Article

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

09/2007;

Source: arXiv

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**ABSTRACT:**We introduce notions of 𝒪-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of Rota–Baxter operators. The invertible 𝒪-operators give a sufficient and necessary condition on the existence of the 2 operations on an algebra with the 2 operations in an associative cluster. The analogues of the classical Yang–Baxter equation in these algebras can be understood as the 𝒪-operators associated to certain dual bimodules. As a byproduct, the constraint conditions (invariances) of nondegenerate bilinear forms on these algebras are given.Communications in Algebra 01/2010; 38(11):4277-4321. · 0.36 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this article, we commence to study the real (simple) left-symmetric algebras. From the known classification of certain complex (semi)simple left-symmetric algebras, we classify their corresponding real forms. We not only obtain the classification of real simple left-symmetric algebras in low dimensions, but also find certain examples of real simple left-symmetric algebras in higher dimensions. In particular, there exists a complex simple left-symmetric algebra without any real form. We also give a geometric construction for a class of real simple left-symmetric algebras. At last, we apply the classification results to study some structures related to geometry.Communications in Algebra 01/2012; 40(5):1641-1668. · 0.36 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The study of the Lie groups with a left invariant flat pseudo-metric is equivalent to the study of the left-symmetric algebras with a nondegenerate left invariant bilinear form. In this paper, we consider such a structure satisfying an additional condition that there is a decomposition into a direct sum of the underlying vector spaces of two isotropic subalgebras. Moreover, there is a new underlying algebraic structure, namely, a special L-dendriform algebra and then there is a bialgebra structure which is equivalent to the above structure. The study of coboundary cases leads to a construction from an analogue of the classical Yang–Baxter equation. KeywordsLeft invariant pseudo-metric–Left-symmetric algebra–L-dendriform algebraMonatshefte für Mathematik 01/2010; 164(3):243-269. · 0.70 Impact Factor

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