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Lie theory and the Chern-Weil homomorphism

Meinrenken, E.
Published in Annales scientifiques de l'Ecole normale supérieure. 2005, vol. 38, no. 2, p. 303-338
Abstract We introduce a canonical Chern-Weil map for possibly non-commutative g-differential algebras with connection. Our main observation is that the generalized Chern-Weil map is an algebra homomorphism ``up to g-homotopy''. Hence, the induced map from invariant polynomials to the basic cohomology is an algebra homomorphism. As in the standard Chern-Weil theory, this map is independent of the choice of connection. Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouviere's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott-Shulman complex.
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ALEKSEEV, Anton, MEINRENKEN, E. Lie theory and the Chern-Weil homomorphism. In: Annales scientifiques de l'Ecole normale supérieure, 2005, vol. 38, n° 2, p. 303-338. https://archive-ouverte.unige.ch/unige:12225

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Deposited on : 2010-10-22

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