3-Lie CYBE And Eight-dimensional Local Cocycle 3-Lie Bialgebras

In this thesis,skew-symmetric solutions of the 3-Lie CYBE,local cocycle 3-Lie bialgebras and 3-pre Lie algebras are mainly studied.Moreover,a kind of infinite dimensional unital 3-Lie Poisson algebra is constructed and its structure is discussed.The sufficient and necessary conditions of Rota-Baxter operators（with weight zero）on the finite dimensional 3-Lie algebra over complex field F are given.For Rota-Baxter operators（with weight zero）on a 3-Lie algebra,corresponding skew-symmetric solutions of the 3-Lie CYBE and local cocycle 3-Lie bialgebras are constructed.Especially,Rota-Baxter operators（with weight zero）on finite simple 3-Lie algebraA₄are discussed.Skew-symmetric solutions of the 3-Lie CYBE on A₄（?）A^*and 8-dimensional local cocycle 3-Lie bialgebras are constructed.Structure of 3-pre Lie algebras induced by 3-Lie Rota-Baxter operators（with weight zero）over complex field F are provided.6 classes of the 3-pre Lie algebras（A,{,,}^t）（1≤t≤6）and its sub-adjacent 3-Lie algebras（A,[,,]^t_C）are constructed by using Rota-Baxter operators（with weight zero）on finite simple 3-Lie algebraA₄.There exists compatible 3-pre-Lie algebra（A,{,,}）of Rota-Baxter 3-Lie algebra（with weight zero）.And the structure of（A,{,,}）is given.Finally,an infinite dimensional unital 3-Lie Poisson algebra L is constructed.The minimal generating set and derivation of L are studied.It is proved that L is a canonical Nambu 3-Lie algebra and A/C₀is an infinite dimensional simple 3-Lie algebra.There are four important 3-Lie algebras 3-Virasoro-Witt Algebra,A^δ_ω、A_ω,and 3-W_∞algebra which can be embed in 3-Lie subalgebras of L.