Lie theory and the Chern–Weil homomorphism

Abstract

Let P → B be a principal G-bundle. For any connection θ on P, the Chern–Weil construction of characteristic classes defines an algebra homomorphism from the Weil algebra W g = S g ∗ ⊗ ∧ g ∗ into the algebra of differential forms A = Ω ( P ) . Invariant polynomials ( S g ∗ ) inv ⊂ W g map to cocycles, and the induced map in cohomology ( S g ∗ ) inv → H ( A basic ) is independent of the choice of θ. The algebra Ω ( P ) is an example of a commutative g -differential algebra with connection, as introduced by H. Cartan in 1950. As observed by Cartan, the Chern–Weil construction generalizes to all such algebras. In this paper, we introduce a canonical Chern–Weil map W g → A 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 ( S g ∗ ) inv → H basic ( A ) 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 Rouvière's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott–Shulman complex.