Reservation About The Contemporary Number Of Bialgebra And Loday Algebra Is Similar

In this thesis, we mainly give a study on the following three topics on Jordan alge-bras:(1) Jordan D-bialgebras and Jordan Yang-Baxter equation; (2) pre-Jordan bialge-bras; (3) Jordan analogues of Loday algebras. And the thesis is organized as follows.Chapter 1 is introduction, in which we recall the background needed in our study, give the problems we will focus on in this thesis, and layout the main results we ob-tained. We also give some notations and symbols so that the readers can understand more easily.In Chapter 2, we give a new interpretation to Jordan D-bialgebras which intro-duced by Zhelyabin in terms of the matched pair of Jordan algebras and thus we give an explicit proof of the equivalence between Jordan D-bialgebras and double constructions of pseudo-euclidean Jordan algebras. Moreover, we show that Jordan D-bialgebras lead to the Jordan Yang-Baxter equation under the coboundary condition and an anti-symmetric solution of Jordan Yang-Baxter equation can induce a Jordan D-bialgebra structure naturally. We also give a discussion on the properties of Jordan Yang-Baxter equation by O-operators of Jordan algebras.In Chapter 3, we prove that a double construction of symplectic Jordan algebras is equivalent to a new bialgebra structure called pre-Jordan bialgebras, and construct pre-Jordan bialgebras by the matched pair of Jordan algebras. We also show that the pre-Jordan bialgebras have many similar properties as of Lie bialgebras and Jordan D-bialgebras. In particular, there are also the so-called "coboundary cases" which also lead to a construction from an algebraic equation (JP-equation) in pre-Jordan algebras, which can be regarded as an analogue of the classical Yang-Baxter equation in Lie algebras. Moreover, we also present the properties of JP-equation by O-operators of Jordan algebras and O-operators of pre-Jordan algebras. Furthermore, we study the properties of pre-Jordan algebras and give some symmetric or antisymmetric bilinear forms satisfying certain conditions on pre-Jordan algebras.In Chapter 4, we introduce the concept of a J-dendriform algebra which is the corresponding algebraic structure of a Jordan analogue of Loday algebras with 2 oper-ations. Firstly, we study the relationship between J-dendriform algebras and pre-Jordan algebras. Also, we introduce the notion of an O-operator of a J-dendriform algebra. Then, an analogue of the classical Yang-Baxter equation in J-dendriform algebras and some symmetric or antisymmetric bilinear forms on J-dendriform algebras satisfying certain conditions are given.In Chapter 5, we give a similar study on another Jordan analogue of Loday al-gebras called J-quadri-algebras which have 4 operations. In the last section of this chapter, we introduce the last Jordan analogue of Loday algebras appearing in this pa-per, namely, J-octo-algebras. We also layout their main properties by O-operators of J-quadri-algebras.In Chapter 6, we summarize the study in the previous sections.