AI 中文总结
该研究证明后李与后霍普夫代数的伴随关系与无穷小结构相容，推广了Cartier–Milnor–Moore定理，完成了相关结构的分类并证明了无穷小后李代数operad的Koszul性。
AI 中文摘要
我们描述了后李代数和后霍普夫代数的无穷小形变，并证明了由泛包络代数与本原元素函子给出的伴随关系与该无穷小结构相容。当限制到连通余交换的无穷小后霍普夫代数时，这成为范畴等价，构成了Cartier–Milnor–Moore定理的推广。我们对𝔰𝔩(2)上的无穷小后李结构进行分类，并讨论一类来自具有协变常挠率的平坦联络的无穷小后李代数。此外，我们对Sweedler霍普夫代数上的无穷小后霍普夫结构进行分类。余交换的无穷小后霍普夫代数在其关联的相邻霍普夫代数上诱导出Hochschild 2-上循环。最后，我们利用李代数与双 magma operad 之间的过滤分配律，证明无穷小后李代数的二次operad是Koszul的。
英文摘要
We describe infinitesimal deformations of post-Lie algebras and post-Hopf algebras and prove that the adjunction given by the universal enveloping algebra and primitive elements functors is compatible with the infinitesimal structure. When restricted to connected and cocommutative infinitesimal post-Hopf algebras, this becomes an equivalence of categories, which constitutes an extension of the Cartier--Milnor--Moore theorem. We classify infinitesimal post-Lie structures on $\mathfrak{sl}(2)$, and discuss a class of infinitesimal post-Lie algebras emerging from flat connections with covariantly-constant torsion. Moreover, we classify infinitesimal post-Hopf structures on Sweedler's Hopf algebra. Cocommutative infinitesimal post-Hopf algebras induce a Hochschild 2-cocycle on the associated subadjacent Hopf algebra. Finally, we prove that the quadratic operad of infinitesimal post-Lie algebras is Koszul, by using a filtered distributive law between the operads of Lie algebras and bi-magmas.
Comments31 pages, comments are welcome
