Researches On Operad And Associated Algebras

The main results of this paper are divided into four parts.Firstly, we introduce the definition and equivalent definitions of an algebraic op-erad. In particular, we give the definitions of symmetric operad and nonsymmetric operad and show that they can be derived from each other. For the combinatorial defi-nition of the operad, we take the nonsymmetric operad for example. Moreover, we also introduce the relation between other types of operads and the trees in the combinatorial definition, such as cyclic operads and shuffle operads and so on. Next, we give the rela-tionship between the operad and homotopy theory and mainly show that how to obtain the homotopy algebras in operad theory. At last, we give some examples to introduce the new developments of the operad.Secondly, we discuss the concepts of a totally compatible dialgebra and a totally compatible Lie dialgebra, defined to be a vector space with two binary operations that satisfy individual and mixed associativity conditions and Lie algebra conditions respec-tively. We show that totally compatible dialgebras are closely related to bimodule alge-bras and semi-homomorphisms. More significantly, Rota-Baxter operators on totally compatible dialgebras provide a uniform framework to generalize known results that Rota-Baxter related operators give tridendriform algebras. We also show that a Rota-Baxter operator on a totally compatible Lie dialgebra gives rise to a PostLie algebra, generalizing the fact that a Rota-Baxter operator on a Lie algebra gives rise to a PostLie algebra. In our paper, both free totally compatible dialgebras and free Rota-Baxter to-tally compatible dialgebras are constructed guided by the general principle of universal algebra.Thirdly, we study the operad of linearly compatible di-algebras, denoted by As2, which is a nonsymmetric operad encoding the algebras with two binary operations that satisfy individual and sum associativity conditions. We also prove that the operad As2is exactly the Koszul dual operad of the operad2As encoding totally compatible di-algebras. We show that the operads As2and2As are Koszul by rewriting method. We make explicit the Homotopy Transfer Theorem for As2-algebras. Finaly, we give a systematic study of matching dialgebras corresponding to the op-erad As(2) in [91] as the only Koszul self dual operad there other than the operads of as-sociative algebras and Poisson algebras. The close relationship of matching dialgebras with semi-homomorphisms and matched pairs of associative algebras are established. By anti-symmetrizing, matching dialgerbas are also shown to give compatible Lie al-gebras, pre-Lie algebras and PostLie algebras. By the rewriting method, the operad of matching dialgebras is shown to be Koszul and the free objects are constructed in terms of tensor algebras. At last, we generalize the Koszul self-duality of the operad As(2) Moreover, the operadic complex computing the homology of the matching dialgebras is made explicit.