Rota-Baxter Lie bialgebras, classical Yang-Baxter equations and special L-dendriform bialgebras

This paper extends the well-known fact that a Rota-Baxter operator of weight 0 on a Lie algebra induces a pre-Lie algebra, to the level of bialgebras. We first show that a nondegenerate symmetric bilinear form that is invariant on a Rota-Baxter Lie algebra of weight 0 gives such a form that is left-invariant on the induced pre-Lie algebra and thereby gives a special L-dendriform algebra. This fact is obtained as a special case of Rota-Baxter Lie algebras with an adjoint-admissible condition, for a representation of the Lie algebra to admit a representation of the Rota-Baxter Lie algebra on the dual space. This condition can also be naturally formulated for Manin triples of Rota-Baxter Lie algebras, which can in turn be characterized in terms of bialgebras, thereby extending the Manin triple approach to Lie bialgebras. In the case of weight 0, the resulting Rota-Baxter Lie bialgebras give rise to special L-dendriform bialgebras, lifting the aforementioned connection that a Rota-Baxter Lie algebra induces a pre-Lie algebra to the level of bialgebras. The relationship between these two classes of bialgebras is also studied in terms of the coboundary cases, classical Yang-Baxter equations and O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}$$\end{document}-operators.