Extensions of n-Hom Lie algebras

Abstract

n-Hom Lie algebras are twisted by n-Lie algebras by means of twisting maps. n-Hom Lie algebras have close relationships with statistical mechanics and mathematical physics. The paper main concerns structures and representations of n-Hom Lie algebras. The concept of n_ρ-cocycle for an n-Hom Lie algebra (G, [,… , ], α) related to a G-module (V, ρ, β) is proposed, and a sufficient condition for the existence of the dual representation of an n-Hom Lie algebra is provided. From a G-module (V, ρ, β) and an n_ρ-cocycle θ, an n-Hom Lie algebra (T_θ(V ), [, … , ]θ, γ) is constructed on the vector space T_θ(V ) = G⊕V, which is called the T_θ-extension of an n-Hom Lie algebra (G, [, … , ], α) by the G-module (V, ρ, β).

Key words： n-Hom Lie algebra; representation; extension; n_ρ-cocycle

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