Supergroupoids, double structures, and equivariant cohomology

2006/05/14 by Rajan Amit Mehta, Mehta, Rajan Amit · 6 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Topics in Algebra #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.math/0605356

Abstract

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.

Citations

• BRST model applied to symplectic geometry

Cited by

• Graded Contact Geometry and the AKSZ Formalism
• Three Examples of Graded Lie Groups
• Cartan's theorem for some topological generalized groups
• Atiyah class of DG manifolds of positive amplitude
• The Weil algebra and the Van Est isomorphism
• LA-Courant algebroids and their applications

Related

• Q-manifolds and Mackenzie theory: an overview
• Homological vector fields over differentiable stacks
• Mackenzie theory and Q-manifolds
• Double Lie algebroids and the double of a Lie bialgebroid
• Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids