Some Studies On Rota-Baxter Family Algebras And(tri) Dendriform Family Algebras

This thesis mainly studies the free Rota-Baxter family algebra,free(tri)dendriform family algebras and free pre-Lie family algebras and operads.Chapter one introduces the background and its recent development,and gives some basic notions and related notations,and then analyzes the motivations of this thesis.Chapter two first constructs free commutative Rota-Baxter family algebras.Then we construct free noncommutative Rota-Baxter family algebras with the help of bracket words,by using the method of Gr(?)bner-Shirshov bases.Finally,we introduce the concept of dendriform(resp.tridendriform)family algebras,and prove that RotaBaxter family algebras of weight zero(resp.)induce dendriform(resp.tridendriform)family algebras.The free commutative dendriform(resp.tridendriform)family algebras are also constructed.Chapter three first proves that a Rota-Baxter family algebra indexed by a semigroup amounts to an ordinary Rota-Baxter algebra structure on the tensor product with the semigroup algebra.Then we show that the same phenomenon arises for dendriform and tridendriform family algebras.Then we construct free dendriform family algebras in terms of typed decorated planar binary trees.Finally,we generalize typed decorated rooted trees to typed valently decorated Schr(?)der trees and use them to construct free tridendriform family algebras.Chapter four constructs the free Rota-Baxter family algebra on typed angularly decorated planar rooted trees.As an application,we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees(not forests),which is quite different from the known construction via angularly decorated planar rooted forests by Guo and Ebrahimi-Fard.We then embed the free dendriform(resp.tridendriform)family algebra into the free Rota-Baxter family algebra of weight zero(resp.one).Finally,we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free(tri)dendriform family algebra.Chapter five first proves the pre-Lie family algebra induced by a dendriform family algebra in the case of a commutative semigroup.Then we construct a pre-Lie family algebra via typed decorated rooted trees,and we prove the freeness of this preLie family algebra.We also construct pre-Lie family operad in terms of typed labeled rooted trees,and we obtain that the operad of pre-Lie family algebras is isomorphic to the operad of typed labeled rooted trees,which generalizes the result of F.Chapoton and M.Livernet.In the end,we construct Zinbiel and pre-Poisson family algebras and generalize results of M.Aguiar.