Lie theory and the Chern–Weil homomorphism

By A. Alekseev, E. Meinrenken. 2003

58 citations recorded by the source corpus.

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 Rouviere's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott–Shulman complex.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b30edf80542bf438ec08f8becc109e3b184cdc9d

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In Algebra and Number Theory

1. R. Mehta
2. C. Rossi
3. Damien Calaque
4. A. Gracia-Saz
5. C. A. Abad
6. M. Crainic

In Geometry and Topology

1. R. Mehta
2. Damien Calaque
3. A. Gracia-Saz
4. C. A. Abad
5. M. Crainic

In Mathematical Physics

1. R. Mehta
2. C. Rossi
3. Damien Calaque
4. A. Gracia-Saz
5. C. A. Abad
6. M. Crainic

Citations in the corpus

• Lectures on Duflo isomorphisms in Lie algebra and complex geometry (2011, background): “Notice that the mentioned series also appears in the wheeling theorem by Bar-Natan, Le and Thurston [4] which shows that two spaces of graph homology are isomorphic as algebras (see also [23] for a completely combinatorial proof of the wheeling theorem, based on Alekseev and Meinrenken’s proof [1, 2”
• Supergroupoids, double structures, and equivariant cohomology (2006, methodology): “dA is a degree 1 derivation of S[1]Γ(A∗) 2.”