{Der}_K(\mathcal{G})$ by $ad_{\mathcal{G}}(x_1 \wedge \cdots \wedge x_{n-1})(y) = \{x_1, \dots, x_{n-1}, y\}$.
• Show that $ad_{\mathcal{G}}$ is a Lie algebra homomorphism from $\Lambda_K^{n-1}(\mathcal{G})$ to $\text{Der}_K(\mathcal{G})$, factoring through the quotient $\Lambda_K^{n-1}(\mathcal{G}) / \mathcal{V}_K(\mathcal{G})$.
• Use the induced representation $\widetilde{ad_{\mathcal{G}}}: \Lambda_K^{n-1}(\mathcal{G}) / \mathcal{V}_K(\mathcal{G}) \to \text{Der}_K(\mathcal{G})$ to define cohomology with coefficients in $\mathcal{G}$.
• Define the cohomology complex $\left(\mathcal{L}_{sks}(\Lambda_K^{n-1}(\mathcal{G}) / \mathcal{V}_K(\mathcal{G}), \mathcal{G}), d_n\right)$ with differential $d_n$ induced by the representation.
• Prove that the differential $d_n$ satisfies $d_n^2 = 0$, ensuring a well-defined cohomology theory.
• Establish that $H_n^0(\mathcal{G}) = \text{Inv}(\mathcal{G})$, the space of $n$-Lie algebra invariants.

Experimental results

Research questions

• RQ1 How can a Lie algebra be naturally constructed from an $n$-Lie algebra for $n \geq 2$?
• RQ2 What is the cohomology theory for $n$-Lie algebras, and how does it relate to the derived Lie algebra of derivations?
• RQ3 How do classical Lie algebra concepts like ideals, solvability, and Cartan subalgebras extend to $n$-Lie algebras via this construction?
• RQ4 What is the role of the invariant subspace $\text{Inv}(\mathcal{G})$ in the cohomology of an $n$-Lie algebra?
• RQ5 Is the cohomology of an $n$-Lie algebra isomorphic to the cohomology of its associated Lie algebra under the derived representation?

Key findings

• The map $ad_{\mathcal{G}}: \Lambda_K^{n-1}(\mathcal{G}) \to \text{Der}_K(\mathcal{G})$ is a Lie algebra homomorphism, allowing the construction of a Lie algebra from an $n$-Lie algebra.
• The quotient $\Lambda_K^{n-1}(\mathcal{G}) / \mathcal{V}_K(\mathcal{G})$ carries a natural Lie algebra structure, and the image of $\widetilde{ad_{\mathcal{G}}}$ is a Lie subalgebra of $\text{Der}_K(\mathcal{G})$.
• The space $\text{Inv}(\mathcal{G})$ of $n$-Lie algebra invariants is isomorphic to the 0th cohomology group $H_n^0(\mathcal{G})$.
• The cohomology complex $\left(\mathcal{L}_{sks}(\Lambda_K^{n-1}(\mathcal{G}) / \mathcal{V}_K(\mathcal{G}), \mathcal{G}), d_n\right)$ is well-defined with $d_n^2 = 0$, confirming a valid cohomology theory.
• Subspaces stable under the representation $\widetilde{ad_{\mathcal{G}}}$ correspond to ideals in the $n$-Lie algebra, linking representation theory to ideal structure.
• The center of the Lie algebra $\Lambda_K^{n-1}(\mathcal{G}) / \mathcal{V}_K(\mathcal{G})$ contains the image of $\Lambda_K^{n-1}(\text{Inv}(\mathcal{G}))$, showing invariants act trivially on the derived Lie algebra.