\loadeufm Linear connections on a vector bundle \(E\) over \(M\) can be viewed as vector bundle morphisms \(\nabla:TM\rightarrow\mathfrak D(E)\), where \(\mathfrak D(E)\) is the Lie algebroid over \(M\) whose sections are derivative endomorphisms of \(\Gamma E\), i.e. \(\mathbb R\)-linear maps \(D:\Gamma E\rightarrow\Gamma E\) for which there exist vector fields \(D_M\) such that \(D(f\psi)=fD(\psi)+D_M(f)\psi\). The connection is flat exactly when \(\nabla\) is actually a Lie algebroid morphism: \(\nabla_{[X,Y]}=[\nabla_X,\nabla_Y]\). In general, a representation of a Lie algebroid \(A\) over \(M\) on the vector bundle \(E\) is a Lie algebroid morphism from \(A\) to \(\mathfrak D(E)\).
It is easy to see that derivative representations of a Lie algebra \(\mathfrak g\) on \(E\), i.e. morphisms of \(\mathfrak g\) into the Lie algebra \(\Gamma\mathfrak D(E)\), which are associated with a given \(\mathfrak g\)-action on \(M\) are in a one-to-one correspondence with representations of the corresponding action Lie algebroid \(M\times\mathfrak g\) on \(E\).
In the paper the concepts of derivative representation of a Lie algebroid \(A\) on a vector bundle \(E\) over a fibered manifold \(F\), \(\tau:F\rightarrow M\), associated with an infinitesimal action of \(A\) on the fibration, are introduced and the above correspondence is generalized to a correspondence between derivative representations of a Lie algebroid \(A\) on \(E\), which are associated with a given infinitesimal action of \(A\) on \(\tau\), and the representations of the corresponding action Lie algebroid on \(E\). Analogous concepts of actions and representations are discussed also on the level of Lie groupoids.
For the entire collection see [Zbl 1007.53002].