The Category $\mathcal{O}$ for Lie Algebras of Vector Fields (II): Lie–Cartan Modules and Cohomology

Abstract

As a sequel to Duan, Shu, and Yao [Publ. Res. Inst. Math. Sci. 56 (2020), 743–760], we introduce here a category $\mathcal{L}\mathcal{C}$ arising from the BGG category $\mathcal{O}$ defined in Duan, Shu, and Yao [Publ. Res. Inst. Math. Sci. 56 (2020), 743–760] for Lie algebras of polynomial vector fields. The objects of $\mathcal{L}\mathcal{C}$ are so-called Lie–Cartan modules which admit both Lie-module structure and compatible $\mathcal{R}$-module structure ($\mathcal{R}$ 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 $\mathcal{L}\mathcal{C}$ 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 $\mathcal{L}\mathcal{C}$. We then establish the cohomology for the category of universal Lie–Cartan modules (called the ${}^u\mathcal{L}\mathcal{C}$-cohomology), extending Chevalley–Eilenberg cohomology theory. Another notable result says that in the fundamental case $\mathfrak{g}=W(n)$, the extension ring ${\mathrm{Ext}}_{{}^u\mathcal{L}\mathcal{C}}^{\bullet }(\mathcal{R},\mathcal{R})$ for the polynomial algebra $\mathcal{R}$ in the ${}^u\mathcal{L}\mathcal{C}$-cohomology is isomorphic to the usual cohomology ring $H^{\bullet }(\mathfrak{gl}(n))$ of the general linear Lie algebra $\mathfrak{gl}(n)$.