Yang-Baxter Operators, Cohomology Groups And Automorphism Groups Of The Nilpotent Lie (Super) Algebras Of Low Dimensions

Firstly, in the first part of Chapter 3, let F be an algebraically closed field of characteristic 0, employing the classification of nilpotent Lie algebras of low dimensions, we find out all the Yang-Baxter operators of any weight on the nilpotent Lie algebras of dimensions 3 or 4. Secondly, in the second part of Chapter 3, let C be the complex field, in the same way, employing the classification of Lie suporalgcbras of dimension 3, we find out all the Yang-Baxter operators on the nilpotent Lie superalgebras of dimensions 3. Due to the classification problem of the lower nilpotent Lie algebras has not been solved, it is important to find out all the 2-cocycles of the nilpotent Lie algebras of dimensions 6 on the complex field which is based on the research of De Graff; what’s more, studying about their lower cohomology groups can provide references for discussing the structures and representatives of the universal nilpotent Lie algebras, let C be the complex field, we find out the cohomology groups and the automorphism groups of a nilpotent Lie algebras of dimension 6 in Chapter 4.