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Koszul duality for modules over Lie algebras. (English) Zbl 1014.17018

Let \(G\) be a compact connected Lie group and set \(\Lambda_{\bullet}=H_*(G)\) and \(S^{\bullet}=H^*(BG)\) where cohomology is taken with coefficients in \(\mathbb R\). Let us recall what kind of algebra \(\Lambda_{\bullet}\) and \(S^{\bullet}\) are. \(H^*(G)\) is calculated by the cohomology of its Lie algebra \({\mathfrak g}\) (or equivalently by the \({\mathfrak g}\)-invariants \((\Lambda{\mathfrak g}^*)^{\mathfrak g}\) of \((\Lambda{\mathfrak g}^*)\)), which is an exterior algebra on the primitive elements, and the same holds for \(H_*(G)\) using the Kronecker pairing. On the other hand, the essence of Chern-Weil theory is that \(H^*(BG)\) is the algebra of polynomials on \({\mathfrak g}^*\) which are invariant under the natural \({\mathfrak g}\)-action (or equivalently it can be described as the symmetric algebra on some space of primitive elements).
Koszul duality in the sense of I. N. Bernstein, I. M. Gelfand and S. I. Gelfand [Funkt. Anal. Prilozh. 12, No. 3, 66-67 (1978; Zbl 0402.14005)] and further developed by M. Goresky, R. Kottwitz and R. MacPherson [Invent. Math. 131, 25-83 (1998; Zbl 0897.22009)] establishes an equivalence between the derived categories (of bounded below complexes of) \(\Lambda_{\bullet}\)-modules \(D_+(\Lambda_{\bullet})\) and \(S^{\bullet}\)-modules \(D_+(S^{\bullet})\). Denoting more generally by \(k\) an algebraically closed field, it is induced by the functors \(h= \operatorname{Hom}_k(\Lambda_{\bullet},-):K_+(S^{\bullet})\to K_+(\Lambda_{\bullet})\) and \(t=S^{\bullet}\otimes_k-:K_+(\Lambda_{\bullet})\to K_+(S^{\bullet})\) on the corresponding homotopy categories.
In the special case where \(G\) acts on a manifold \(X\), Koszul duality is interpreted as a link between the equivariant cohomology of \(X\), i.e. the cohomology of its Borel construction \(EG\times_G X\) for some space \(EG\) modeling the total space of the universal \(G\)-bundle \(EG\to BG\), and the invariant cohomology of \(X\). The latter is the cohomology of the \({\mathfrak g}\)-invariants \(\Omega^{\bullet}(X)^{\mathfrak g}\) of the de Rham complex \(\Omega^{\bullet}(X)\) of \(X\); \(\Omega^{\bullet}(X)^{\mathfrak g}\) is quasi-isomorphic to \(\Omega^{\bullet}(X)\) by an averaging argument. Observing that \(\Omega^{\bullet}(X)^{\mathfrak g}\) gives rise to an object of \(D_+(\Lambda_{\bullet})\) and \(\Omega^{\bullet}(EG\times_G X)\) (if one can give a meaning to the de Rham complex) gives rise to an object of \(K_+(S^{\bullet})\), Koszul duality in this context is a functor on cochain level reconstructing equivariant from ordinary cohomology.
In the article under review, the authors show how to formulate Koszul duality in a pure Lie algebraic framework. Namely, they define a category \(K({\mathfrak g})\) consisting of differential graded vector spaces \(M^{\bullet}=(M^{\bullet},d)\) with a “contraction” map \(i_{\lambda}:M^{\bullet}\to M^{\bullet-1}\) for each \(\lambda\in{\mathfrak g}\) such that \(i_{\lambda}\circ i_{\mu}=-i_{\mu}\circ i_{\lambda}\) and \([{\mathcal L}_{\lambda},i_{\mu}]=i_{[\lambda,\mu]}\) for all \(\lambda,\mu\in{\mathfrak g}\) where \({\mathcal L}_{\lambda}=d\circ i_{\mu}+i_{\lambda}\circ d\). \({\mathfrak g}\) acts on \(M^{\bullet}\) by means of this “Lie derivative” \({\mathcal L}\). For \(M^{\bullet}\in ob(K({\mathfrak g}))\), \(\Lambda{\mathfrak g}\) acts on \(M^{\bullet}\) via the contraction map and the \({\mathfrak g}\)-invariants \((M^{\bullet})^{\mathfrak g}\) are thus a differential graded module over \(\Lambda_{\bullet}\), defining on object of the derived category \(D(\Lambda_{\bullet})\), called the invariant cohomology of \(M^{\bullet}\). On the other hand, set \((M^{\bullet})_{\mathfrak g}=(S^{\bullet}{\mathfrak g}^*\otimes M^{\bullet})^{\mathfrak g}\). \((M^{\bullet})_{\mathfrak g}\) defines an object of \(D(S^{\bullet})\) called the equivariant cohomology of \(M^{\bullet}\). In the particular case where \(G\) acts on \(X\), it gives the Cartan model of \(\Omega^{\bullet}(EG\times_G X)\) for the equivariant cohomology of \(X\). In their main theorem, the authors show that \(h((M^{\bullet})_{\mathfrak g})\) is quasi-isomorphic to \((M^{\bullet})^{\mathfrak g}\), thus formulating Goresky, Kottwitz and MacPherson’s Koszul duality in Lie algebraic terms.
The proof consists in constructing a quasi-isomorphism \(\psi\) between \[ (\Lambda^{\bullet}{\mathfrak g}^*)^{\mathfrak g}\otimes(S^{\bullet}{\mathfrak g}^*\otimes M^{\bullet})^{\mathfrak g}\quad\text{ and }\quad(S^{\bullet}{\mathfrak g}^*\otimes\Lambda^{\bullet}{\mathfrak g}^*\otimes M^{\bullet})^{\mathfrak g} \] (because the latter is \((W({\mathfrak g})\otimes M^{\bullet})^{\mathfrak g}\) with the Weil algebra \(W({\mathfrak g})\) of \({\mathfrak g}\), which is quasi-isomorphic to \((M^{\bullet})^{\mathfrak g}\) for reductive \({\mathfrak g}\).) In order to do this, the authors extend an explicit isomorphism \((M^{\bullet})_{\mathfrak g}\to (W({\mathfrak g})\otimes M^{\bullet})_{\text{ basic}}\) (linking Cartan and Weil model of equivariant cohomology) to \(\psi\) using distinguished transgression elements. Thereby they correct a small inaccuracy in [M. Goresky, et al. (loc. cit.)]. Filtering by the degree in the symmetric algebra on both sides, one deduces quasi-isomorphy by a spectral sequence argument.
Useful references in order to understand this densely written article are [M. Goresky, et al. (loc. cit.) and V. W. Guillemin and S. Sternberg, Supersymmetry and equivariant de Rham theory. Math. Past Present, Springer Berlin (1999; Zbl 0934.55007)].

MSC:

17B55 Homological methods in Lie (super)algebras
55N91 Equivariant homology and cohomology in algebraic topology
17B20 Simple, semisimple, reductive (super)algebras
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References:

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