Rota-Baxter Operators Of Weight Zero And Phase Spaces Associated To Left-symmetric Algebras And Hom-pre-anti-flexible Algebras

Left-symmetric algebras(also called pre-lie algebras)are a kind of very important non-associative algebras.It is closely related to many fields of mathematics and mathematical physics.In this context,this paper draws the following conclusions:At first,we compute the Rota-Baxter operators of weight zero on some four dimen-sional pseudo-Riemannian left-symmetric algebras.Furthermore we construct some new left-symmetric algebras by these Rota-Baxter operators.The second,we construct some non-abelian ten dimensional symplectic Lie algebras which called phase spaces by the symmetric solution of S-equation in a five dimensional left-symmetric algebras.At last,we maninly study the bimodules and Hom-O-operators of Hom-pre-anti-flexible algebras.Moreover,we introduce a new algebra structure called Hom-anti-flexible quadri-algebras which have closed relation to Hom-pre-anti-flexible algebras.