Some Classical Yang-baxter Equations,Derivatives And Triple Derivatives Of Some Nilpotent Lie Algebras

Lie algebras were put forward in the late nineteenth century.As a kind of very important non-associative algebras,Lie algebras play an important role in mathematics and mathematical physics.In the study of Lie algebras,an important class of Lie algebras,that is,nilpotent Lie algebras was found.The related properties of nilpotent Lie algebras have not been studied completely because of the complexity of their structures.It's well known that it's an important way to study some algebras through their derivation structures,but it's not always possible to calculate directly by the definition.Therefore,by calculating the results of the linear transformation on the special basis elements,we obtained all the solutions of the classical Yang-Baxter equation on the three-dimensional Heisenberg Lie algebra and the corresponding left-symmetric algebraic structures.We also obtained the matrix forms of the derivation and the automorphism of the special four-dimensional nilpotent left-symmetric algebra L₁ and its sub-adjacent Lie algebra by calculating the linear transformation of some generators of the algebras.And we also obtained the matrix forms of the triple derivations of the filiform Lie algebras R_n and found that the triple derivatives algebra is a?2n-1?-dimensional solvable Lie algebra.