Let g be a reductive Lie algebra over a field of characteristic zero. Suppose that g acts on a complex of vector spaces M • by i λ and L λ , which satisfy the same identities that contraction and Lie derivative do for differential forms. Out of this data one defines the cohomology of the invariants and the equivariant cohomology of M • . We establish Koszul duality between them. IntroductionLet G be a compact Lie group. Set • = H * (G) and S • = H * (BG). The coefficients are in R or C. Suppose that G acts on a reasonable space X . In the paper [GKM], M. Goresky, R. Kottwitz, and R. MacPherson established a duality between the ordinary cohomology that is a module over • and the equivariant cohomology that is a module over S • . This duality is on the level of chains, not on the level of cohomology. Koszul duality says that there is an equivalence of derived categories of • -modules and S • -modules. One can lift the structure of an S • -module on H * G (X ) and the structure of a • -module on H * (X ) to the level of chains in such a way that the obtained complexes correspond to each other under Koszul duality. Equivariant coefficients in the sense of [BL] are also allowed. Later, C. Allday and V. Puppe [AP] gave an explanation for this duality based on the minimal Brown-Hirsh model of the Borel construction. One should remark that Koszul duality is a reflection of a more general duality: the one described by D. Husemoller, J. Moore, and J. Stasheff in [HMS].Our goal is to show that this duality phenomenon is a purely algebraic affair. We construct it without appealing to topology. We consider a reductive Lie algebra g and a complex of vector spaces M • on which g acts via two kinds of actions: i λ and L λ . These actions satisfy the same identities that contraction and Lie derivative do in the case of the action on the differential forms of a G-manifold. Such differential gmodules were already described by H. Cartan [Ca] (see also [AM], [GS]). We do not assume that M • is finite-dimensional or semisimple. We also wish to correct a small

Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the same finitedimensional representation of the structural quantum group. On the level of K 0 -groups, we realize the induced map by the pullback of explicit matrix idempotents. We also show how to extend our result to the case when the quantum-group representation is infinite dimensional, and then apply it to the Ehresmann-Schauenburg quantum groupoid. Finally, using noncommutative Milnor's join construction, we define quantum quaternionic projective spaces together with noncommutative tautological quaternionic line bundles and their duals. As a key application of the main theorem, we show that these bundles are stably non-trivial as noncommutative complex vector bundles. Contents 1. Pushing forward modules associated with Galois-type coactions 4 1.1. Faithfully flat coalgebra-Galois extensions 5 1.2. Principal coactions 8 1.3. The Hopf-algebraic case revisited 10 2. Pulling back noncommutative vector bundles associated with free actions of compact quantum groups 11 2.1. Iterated equivariant noncommutative join construction 12 2.2. Noncommutative tautological quaternionic line bundles and their duals 13Acknowledgments 16 References 16Our result is motivated by the search of K 0 -invariants. The main idea is to use equivariant homomorphisms to facilitate computations of such invariants by moving them from more complicated to simpler algebras. This strategy was recently successfully applied in [19] to distinguish the K 0 -classes of noncommutative line bundles over two different types of quantum complex projective spaces. Herein we 1 1 Keywords: K-theory of C*-algebras, free actions of compact quantum groups, Hopf algebras and faithful flatness, Chern-Galois character, noncommutative join construction and Borsuk-Ulam-type conjecture, quantum quaternionic projective spaces.

The general formula for the variation of the Godbillon-Vey class is given in terms of the obstruction to the existence of a projective transversal structure (when a foliation arises by gluing of level sets of local functions with fractional linear transition maps). Using the above formula one obtains (under the technical condition of separability of some topological space of cohomology) that the Godbillon-Vey number of a foliation F of codimension one on a compact orientable 3-fold is topologically rigid (i.e. constant under infinitesimal singular deformations) iff F admits a projective transversal structure.

Abstract. A natural isomorphism between the cyclic object computing the relative cyclic homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic object computing the cyclic homology of a Galois coalgebra with SAYD coefficients is presented. The isomorphism can be viewed as the cyclic-homological counterpart of the Takeuchi-Galois correspondence between the left coideal subalgebras and the quotient right module coalgebras of a Hopf algebra. A spectral sequence generalizing the classical computation of Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous quotient-coalgebra-Galois extensions is constructed. A Pontryagin type self-duality of the Takeuchi-Galois correspondence is combined with the cyclic duality of Connes in order to obtain dual results on the invariant cyclic homology, with SAYD coefficients, of algebras of invariants in homogeneous quotient-coalgebra-Galois extensions. The relation of this dual result with the Chern character, Frobenius reciprocity, and inertia phenomena in the local Langlands program, the Chen-Ruan-Brylinski-Nistor orbifold cohomology and the Clifford theory is discussed.

a b s t r a c tWe present a recursive algorithm computing all the genus zero Gromov-Witten invariants from a finite number of initial ones, for Fano manifolds with generically semisimple quantum (and tame semisimple small quantum) (p, p)-type cohomology, whose first Chern class is a strictly positive combination of effective integral basic divisors.