Home > Chern-Weil homomorphism

Denote by either the real field or complex field. Let G be a real or complex Lie group with Lie algebra ; and let

denote the algebra of -valued polynomials on . Let be the subalgebra of fixed points in under the adjoint action of G, so that for instance

for all .

The Chern-Weil homomorphism is a homomorphism of -algebras from to the cohomology algebra . Such a homomorphism exists and is uniquely defined for every principal G-bundle P on M. One can usually think of the bundle P as living inside the K-theory of M, , so that the class of Chern-Weil homomorphisms is parametrized by .

Definition of the homomorphism

Choose any connection form w in P, and let be the associated curvature 2-form. If is a homogeneous polynomial of degree k, let be the 2k-form on P given by

where is the sign of the permutation in the symmetric group on 2k numbers .

(see Pfaffian).

One can then show that is closed , and that the cohomology class of is independent of the choice of connection on P, so it depends only upon the principal bundle.

Thus letting be the cohomology class obtained in this way from f, we obtain an algebra homomorphism .