The notion of Lie–Rinehart pair is an algebraic encoding of the notion of Lie algebroid. It is the pair consisting of the associative algebra of functions on the base space of the Lie algebroid and of the Lie algebra of its global sections. The anchor map of the Lie algebroid is encoded in the action of the Lie algebra on the associative algebra by derivations and the local structure is encoded in the Lie algebra being a module over the associative algebra.
Since in this formulation the base manifold of the Lie algebroid is entirely described dually in terms of its algebra of functions, and since the definition does not refer to this being a commutative algebra, the notion of Lie-Rinehart pair in fact generalizes the notion of Lie algebroid from ordinary differential geometry to noncommutative geometry.
A Lie–Rinehart-pair is a pair consisting of
with both module structures being compatible in the obvious way:
acts as linear transformations of in a way obeying the Leibniz rule: that is, we have an associative algebra homomorphism from , where is the algebra of all linear transformations of , such that
In the case that is the algebra of smooth functions on a smooth manifold , Lie–Rinehart pairs are naturally identified with Lie algebroids over : given the Lie algebroid in its incarnation as a vector bundle morphism
equipped with a bracket
we obtain a Lie–Rinehart pair by setting
the action of on is the obvious multiplication of sections of vector bundles over by functions on
the action of on is given by first applying the anchor map and then using the canonical action of vector fields on functions.
So for all the examples listed at Lie algebroid we obtain an example for Lie–Rinehart pairs.
the Lie–Rinehart pair corresponding to an ordinary Lie algebra is with acting trivially on .
In open-closed string field theory one finds at least one half of the axioms of homotopy Lie-Rinehart pairs.
The original reference is
A brief review in section 1 of
V. Nistor, A. Weinstein, P. Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189, 117–152 (1999)
A notion of universal enveloping algebra of a Lie–Rinehart algebra is discussed in
A connection with BV-theory is made in