Deformations of Lie brackets and representations up to homotopy

Abstract

We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets.

Van Est isomorphism for homogeneous cochains

“…In [16,17], it was shown that VB-groupoids and VB-algebroids provide an intrinsic version of the notion of (2-term) representation up to homotopy, generalizing the example given in the introduction, as well as Examples 2.5 and 2.9 above. In this section, we show how Theorem 2.14, when applied to 1-homogeneous cochains, recovers a Van Est result for the underlying 2-term representations up to homotopy of [4]. We also comment on how this approach realizes the original porposal in [11] for proving a rigidity conjecture.…”

“…Remark 3. 4. The conjecture was originally proved in [4] using a Van Est result for representations up to homotopy.…”

“…where φ u ǫ : M → G is the flow of the right-invariant vector field − → u and the definition R u µ E is analogous. Note that our conventions are different from [4]. One can now check the identities:…”

“…Later, we shall show how VB-groupoids and algebroids provide a useful framework to interpret many of the definitions and to give a proof of Theorem 4. 4.…”

Integration of $2$-term representations up to homotopy via $2$-functors

“…The second item deserves more attention. Note first that, as shown in [3], any 2-term representation up to homotopy of G(A) differentiates to a 2-term representation up to homotopy of A. In this case, one can see that G(A) ⋉ E integrates A ⋉ E, both as a groupoid and as a VB-groupoid [10].…”