Supergroupoids, double structures, and equivariant cohomology

By R. Mehta. 2006

97 citations recorded by the source corpus.

Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double structures of Mackenzie; in particular, we show that Q-groupoids are intermediary objects between Mackenzie's LA-groupoids and double complexes, which include as a special case the simplicial model of equivariant cohomology. There is also a double complex associated to a Q-algebroid, which in the above special case is the BRST model of equivariant cohomology. Other special cases include models for the Drinfel'd double of a Lie bialgebra and Ginzburg's equivariant Poisson cohomology. Finally, a supergroupoid version of the van Est map is used to give a homomorphism from the double complex of a Q-groupoid to that of a Q-algebroid.

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

View Paper: https://www.semanticscholar.org/paper/c314230cbc961cce1889cd3d8db01e6e18cb8223

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Citations in the corpus

Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids (2006, unknown): “Independently, Mehta [43] has given a super formulation of the related notion of LA-groupoid (see §7), and Gracia-Saz and Mehta [12] have given a super formulation of LA-vector bundles, which they call VB-algebroids.”
Q-algebroids and their cohomology (2007, background): “We will follow the notation of [26] (also see [27]) regarding supermanifolds.”
Lie algebroid structures on double vector bundles and representation theory of Lie algebroids (2008, background): “gent to the zero section of E→ M, is naturally isomorphic to E. The linear sections of TEover E are the linear vector ﬁelds, and the core sections are the ﬁbrewise-constant vertical vector ﬁelds. 1In [14], the term vertical was used instead of core. However, it is now apparent that the present termi”
The Weil algebra and the Van Est isomorphism (2009, background): “All these have been carried out in the PhD thesis of R. Mehta [ 17 ].”
VB-groupoids and representation theory of Lie groupoids (2010, methodology): “The relationship between the modular class of a Lie groupoid and that of its associated Lie algebroid is given by the van Est map [2, 10, 13].”