Let $g$ be a Lie algebra over $\mathbb{C}$. Then the equivalence between the derived category of modules over $U(g)$ and the coderived category of comodules over it's Chevalley complex $C_*(g)$ in which $M\rightarrow C_*(g,M)$ is a classical example of Koszul duality. In the case when $g$ is a finite dimensional Lie algebra, it is easier for most humans to dualize everything and discuss modules over the Chevalley cochain complex $C^*(g)$. Most of the time, people are most interested in various subcategories of modules over a given Liealgebra. For example the category of semisimple modules or finite dimensional modules over $g$. I would like to know if for any Liealgebra $g$ if one can understand what happens to the subcategories of the derived category generated by these objects under the Koszul duality. Are the corresponding subcategories equally natural to describe? My intuition is that the semisimple modules are sent to something like (co) perfect modules based upon the observation that $\mathbb{C}$ is sent to $C_*(g) $. But I don't think this is exactly right and I wonder if someone could help me out.
1 Answer
The derived category of finitedimensional $g$modules is not a full subcategory of the derived category of arbitrary $g$modules for a finitedimensional Lie algebra $g$, in general (e.g., for a semisimple Lie algebra $g$). However, one can consider the full subcategory of the derived category of $g$modules consisting of complexes of $g$modules with bounded and finitedimensional cohomology. This triangulated category is equivalent to the homotopy category of DGmodules over $C^\ast(g)$ that are free/projective and finitely generated as graded modules.
The proof (based on the general Koszul duality theorem mentioned in the question) proceeds as follows. The category of DGmodules over $C^\ast(g)$ that are injective/projective as graded modules is a full subcategory of the co/contraderived category of DGmodules. The category of complexes of $g$modules with bounded and finitedimensional cohomology is generated by finitedimensional $g$modules, and these are transformed by the Koszul duality functor into DGmodules that are free and finitely generated as graded modules. This provides a fully faithful functor in one direction.
To prove that it is essentially surjective, consider the functor assigning to a complex of $g$modules the underlying complex of vector spaces. Let us presume that our Koszul duality functor is $M\longmapsto C^\ast(g,M)$. Then this Koszul duality functor transforms our functor of forgetting the action of $g$ into the functor assigning to a DGmodule over $C^\ast(g)$, projective as a graded module, its quotient complex by the action of the augmentation ideal of $C^\ast(g)$. Thus the complex of $g$modules corresponding to a DGmodule that is projective and finitely generated as a graded module has bounded and finitedimensional cohomology.

$\begingroup$ Lenya, in the first sentence, are you saying that for a semisimple Lie algebra g there are nontrivial derived homomorphisms of a module into itself because $H^*(g)$ has positive degree elements? $\endgroup$ Jul 17, 2010 at 19:56

$\begingroup$ Yes. None in the derived category of finitedimensional modules, since it is semisimple (in characteristic zero), but quite some, between a finitedimensional module and itself, in the derived category of arbitrary modules. $\endgroup$ Jul 17, 2010 at 22:17