Lie-Cartan modules and cohomology

As a sequel to [Duan-Shu-Yao], we introduce here a category ℒ𝒞 arising from the BGG category 𝒪 defined in [Duan-Shu-Yao] for Lie algebras of polynomial vector fields. The objects of 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 ℒ𝒞 is abelian, and a “highest weight category" with depths. Notably, the set of co-standard objects in the category 𝒪 turns out to represent the isomorphism classes of simple objects of ℒ𝒞. We then establish the cohomology for this category (called the 𝓊ℒ𝒞-cohomology), extending Chevalley-Eilenberg cohomology theory. Another notable result says that in the fundamental case 𝔤= W(n), the extension ring Ext^∙_𝓊ℒ𝒞(R,R) for the polynomial algebra R in the 𝓊ℒ𝒞-cohomology is isomorphic to the usual cohomology ring H^∙(𝔤𝔩(n)) of the general linear Lie algebra 𝔤𝔩(n).