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 \(\mathcal {O}\)-operators.

本文将李代数上权重为 0 的 Rota-Baxter 算子导出前李代数这一众所周知的事实扩展到双代数的水平。我们首先证明，在权重为 0 的 Rota-Baxter 李代数上不变的非简并对称双线性形式给出了在诱导前李代数上左不变的形式，从而给出了特殊的 L-树状代数。这一事实是作为 Rota-Baxter 李代数的特例而获得的，其具有伴随容许条件，即李代数的表示承认对偶空间上 Rota-Baxter 李代数的表示。该条件也可以自然地表述为 Rota-Baxter 李代数的 Manin 三元组，而 Rota-Baxter 李代数的 Manin 三元组又可以用双代数来表征，从而将 Manin 三元组方法扩展到李双代数。在权重为 0 的情况下，所得的 Rota-Baxter Lie 双代数产生特殊的 L-树状双代数，将上述 Rota-Baxter Lie 代数将前李代数导出到双代数的水平。这两类双代数之间的关系也根据共界情况、经典的 Yang-Baxter 方程和\(\mathcal {O}\)算子进行了研究。