nLab Lie-Rinehart pair

Idea

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.

Definition

A Lie–Rinehart-pair $(A,\mathfrak{g})$ is a pair consisting of

1. an associative algebra $A$

2. a Lie algebra $\mathfrak{g}$

such that

1. $A$ is a $\mathfrak{g}$-module

2. $\mathfrak{g}$ is an $A$-module

with both module structures being compatible in the obvious way:

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

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

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

Examples

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

$$\array{ E &&\stackrel{\rho}{\to}&& T X \\ & \searrow && \swarrow \\ && X }$$

equipped with a bracket

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

we obtain a Lie–Rinehart pair by setting

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

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

• the action of $\mathfrak{g}$ on $C^\infty(X)$ 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 $(C^\infty(X), \Gamma(T X))$ with the obvious action on each other.

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

• the Lie–Rinehart pair corresponding to a Poisson Lie algebroid on a Poisson manifold $X$ is $(C^\infty(X), MultVect(X))$, where the Lie algebra is the the space of multivector fields on $X$ equipped with the Schouten bracket.

In open-closed string field theory one finds at least one half of the axioms of homotopy Lie-Rinehart pairs.

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 called Courant-Dorfman algebras there.

Related concepts

• bisections of a Lie groupoid

References

The original reference:

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

On the history of this and related concepts:

• Johannes Huebschmann, On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras [arXiv:2208.02539]

Review:

• Johannes Huebschmann, Sec 1 of: Lie–Rinehart algebras, descent, and quantization (arXiv)

• M. Kapranov, Free Lie algebroids and the space of paths, Sel. Math. (N.S.) 13, n. 2 277–319 (2007), arXiv:math.AG/0702584, doi

• V. Nistor, Alan Weinstein, Ping 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

• Ieke Moerdijk, Janez Mrčun, On the universal enveloping algebra of a Lie–Rinehart algebra, Proc. Amer. Math. Soc. in press, (pdf, arXiv/0801.3929)

A connection with BV-theory and L-infinity algebra is made in

• Lars Kjeseth, Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs, Homology Homotopy Appl. Volume 3, Number 1 (2001), 139-163. (Euclid)

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

Discussion of crossed modules internal to the category of Lie-Rinehart algebras:

• Joel Couchman, Crossed modules and internal categories of Lie-Rinehart algebras, Master’s thesis, Macquarie University (2017) [doi:10.25949/19439645.v1, pdf]