Brylinski, Jean-Luc Non-commutative Ruelle-Sullivan type currents. (English) Zbl 0738.58006 The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. I, Prog. Math. 86, 477-498 (1990). For a manifold \(M\) there is an obvious embedding \(A^*(M)\to C^*(\hbox{Vect}(M),C^ \infty(M))\) of the graded differential algebra of differential forms into the complex of continuous Lie algebra cochains from the Lie algebra of vector fields on \(M\) with values in smooth functions. It played some role in Gel’fand-Fuks cohomology. The author considers an obvious sort of dual mapping of complexes \(C_ *(\hbox{Vect}(M),A^ m_{cl}(M))\to A^{m-*}(M)\), where \(A_{cl}(M)\) denotes the closed forms on \(M\). One may restrict this to the fundamental vector fields of some Lie group action on \(M\) to get an induced mapping from the homology of the Lie algebra into the cohomology of \(M\), if there exists a closed form on \(M\) which is invariant under the infinitesimal action. This reproduces some well known results for symplectic and Poisson actions.For the entire collection see [Zbl 0737.14007]. Reviewer: P.Michor (Wien) Cited in 2 Documents MSC: 58B25 Group structures and generalizations on infinite-dimensional manifolds 46L85 Noncommutative topology 46L87 Noncommutative differential geometry 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties 58A10 Differential forms in global analysis 17B66 Lie algebras of vector fields and related (super) algebras Keywords:graded differential algebra; differential forms; complex of continuous Lie algebra cochains; Lie algebra of vector fields; homology; cohomology PDFBibTeX XMLCite \textit{J.-L. Brylinski}, Prog. Math. None, 477--498 (1990; Zbl 0738.58006)