Bibliographic record
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. Soit P → B un G -fibré principal. Pour toute connexion θ sur P la construction de classes caractéristiques de Chern–Weil définit un homomorphisme d'algèbres de l'algèbre de Weil W g = S g ∗ ⊗ ∧ g ∗ dans l'algèbre des formes différentielles A = Ω ( P ) . Les polynômes invariants ( S g ∗ ) inv ⊂ W g s'envoient dans l'espace des cocyles et l'application induite en cohomologie ( S g ∗ ) inv → H ( A basic ) est indépendante du choix de θ . L'algèbre Ω ( P ) est un exemple d'une algèbre g -différentielle commutative (définie par H. Cartan en 1950). Dans cet article nous définissons l'application canonique de Chern–Weil W g → A pour les algèbres g -différentielles qui possèdent une connexion et qui ne sont pas commutatives. Le résultat principal est que l'application de Chern–Weil généralisée est un homomorphisme d'algèbre à g -homotopie près. Alors, l'application induite ( S g ∗ ) inv → H basic ( A ) est un homomorphisme d'algèbres. Comme dans la théorie standard, cette application est indépendante du choix de la connexion. Parmi les applications de nos résultats on trouve : une démonstration facile de l'isomorphisme de Duflo pour les algèbres de Lie quadratiques, une démonstration courte de la conjecture de Vogan sur la cohomologie de Dirac, des projections de Harish-Chandra généralisées pour les algèbres de Lie quadratiques, une extension du théorème de Rouvière sur les paires symétriques, et une nouvelle construction des formes caractéristiques universelles dans le complexe de Bott–Shulman.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".