标题： Lie-Cartan 模块和上同调 显示英文标题

标题： Lie-Cartan modules and cohomology

摘要： 作为[Duan-Shu-Yao]的后续，我们在这里引入一个范畴$\mathscr{LC}$，该范畴来源于在[Duan-Shu-Yao]中为多项式向量场李代数定义的BGG范畴$\mathcal{O}$。 $\mathscr{LC}$的对象是所谓的李-卡当模，它们同时具有李模结构和相容的$R$模结构（$R$表示相应的多项式环）。 这个术语是自然的，它来自微分几何中的仿射联络，通过这种方式，拓扑中的结构层和几何中的向量场被整合到微分流形中。 在本文中，我们研究李-卡当模及其范畴和上同调性质。 范畴$\mathscr{LC}$是阿贝尔的，并且是一个具有深度的“最高权范畴”。 显然，范畴$\mathcal{O}$中的共标准对象集合实际上代表了范畴$\mathscr{LC}$中简单对象的同构类。 我们随后建立了该范畴的上同调（称为$\mathscr{uLC}$-上同调），扩展了Chevalley-Eilenberg上同调理论。 另一个显著的结果指出，在基本情况下$\mathfrak{g}= W(n)$，多项式代数$R$在$\mathscr{uLC}$-上同调中的扩张环$\text{Ext}^\bullet_{\mathscr{uLC}}(R,R)$同构于一般线性李代数$\mathfrak{gl}(n)$的通常上同调环$H^\bullet(\mathfrak{gl}(n))$。 显示英文摘要

摘要： As a sequel to [Duan-Shu-Yao], we introduce here a category $\mathscr{LC}$ arising from the BGG category $\mathcal{O}$ defined in [Duan-Shu-Yao] for Lie algebras of polynomial vector fields. The objects of $\mathscr{LC}$ are so-called Lie-Cartan modules which admit both Lie-module structure and compatible $R$-module structure ($R$ denotes the corresponding polynomial ring). This terminology is natural, coming from affine connections in differential geometry through which the structure sheaves in topology and the vector fields in geometry are integrated for differential manifolds. In this paper, we study Lie-Cartan modules and their categorical and cohomology properties. The category $\mathscr{LC}$ is abelian, and a ``highest weight category" with depths. Notably, the set of co-standard objects in the category $\mathcal{O}$ turns out to represent the isomorphism classes of simple objects of $\mathscr{LC}$. We then establish the cohomology for this category (called the $\mathscr{uLC}$-cohomology), extending Chevalley-Eilenberg cohomology theory. Another notable result says that in the fundamental case $\mathfrak{g}= W(n)$, the extension ring $\text{Ext}^\bullet_{\mathscr{uLC}}(R,R)$ for the polynomial algebra $R$ in the $\mathscr{uLC}$-cohomology is isomorphic to the usual cohomology ring $H^\bullet(\mathfrak{gl}(n))$ of the general linear Lie algebra $\mathfrak{gl}(n)$.