Structure of n-Lie Algebras with Involutive Derivations

Ruipu Bai,Yansha Gao,Shuai Hou

doi:10.1155/2018/7202141

Ruipu Bai, Yansha Gao + Show 1 more

Open Access

https://doi.org/10.1155/2018/7202141

Copy DOI

Abstract

We study the structure of n-Lie algebras with involutive derivations for n≥2. We obtain that a 3-Lie algebra A is a two-dimensional extension of Lie algebras if and only if there is an involutive derivation D on A=A1 ∔ A-1 such that dim A1=2 or dim A-1=2, where A1 and A-1 are subspaces of A with eigenvalues 1 and -1, respectively. We show that there does not exist involutive derivations on nonabelian n-Lie algebras with n=2s for s≥1. We also prove that if A is a (2s+2)-dimensional (2s+1)-Lie algebra with dim A1=r, then there are involutive derivations on A if and only if r is even, or r satisfies 1≤r≤s+2. We discuss also the existence of involutive derivations on (2s+3)-dimensional (2s+1)-Lie algebras.

Highlights

Derivation is an important tool in studying the structure of n-Lie algebras [1]
We investigate in this paper the existence of involutive derivations on finite dimensional n-Lie algebras
The linear mapping D : A 󳨀→ A defined by D(ei) = ei, 1 ≤ i ≤ s + 1, D(ej) = −ej, s + 2 ≤ j ≤ 2s + 1 is an involutive derivation on A

Summary

Introduction

Derivation is an important tool in studying the structure of n-Lie algebras [1]. The derivation algebra Der(A) of an n-Lie algebra A over the field of real numbers is the Lie algebra of the automorphism group Aut(A), which is a Lie group if dim A < ∞ [2]. Let A be a finite dimensional n-Lie algebra with n = 2s + 1, s ≥ 1, and D be an involutive derivation on A. We study involutive derivations on (n + 1)dimensional n-Lie algebras over F.

Results

Conclusion