Lie theory and the Chern–Weil homomorphism

Abstract

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 = Sg * ⊗ ∧g * into the algebra of differential forms A = Ω(P ). Invariant polynomials (Sg * )inv ⊂ W g map to cocycles, and the induced map in cohomology (Sg * )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 noncommutative 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 (Sg * )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.

Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry

“…The component P (2,0) consists of a family of holomorphic bidifferential operators on X for any open subset U α in U: Identity (A.6) can be written more explicitly as…”

Principal series representations and harmonic spinors

“…In order to obtain the most natural results, this degree three element is a necessary ingredient in the Dirac operator (see (2.6)). For example, the cubic term is needed for the following: (i) a simple formula for the square of the Dirac operator [8], (ii) a generalization of the Borel-Weil-Bott theorem [9], (iii) a strong connection with infinitesimal character [7, Section 7]; [1,10] and (iv) Lemma 2.5 below.…”

“…There are also interesting recent results in terms of (the algebraic version of) Kostant's cubic Dirac operator. See, for example, [1,7,10].…”

Universal Coefficient Theorem and Quantum Field Theory

“…Their cohomology classes do not depend on the connection. Thirdly and finally we construct the algebra Weil homomorphism I(G) → H even DR (M ) [145] which associates de Rham cohomology classes with the closed forms on M previously obtained.…”