In the paper "Deformations of Lie brackets"- by I. Moerdijk and M. Crainic, they define deformations of a Lie algebroid as follows:

Let $A$ be a fixed vector bundle, and $I\subset \mathbb{R}$ and interval.

  1. A family of Lie algebroids over $I$ is a collection $(A_t)_{t \in I}$ of Lie algebroids $A_t= (A, [.,.]_t, \rho_t)$ varying smoothly with respect to $t$. (Here $\rho_t$ denotes the anchor map.)\
  2. A deformation of a Lie algebroid $(A, [.,], \rho)$ is a family $(A_t)_{t \in I}$ of Lie algebroids over an interval containing the origin with $A_0=A$.

My question is: what is meant by "varying smoothly with respect to $t$" in this context?

A Lie algebroid over a manifold $M$ is

  • a vector bundle $\pi:E\rightarrow M $

  • a Lie bracket $[\cdot,\cdot]:\Gamma(E)\times \Gamma(E)\rightarrow \Gamma(E)$

  • a map $\rho:E\rightarrow TM$ (called the anchor map)

such that $$[X,fY]=f[X,Y]+\rho(X)(f).Y$$ for all $X,Y\in\Gamma(E)$ and $f\in C^{\infty}(M)$.

  • $\begingroup$ I am trying to understand your question... You want to define what it means by a family of Lie algebroids over an interval.. Right? $\endgroup$ – Praphulla Koushik Dec 26 '18 at 9:36

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