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Full-Text Articles in Physical Sciences and Mathematics
Lie Algebroid Modules And Representations Up To Homotopy, Rajan Amit Mehta
Lie Algebroid Modules And Representations Up To Homotopy, Rajan Amit Mehta
Mathematics Sciences: Faculty Publications
We establish a relationship between two different generalizations of Lie algebroid representations: representation up to homotopy and Vaĭntrob’s Lie algebroid modules. Specifically, we show that there is a noncanonical way to obtain a representation up to homotopy from a given Lie algebroid module, and that any two representations up to homotopy obtained in this way are equivalent in a natural sense. We therefore obtain a one-to-one correspondence, up to equivalence.
L ∞-Algebra Actions, Rajan Amit Mehta, Marco Zambon
L ∞-Algebra Actions, Rajan Amit Mehta, Marco Zambon
Mathematics Sciences: Faculty Publications
We define the notion of action of an L -algebra g on a graded manifold M, and show that such an action corresponds to a homological vector field on g[1]×M of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particular, we consider actions of g on a second L -algebra, leading to a notion of "semidirect product" of L -algebras more general than those we found in the literature.
Lie Algebroid Structures On Double Vector Bundles And Representation Theory Of Lie Algebroids, Alfonso Gracia-Saz, Rajan Amit Mehta
Lie Algebroid Structures On Double Vector Bundles And Representation Theory Of Lie Algebroids, Alfonso Gracia-Saz, Rajan Amit Mehta
Mathematics Sciences: Faculty Publications
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.