Article

# Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation

(Impact Factor: 0.84). 09/2007; 10(02). DOI: 10.1142/S0219199708002752
Source: arXiv

ABSTRACT

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parak\"ahler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "$S$-equation", which is an analogue of the classical Yang-Baxter equation. In a certain sense, the $S$-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the $S$-equation gives a parak\"ahler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "${\cal O}$-operator", whereas a skew-symmetric solution of the classical Yang-Baxter equation corresponds to the skew-symmetric part of an ${\cal O}$-operator. Thus a method to construct symmetric solutions of the $S$-equation (hence parak\"ahler Lie algebras) from ${\cal O}$-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parak\"ahler Lie groups correspond to Poisson-Lie groups in this sense.

0 Followers
·
• Source
• "In this section, we give the definition of the center -symmetric algebra, provide their basic properties and deduce relevant algebraic consequences, similarly to known framework of left symmetric algebras [1]. "
##### Article: Center-symmetric algebras and bialgebras: relevant properties and consequences
[Hide abstract]
ABSTRACT: Lie admissible algebra structures, called center-symmetric algebras, are defined. Main properties and algebraic consequences are derived and discussed. Bimodules are given and used to build a center-symmetric algebra on the direct sum of underlying vector space and a finite dimensional vector space. Then, the matched pair of center-symmetric algebras is established and related to the matched pair of sub-adjacent Lie algebras. Besides, Manin triple of center-symmetric algebras is defined and linked with their associated matched pairs. Further, center-symmetric bialgebras of center-symmetric algebras are investigated and discussed. Finally, a theorem yielding the equivalence between Manin triple of center-symmetric algebras, matched pairs of Lie algebras and center-symmetric bialgebras is provided.
• Source
• "So pre-Lie algebras naturally play important roles in the study involving the representations of Lie algebras on the underlying spaces of the Lie algebras themselves or their dual spaces. For example, they are the underlying algebraic structures of the non-abelian phase spaces of Lie algebras [5] [17], which lead to a bialgebra theory of pre-Lie algebras [6]. They are also regarded as the algebraic structures " behind " the classical Yang-Baxter equations (CYBE) and they provide a construction of solutions of CYBE in certain semidirect product Lie algebras (that is, over the " double " spaces) induced by pre-Lie algebras [4] [18]. "
##### Article: Categorification of Pre-Lie Algebras and Solutions of 2-graded Classical Yang-Baxter Equations
[Hide abstract]
ABSTRACT: In this paper, we introduce the notion of a pre-Lie 2-algebra, which is a categorification of a pre-Lie algebra. We prove that the category of pre-Lie 2-algebras and the category of 2-term pre-Lie$_\infty$-algebras are equivalent. We classify skeletal pre-Lie 2-algebras by the third cohomology of a pre-Lie algebra. We prove that crossed modules of pre-Lie algebras are in one-to-one correspondence with strict pre-Lie 2-algebras. $\mathcal O$-operators on Lie 2-algebras are introduced, which can be used to construct pre-Lie 2-algebras. As an application, we give solutions of 2-graded classical Yang-Baxter equations in some semidirect product Lie 2-algebras.
• Source
• "Let introduce now a key notion in our work, namely, the notion of quasi S -matrix as a generalization of the one of S -matrix appeared first in [4]. Let U be a left symmetric algebra. "
##### Dataset: On para-Kähler and hyper-para-Kähler Lie algebras
[Hide abstract]
ABSTRACT: We study Lie algebras admitting para-Kähler and hyper-para-Kähler structures. We give new characterizations of these Lie algebras and we develop many methods to build large classes of examples. Bai considered para-Kähler Lie algebras as left symmetric bialgebras. We reconsider this point of view and improve it in order to obtain some new results. The study of para-Kähler and hyper-para-Kähler is intimately linked to the study of left symmetric algebras and, in particular, those admitting invariant symplectic forms. In this paper, we give many new classes of left symmetric algebras and a complete description of all associative algebras admitting an invariant symplectic form. We give also all four dimensional hyper-para-Kähler Lie algebras.