# deformations of Lie algebroids

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 $$f\in C^{\infty}(M)$$.

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