Mathematica Eterna
Open Access

Abstract

Structures of n-Lie algebra An

BAI Ruipu, Zhang Yan, Lin Lixin and Guo Weiwei

In this paper, we discuss the structure of the exterior direct sum nLie algebra (An , [, · · · , , ]k) of an n-Lie algebra A. And it is proved that, (1) if I1, · · · , In−1 are ideals of an n-Lie algebra A, then the vector space (I1, I2, · · · , Ik−1, I1, Ik+1, · · · , In−1) is also an ideal of (An , [, · · · , , ]k), and if I is a solvable (nilpotent) ideal of A, then I n is also solvable (nilpotent). (2) For a linear mapping δ ∈ End(A), then δ is a derivation of A if and only if fδ ∈ Hom(A, An ) is an n-Lie algebra homomorphism. (3) If (V, ρ) is an A-module, then (V n , ρ¯) is an An -module.