Q-Deformed 3-Lie Algebras

We studied q-deformed 3-Lie algebras and q-representations and its hom structures.The q-deformed 3-Lie algebra is simply called q-3-Lie algebra.The main contents are as following:The definitions of q-3-Lie algebra and Type(I)-q-3-Lie algebra are provided,and the struc-tures and the realizations are studied.And it is proved that if(A,[,,]q,Jq)is a Type(Ⅰ)-q-3-Lie algebra,then(A,adq,Jq)is a q-module of Type(I)-q-3-Lie algebra Aq,which is called the ad-joint representation of Aq.From q-Lie algebras,group algebras,and commutative associative algebras,q-3-Lie algebras and and Type(Ⅰ)-q-3-Lie algebras(L,[..]qf,Jq)and are constructed respectively.The non-trivial one-dimensional central extension of q-3-Lie algebras are investigated,and new q-3-Lie algebras are obtained,whereAt last of the paper,the structures of hom-q-3-Lie algebra are discussed,and from known hom-Lie algebra(H,[,]q,αq),the hom-q-3-Lie algebra(H,[,,]q,[,,]’q,Jq,αq,βq)is construct-ed.