AI 中文总结
研究沿着简单分离曲线拆分戈德曼 - 图拉耶夫李双代数，引入双李双模范畴概念及代数构造，通过几何应用将相关运算组装成双李双模范畴，证明了该李双代数的分解定理。
AI 中文摘要
我们引入双李双模范畴的概念，它由一个李双代数\(L\)、一个李\(L -\)双模范畴\(P\)以及\(P\)上满足适当相容性条件的双李括号组成。我们描述了一种将两个双李双模范畴组合成一个李双代数的代数构造。作为几何应用，我们表明具有边界的曲面的戈德曼 - 图拉耶夫李双代数，连同在成对不同边界点之间路径同伦类集合的线性张成上的川上 - 久野相交运算，可组装成一个双李双模范畴。我们证明了关于定向曲面的戈德曼 - 图拉耶夫李双代数的一个分解定理，该定理是根据沿着一条简单分离闭曲线切割得到的两个具有边界的曲面所对应的双李双模范畴给出的。
英文摘要
We introduce the notion of a double Lie bimodule, consisting of a Lie bialgebra L, a Lie L-bimodule P, and a double Lie bracket on P satisfying suitable compatibility conditions. We describe an algebraic construction that combines two double Lie bimodules into a Lie bialgebra. As a geometric application, we show that the Goldman-Turaev Lie bialgebra of a surface with boundary, together with the Kawazumi-Kuno intersection operations on the linear span of the set of homotopy classes of paths between pairwise distinct boundary points, may be assembled into a double Lie bimodule. We prove a decomposition theorem for the Goldman-Turaev Lie bialgebra of an oriented surface in terms of the double Lie bimodules corresponding to the two surfaces with boundary obtained by cutting along a simple separating closed curve.
