lie algebras - What is a Module over a Lie algebroid?

A $mathfrak{g}_A$-module $M$ is a $k$-module endowed with structures of $A$-module and $mathfrak{g}_A$-module satisfying the compatibility equations $(ax)m = a(xm)$ and $x(am) = x(a)m + a(xm)$ for any $ain A$, $xinmathfrak{g}_A$, and $min M$. Here $x(a)$ denotes the action of $mathfrak{g}_A$ in $A$, while the three other actions are denoted by $ax$, $am$, and $xm$.

A $mathfrak{g}_A$-module is the same that a module over the enveloping algebra $U_A(mathfrak{g}_A)$.