Lie algebroid structures on double vector bundles and representation theory of Lie algebroids
By A. Gracia-Saz, R. Mehta. 2008
117 citations recorded by the source corpus.
Abstract A VB -algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB -algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB -algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/967e029b289170e4e897ef089c67578b337e90bb
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In Algebra and Number Theory
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Vector bundles over Lie groupoids and algebroids (2014, methodology): “A VB-algebroid [23, 13] consists of a double vector bundle (Ω, E,A,M) equipped with a Lie-algebroid structure Ω ⇒ E that is compatible with the second vector-bundle structure on Ω in the following sense: we require the linear Poisson structure on Ω∗E, corresponding to Ω ⇒ E, to be also linear with r”
- The Atiyah class of a dg-vector bundle (2015, background): “1 The example above is a special case of a general fact [10], that LA-vector bundles [6, 7, 8] (also known as VB-algebroids [2]) give rise to dg-vector bundles.”
- VB-groupoids and representation theory of Lie groupoids (2010, background): “It is known [2, 3, 6, 7, 11] that the modular class is only the first of a sequence of “secondary” characteristic classes associated to Lie algebroid representations up to homotopy.”