  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
$\mathfrak{gl}(n)$.
