Representations up to homotopy of Lie algebroids

By C. A. Abad, M. Crainic. 2009

148 citations recorded by the source corpus.

Abstract We introduce and study the notion of representation up to homotopy of a Lie algebroid, paying special attention to examples. We use representations up to homotopy to define the adjoint representation of a Lie algebroid and show that the resulting cohomology controls the deformations of the structure. The Weil algebra of a Lie algebroid is defined and shown to coincide with Kalkman's BRST model for equivariant cohomology in the case of group actions. The relation of this algebra with the integration of Poisson and Dirac structures is explained in [3].

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

View Paper: https://www.semanticscholar.org/paper/5b0e7f63adc54b2a07b7ebfa013e2f84b4326a9f

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Lie algebroid structures on double vector bundles and representation theory of Lie algebroids (2008, methodology): “lgebroids and equivalence classes of 2-term ﬂat Lie algebroid superconnections. Given a Lie algebroidA→ M, a decomposition of the tangent prolongationVB–algebroid TAyields a ﬂat A-superconnection on A[1]⊕TM, the diagonal components of which form a basic connection in the sense of Fernandes [6]. In f”
Vector bundles over Lie groupoids and algebroids (2014, background): “f semi-direct products, and analogously for Lie algebroids. More generally, it is shown in [13, 14] that VB-groupoids and VB-algebroids provide an intrinsic approach to representations up to homotopy [1, 2], a “higher” notion of representation needed to make sense e.g. of the adjoint representation ”
Multiplicative forms and Spencer operators (2012, unknown): “in [2, 6]): R ∇ (α, β)(X) := ∇X([α, β])−[∇X (α), β]−[α,∇X (β)]−∇∇bas β X(α)+∇∇bas α X(β), where α, β are sections of A and X,Y are vector fields on M and ∇ α (X) = ρ(∇X(α)) + [ρ(α), X ].”
The Weil algebra and the Van Est isomorphism (2009, methodology): “However, the associated operators d∇, d v ∇ acting on W∇(A) are more involved and are computed in [2].”
Representations up to homotopy and Bott's spectral sequence for Lie groupoids (2009, methodology): “ntation up to homotopy); up to isomorphism, the adjoint representation is be independent of this choice. The discussion is completely parallel to the one from the inﬁnitesimal case (Subsection 2.2 of [1]). However, the presentation here is self-contained. One reason comes from the fact that Ehresman”