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Comparison of categorical characteristic classes of a transitive Lie algebroid with the Chern-Weil homomorphism
- Source :
- Russian Journal of Mathematical Physics. 22:328-335
- Publication Year :
- 2015
- Publisher :
- Pleiades Publishing Ltd, 2015.
-
Abstract
- This paper is devoted to analyzing two approaches to characteristic classes of transitive Lie algebroids. The first approach is due to Kubarski [5] and is a version of the Chern-Weil homomorphism. The second approach is related to the so-called categorical characteristic classes (see, e.g., [6]). The construction of transitive Lie algebroids due to Mackenzie [1] can be considered as a homotopy functor T LAg from the category of smooth manifolds to the transitive Lie algebroids. The functor T LAg assigns to every smooth manifold M the set T LAg(M) of all transitive algebroids with a chosen structural finite-dimensional Lie algebra g. Hence, one can construct [2, 3] a classifying space Bg such that the family of all transitive Lie algebroids with the chosen Lie algebra g over the manifold M is in one-to-one correspondence with the family of homotopy classes of continuous maps [M, Bg]: T LAg(M) ≈ [M, Bg]. This enables us to describe characteristic classes of transitive Lie algebroids from the point of view of a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles and to compare them with those derived from the Chern-Weil type homomorphism by Kubarski [5]. As a matter of fact, we show that the Chern-Weil type homomorphism by Kubarski does not cover all characteristic classes from the categorical point of view.
- Subjects :
- Discrete mathematics
Lie algebroid
Transitive relation
Classifying space
Pure mathematics
Chern–Weil homomorphism
Functor
Statistical and Nonlinear Physics
Characteristic class
Mathematics::K-Theory and Homology
Lie algebra
Homomorphism
Mathematics::Symplectic Geometry
Mathematical Physics
Mathematics
Subjects
Details
- ISSN :
- 15556638 and 10619208
- Volume :
- 22
- Database :
- OpenAIRE
- Journal :
- Russian Journal of Mathematical Physics
- Accession number :
- edsair.doi...........2f0c9f64a8bc9adec00e33a2367866c8