Semiinfinite Cohomology of Lie-* Algebras

This paper is a natural extension of the previous note [Ar2]. Semiinfinite cohomology of Tate Lie algebra was defined in that note in terms of some duality resembling Koszul duality. The language of differential graded Lie algebroids was the main technical tool of the note. The present note is devoted to globalization of the main construction from [Ar2] in the following sense. The setup in [Ar2] included a suitably chosen module over a Tate Lie algebra g with a fixed Lie subalgebra b being a c-lattice in g. The rough global analogue of this picture is as follows. Consider a compact curve X over a field of characteristic zero. Denote by mod-DX the category of (right) D-modules on X. We fix a Lie algebra G in the category mod-DX . This data can be viewed as a family of Lie algebras gx along the curve X. Another part of the data includes a Lie subalgebra B ⊂ G. So the problem is to define semiinfinite cohomology complex of such pair. In fact we need some additional constraints on the pair B ⊂ G. So the formal picture starts from a different notion of a Lie-* algebra L on X (see the precise definition in Section 2). Roughly spaking a DX -locally free Lie-* algebra is a Dmodule incarnation of a Lie algebra in the category of vector bundles on X with the bracket given by a differential operator. We define two types of modules over a Lie-* algebra (see 2.2.1). The first one called a Lie-* module is just a D-module incarnation of the module over the Lie algebra in the category of vector bundles, like above, with the action given by a differential operator. Still we will be more interested in the second type of modules over a Lie-* algebra called chiral modules (see 2.2.1 for the definition). So starting from a Lie-* algebra L and a chiral module M we perform the main construction more or less parallel to the one from [Ar2]. Namely we define the Lie algebra G = G(L) in the category mod-DX with the Lie subalgebra B = B(L) ⊂ G(L). We show that a L-chiral module M becomes a G(L)-module. Next, imitating the construction of [Ar2] Section 4, we define a DG Lie algebroid