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Lie theory and the Chern-Weil homomorphism. (English) Zbl 1105.17015
Let \(\mathbb{F}\) be a field of characteristic \(0\), \(\mathfrak{g}\) a Lie algebra and \(W\mathfrak{g}=S\mathfrak{g}^{\ast}\otimes\Lambda\mathfrak{g} ^{\ast}\) the Weil algebra. The authors introduce a canonical Chern-Weil map \(W\mathfrak{g}\rightarrow\mathcal{A}\) for possibly noncommutative \(\mathfrak{g}\)-differential algebras \(\mathcal{A}\) with a connection. This generalizes to the noncommutative case an algebraic construction of H. Cartan [Centre Belge Rech. Math., Colloque Topologie, Bruxelles, 1950, 1, 15–27 (1951; Zbl 0045.30601)]. This generalized Chern-Weil map is an algebra homomorphism “up to \(\mathfrak{g}\)-homotopy” and induces an algebra homomorphism \((S\mathfrak{g}^{\ast})_{\mathrm{inv}}\rightarrow H_{\mathrm{basic}}(\mathcal{A})\), which does not depend on the choice of the connection. The following applications are worked out: for quadratic Lie algebras (Lie algebras with an invariant non-degenerate symmetric bilinear form), a description as arising from a symmetrization map for some superalgebra containing \(U(\mathfrak{g})\), of the Duflo isomorphism \(S\mathfrak{g} \rightarrow U\mathfrak{g}\); a short proof of a conjecture of Vogan on Dirac cohomology; generalized Harish-Chandra projections for quadratic Lie algebras; an extension of Rouvière’s theorem for symmetric pairs; a new construction of universal characteristic forms in the Bott-Shulman complex.

MSC:
17B99 Lie algebras and Lie superalgebras
53C05 Connections (general theory)
57R20 Characteristic classes and numbers in differential topology
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