论文标题
Rota-Baxter Like代数的整合和几何化
Integration and geometrization of Rota-Baxter Lie algebras
论文作者
论文摘要
本文首先介绍了一个Lie组的Rota-Baxter操作员（重量$ 1 $）的概念，以便其差异化为相应的Lie代数提供了Rota-Baxter操作员。谎言组的直接产物，包括伊瓦泽（Iwasawa）和兰兰（Langlands）的分解，载有天然的rota-baxter操作员。 Rota-baxter操作员在谎言组上的正式逆恰恰是Lie组的交叉同构，其切线图是Lie代数上的重量$ 1 $的差分操作员。事实证明，Rota-Baxter Lie组的分解定理，直接衍生在Lie组级别，这是Semenov-Tian-Shansky的众所周知的全球分解定理，在他对综合系统的研究中。当几何化时，引入了rota-baxter的概念为代数和rota-baxter lie clastoids，而后者则是后者的分化。此外，Rota-baxter Lie代数自然会引起lie后代数，从而推广了rota-baxter lie代数和lie后代数的众所周知的事实。结果表明，可以通过其对歧管的作用来实现Rota-Baxter Lie代数或Rota-Baxter Lie组的几何化。为这些新概念提供了示例和应用程序。
This paper first introduces the notion of a Rota-Baxter operator (of weight $1$) on a Lie group so that its differentiation gives a Rota-Baxter operator on the corresponding Lie algebra. Direct products of Lie groups, including the decompositions of Iwasawa and Langlands, carry natural Rota-Baxter operators. Formal inverse of the Rota-Baxter operator on a Lie group is precisely the crossed homomorphism on the Lie group, whose tangent map is the differential operator of weight $1$ on a Lie algebra. A factorization theorem of Rota-Baxter Lie groups is proved, deriving directly on the Lie group level, the well-known global factorization theorems of Semenov-Tian-Shansky in his study of integrable systems. As geometrization, the notions of Rota-Baxter Lie algebroids and Rota-Baxter Lie groupoids are introduced, with the former a differentiation of the latter. Further, a Rota-Baxter Lie algebroid naturally gives rise to a post-Lie algebroid, generalizing the well-known fact for Rota-Baxter Lie algebras and post-Lie algebras. It is shown that the geometrization of a Rota-Baxter Lie algebra or a Rota-Baxter Lie group can be realized by its action on a manifold. Examples and applications are provided for these new notions.
