摘要
n-Hom Lie algebras are twisted by n-Lie algebras by means of twisting maps. n-Horn Lie algebras have close relationships with statistical mechanics and mathematical physics. The paper main concerns structures and representations of n-Horn Lie algebras. The concept of nρ-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 nρ-cocycle θ, an n-Horn Lie algebra （Tθ（V）, [,..., ]θ, γ） is constructed on the vector space Tθ（V） = G＋V, which is called the Tθ-extension of an n-Horn Lie algebra （G, [,..., ], α） by the G-module （V, ρ,β）.
n-Hom Lie algebras are twisted by n-Lie algebras by means of twisting maps. n-Horn Lie algebras have close relationships with statistical mechanics and mathematical physics. The paper main concerns structures and representations of n-Horn Lie algebras. The concept of nρ-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 nρ-cocycle θ, an n-Horn Lie algebra （Tθ（V）, [,..., ]θ, γ） is constructed on the vector space Tθ（V） = G＋V, which is called the Tθ-extension of an n-Horn Lie algebra （G, [,..., ], α） by the G-module （V, ρ,β）.
基金
Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant No. 11371245） and the Natural Science Foundation of Hebei Province, China （Grant No. A2014201006）.
