nLab
Lie-Rinehart pair

Idea

A Lie–Rinehart pair is an algebraic structure that generalizes the notion of Lie algebroid from the case where the base is a manifold to that where it is an arbitrary noncommutative space.

Definition

A Lie–Rinehart-pair $(A,𝔤)$ is a pair consisting of an associative algebra $A$ and a Lie algebra $𝔤$ such that $A$ is a $𝔤$-module and $𝔤$ is an $A$-module with both module structures being compatible in the obvious way:

• $𝔤$ acts as derivations of $A$: that is, we have a Lie algebra homomorphism $𝔤\to \mathrm{Der}(A)$.

• $A$ acts as linear transformations of $𝔤$ in a way obeying the Leibniz law: that is, we have an associative algebra homomorphism from $A\to \mathrm{End}(𝔤)$, where $\mathrm{End}(𝔤)$ is the algebra of all linear transformations of $𝔤$, such that

  $$[v,aw]=v(a)w+a[v,w].$$[v, a w] = v(a) w + a [v,w].

Examples

In the case that $A=C^{\mathrm{\infty }}(X)$ is the algebra of smooth functions on a smooth manifold $X$, Lie–Rinehart pairs $(C^{\mathrm{\infty }}(X),𝔤)$ are naturally identified with Lie algebroids over $X$: given the Lie algebroid in its incarnation as a vector bundle morphism

$$\begin{array}{ccccc}E& & \stackrel{\rho }{\to }& & TX\\ & \searrow & & \swarrow \\ & & X\end{array}$$\array{ E &&\stackrel{\rho}{\to}&& T X \\ & \searrow && \swarrow \\ && X }

equipped with a bracket

$$[-,-]:\Gamma (E)\otimes \Gamma (E)\to \Gamma (E)$$[-,-] : \Gamma(E) \otimes\Gamma(E) \to \Gamma(E)

we obtain a Lie–Rinehart pair by setting

• $𝔤=\Gamma (E)$ is the Lie algebra of sections of $E$ using the above bracket

• the action of $A$ on $𝔤$ is the obvious multiplication of sections of vector bundles over $X$ by functions on $X$

• the action of $𝔤$ on $C^{\mathrm{\infty }}(X)$ is given by first applying the anchor map $\rho$ 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.

In particular

• the Lie–Rinehart pair coresponding to the tangent Lie algebroid of a manifold $X$ is $(C^{\mathrm{\infty }}(X),\Gamma (TX))$ with the obvious action on each other.

• the Lie–Rinehart pair corresponding to an ordinary Lie algebra $𝔤$ is $(ℝ,𝔤)$ with $𝔤$ acting trivially on $ℝ$.

• the Lie–Rinehart pair corresponding to a Poisson Lie algebroid on a Poisson manifold $X$ is $(C^{\mathrm{\infty }}(X),\mathrm{MultVect}(X))$, where the Lie algebra is the the space of multi vector fields on $X$ equipped with the bracket … (exercise for the reader) …

Generalizations

A little bit is known in the literature to generalizations of the notion of Lie–Rinehart algebras that are to Lie ∞-algebroids as the latter are to Lie algebroids.

In

• Dmitry Roytenberg, Courant–Dorfman algebras and their cohomology (arXiv)

the analogous algebraic structure for Courant algebroids is discussed. These “2-Lie–Rinehart algebras” are ccalled Courant–Dorfman algebras there.

References

The original reference is

• G. Rinehart Differential forms for general commutative algebras Trans. Amer. Math. Soc. 108 (1963), 195-222

A brief review in section 1 of

• Johannes Huebschmann, Lie–Rinehart algebras, descent, and quantization (arXiv)

A notion of universal enveloping algebra of a Lie–Rinehart algebra is discussed in

• Ieke Moerdijk, J. Mrcun, On the universal enveloping algebra of a Lie–Rinehart algebra (arXiv)

A connection with BV-theory is made in

• Johannes Huebschmann, Lie–Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras (journal)