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Lie-Rinehart pair

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

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,𝔤)(A,\mathfrak{g}) is a pair consisting of

  1. an associative algebra AA

  2. a Lie algebra 𝔤\mathfrak{g}

such that

  1. AA is a 𝔤\mathfrak{g}-module

  2. 𝔤\mathfrak{g} is an AA-module

with both module structures being compatible in the obvious way:

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

  2. AA acts as linear transformations of 𝔤\mathfrak{g} in a way obeying the Leibniz rule: that is, we have an associative algebra homomorphism from AEnd(𝔤)A \to End(\mathfrak{g}), where End(𝔤)End(\mathfrak{g}) is the algebra of all linear transformations of 𝔤\mathfrak{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 (X)A = C^\infty(X) is the algebra of smooth functions on a smooth manifold XX, Lie–Rinehart pairs (C (X),𝔤)(C^\infty(X), \mathfrak{g}) are naturally identified with Lie algebroids over XX: given the Lie algebroid in its incarnation as a vector bundle morphism

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

equipped with a bracket

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

we obtain a Lie–Rinehart pair by setting

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

  • the action of AA on 𝔤\mathfrak{g} is the obvious multiplication of sections of vector bundles over XX by functions on XX

  • the action of 𝔤\mathfrak{g} on C (X)C^\infty(X) is given by first applying the anchor map ρ\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

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

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

  • 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, 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

Revised on May 27, 2013 14:24:25 by Urs Schreiber (82.113.98.170)