The O-operators On Lie Algebraic Analogues Of Loday Algebras And The Analogues Of The Classical Yang-Baxter Equation

In this paper, we study the operator forms of the S-equation in a pre-Lie algebra: O-operators of pre-Lie algebras. And we introduce the notions of L-dendriform alge-bras, L-quadri-algebras and L-octo-algebras. We study the O-operators of L-dendriform algebras and L-quadri-algebras and the analogues of the classical Yang-Baxter equation in these algebras. We get Lie algebraic analogues of Loday algebras which contain Lie algebras, pre-Lie algebras, L-dendriform algebras, L-quadri-algebras, L-octo-algebras and so on. The thesis includes three parts:(1) O-operators of pre-Lie algebras:suppose (A,o) is a pre-Lie algebra. The analogue of the classical Yang-Baxter equation in a pre-Lie algebra is called the S-equation. In this paper, we prove that:(Ⅰ) if r∈A⊕A and r is symmetric, then r is a symmetric solution of the S-equation in (A, o) if and only if r is an 0-operator associ-ated with bimodule (L。*-R。*,-R。*,A*); (Ⅱ) if T:A*→A is a linear operator which is symmetric and invertible, then the bilinear form B which is induced by T is a 2-cocycle of (A, o) if and only if T is an 0-operator associated with bimodule (L。* -R。*,-R。*,A*); if T is a linear operator which is skew-symmetric and invertible, then the bilinear form B which is induced by T is an invariant bilinear form of (A, o) if and only if T is an O-operator associated with bimodule (L。* -R。*,O, A*)(2) We introduce the notion of an L-dendriform algebra due to two different mo-tivations. L-dendriform algebras are regarded as the underlying algebraic structures of pseudo-Hessian structures on Lie groups and the algebraic structures behind the 0-operators of pre-Lie algebras and the related S-equation. As a direct consequence, they provide some explicit solutions of S-equations in certain pre-Lie algebras constructed from L-dendriform algebras. Furthermore, we introduce the notion of the LD-equation from two different ways:(Ⅰ) from 0-operators of L-dendriform algebras associated with certain bimodules; (Ⅱ) from L-dendriform bialgebras (or equivalently from the double construction of nondegenerate 2-cocycle of the pre-Lie algebra). We prove these two ways are coincident. So LD-equations in L-dendriform algebras are similar to classical Yang-Baxter equations in Lie algebras and S-equations in pre-Lie algebras.(3) By the study of Lie algebras, pre-Lie algebras and L-dendriform algebras, we consider the Lie algebraic analogues of Loday algebras. We introduce the notions of L-quadri-algebras and L-octo-algebras as examples. By the study of the O-operators of L-quadri-algebras, we get the corresponding bilinear forms and the analogue of the classical Yang-Baxter equation in L-quadri-algebras.