# Complexes of representations with complementary central charges

This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary central charges. This applies to infinite-dimensional Lie algebras of a certain kind, like the Virasoro or Kac-Moody algebras.

Let me describe the Virasoro case. Consider representations on which the central element of Virasoro acts by a constant c, and also representations on which it acts by the constant 26-c. Additional conditions have to be imposed on the representations, namely, that the positive-graded part of Virasoro acts nilpotently, or some variation of this. Then there is a way to assign (contravariantly) complexes of representations with the central charge 26-c to complexes of representations with the central charge c.

The Kac-Moody case is similar, except that there is another number (depending on the Kac-Moody algebra) in place of 26. (Actually, this holds for any Lie algebra graded by the integers with finite-dimensional components, and it can be generalized even further.)

This correspondence functor was not very well defined, classically, because it sometimes assigns acyclic complexes to nonacyclic ones and vice versa. I know how to make it well-defined, and even a (covariant) equivalence of triangulated categories. This is exactly the reason why I am asking for references to any classical expositions where the problem of constructing such a functor or equivalence were, at least, raised. I myself learned about this problem from folklore.

The only references I am presently aware of are some papers by Feigin--Fuchs and Rocha-Caridi--Wallach circa 1984 where the Verma and irreducible modules over the Virasoro algebra were studied. The discussion there is very brief; the very fact that one sometimes obtains acyclic complexes as duals to irreducible modules is never mentioned explicitly. Are there any later and/or more detailed references?

EDIT: Let me add some details about the construction. Feigin and Fuchs explain here and here that the categories of Verma modules over the Virasoro with central charges c and 26-c are anti-equivalent. The next natural step would be extending this anti-equivalence to non-Verma modules by a kind of derived functor construction, using Verma modules as adjusted objects. So one takes the trivial one-dimensional module at c=0, writes its left resolution made of Verma modules, and applies the Feigin-Fuchs anti-equivalence to this complex of Verma modules. What one obtains is a complex of Verma modules at c=26, bounded from below but not from above. This complex turns out to be acyclic.

Thus if one wishes to make this an equivalence of derived categories, one has to explain in what sense can an acyclic complex be a nontrivial object and what kind of derived category this is. My question is whether this predicament have been described anywhere in the literature (before my own monograph have presumably resolved it, among other things).

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I am curious: what is that number for Kac-Moody algebras? –  André Henriques May 14 '13 at 13:10
I don't remember it very well, and don't even quite remember where I've read about it many years ago. But my guess is for an affine Lie algebra it should be minus twice the dual Coxeter number, $-2h^\vee$. See mathoverflow.net/questions/25592/… –  Leonid Positselski May 14 '13 at 14:01