nLab
Lie algebroid

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

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

Definition

A Lie algebroid is the many object version of a Lie algebra. It is the infinitesimal approximation to a Lie groupoid.

There are various equivalent definitions:

In terms of vector bundles with anchor

Definition in terms of vector bundles with anchor map

A Lie algebroid over a manifold XX is

  • a vector bundle EXE \to X;

  • equipped with a Lie brackets [,]:Γ(E)Γ(E)Γ(E)[\cdot,\cdot] : \Gamma(E)\otimes \Gamma(E) \to \Gamma(E) (over the ground field) on its space of sections;

  • a morphisms of vector bundles ρ:ETX\rho : E \to TX, whose tangent map preserves the bracket: (dρ)([ξ,ζ] ΓE)=[dρ(ξ),dρ(ζ)] ΓTX(d\rho)([\xi,\zeta]_{\Gamma E}) = [d\rho(\xi),d\rho(\zeta)]_{\Gamma TX}; (but this property of preserving brackets is implied by the next property, see Y. Kosmann-Schwarzbach and F. Magri. Poisson-Nijenhuis structures. Ann. Inst. H. Poinaré Phys. Théor., 53(1):3581, 1990.)

  • such that the Leibniz rule holds: for all X,YΓ(E)X, Y \in \Gamma(E) and all fC (X)f \in C^\infty(X) we have

    [X,fY]=f[X,Y]+ρ(X)(f)Y. [X, f \cdot Y] = f\cdot [X,Y] + \rho(X)(f) \cdot Y \,.

The CE-algebra of a vector bundle with anchor

Given this data of a vector bundle EXE \to X with anchor map ρ\rho as above, one obtains the structure of a dg-algebra on the exterior algebra C (X) Γ(E) *\wedge^\bullet_{C^\infty(X)} \Gamma(E)^* of smooth sections of the dual bundle by the formula

(dω)(e 0,,e n)= σShuff(1,n)sgn(σ)ρ(e σ(0))(ω(e σ(1),,e σ(n)))+ σShuff(2,n1)sign(σ)ω([e σ(0),e σ(1)],e σ(2),,e σ(n)), (d\omega)(e_0, \cdots, e_n) = \sum_{\sigma \in Shuff(1,n)} sgn(\sigma) \rho(e_{\sigma(0)})(\omega(e_{\sigma(1)}, \cdots, e_{\sigma(n)})) + \sum_{\sigma \in Shuff(2,n-1)} sign(\sigma) \omega([e_{\sigma(0)},e_{\sigma(1)}],e_{\sigma(2)}, \cdots, e_{\sigma(n)} ) \,,

for all ω C (X) nΓ(E) *\omega \in \wedge^n_{C^\infty(X)} \Gamma(E)^* and (e iΓ(E))(e_i \in \Gamma(E)), where Shuff(p,q)Shuff(p,q) denotes the set of (p,q)(p,q)-shuffles σ\sigma and sgn(σ)sgn(\sigma) the signature {±1}\in \{\pm 1\} of the corresponding permutation.

More details on this are at Chevalley-Eilenberg algebra.

Conversely, one finds that every semi-free dga finitely generated in degree 1 over C (X)C^\infty(X) arises this way, so that one may turn this around:

Semi-free dg-algebras

Definition in terms of Chevalley–Eilenberg algebra

A Lie algebroid over a manifold XX is a vector bundle EXE \to X equipped with a degree +1 derivation dd on the free (over C (X)C^\infty(X)) graded-commutative algebra C (X) Γ(E) *\wedge^\bullet_{C^\infty(X)} \Gamma(E)^* (where the dual is over C C^\infty), such that d 2=0d^2 = 0.

This is for Γ(E)\Gamma(E) satisfying suitable finiteness conditions. In general, as the masters well knew, the correct definition is the algebra of alternating multilinear functions from Γ(E)\Gamma(E) to the ground field, assumed of characteristic 0. This can also be phrased in terms of linear maps from the corresponding coalgebra cogenerated by Γ(E)\Gamma(E), but the masters did not have coalgebras in those days.

The differential graded-commutative algebra

CE(𝔤):=( C (X) Γ(E) *,d) CE(\mathfrak{g}) := (\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*, d)

is the Chevalley-Eilenberg algebra of the Lie algebroid (in that for X=ptX = pt it reduces to the ordinary Chevally–Eilenberg algebra for Lie algebras).

In the existing literature this is often addressed just as “the complex that computes Lie algebroid cohomology”.

It is helpful to compare this definition to the general definition of Lie ∞-algebroids, the vertical categorification of Lie algebras and Lie algebroids.

Lie-Rinehart algebras

Definition in terms of commutative Lie–Rinehart pairs

A Lie algebroid over the manifold XX is

  • a Lie algebra 𝔤\mathfrak{g};

  • the structure of a Lie module over 𝔤\mathfrak{g} on C (X)C^\infty(X) (i.e. an action of 𝔤\mathfrak{g} on XX);

  • the structure of a C (X)C^\infty(X)-module on 𝔤\mathfrak{g} (in fact: such that 𝔤\mathfrak{g} is a finitely generated projective module);

  • such that the two actions satisfy two compatibility conditions which are modeled on the standard relations obtained by setting 𝔤=Γ(TX)\mathfrak{g} = \Gamma(T X).

This is the special case of a Lie-Rinehart pair (A,𝔤)(A,\mathfrak{g}) where the associative algebra AA is of the form C (X)C^\infty(X).

Examples

Properties

Lie theory

The extent to which Lie algebroids are to Lie groupoids as Lie algebras are to Lie groups is the content of general Lie theory, in which Lie's theorems have been generalized to Lie algebroids.

Poisson geometry

The fiberwiese linear dual of a Lie algebroid (regarded as a vector bundle) is naturally a Poisson manifold: the Lie-Poisson structure.

algebraic structureoidification
truth valuetransitive relation
magmamagmoid
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
(left,right) cancellative monoid(left,right) cancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
differential ring?differential ringoid?
nonassociative algebralinear magmoid
nonassociative unital algebraunital linear magmoid
nonunital algebralinear semicategory
associative unital algebralinear category
C-star algebraC-star category
differential algebradifferential algebroid
flexible algebraflexible linear magmoid
alternative algebraalternative linear magmoid
Lie algebraLie algebroid
strict monoidal categorystrict 2-category
strict 2-groupstrict 2-groupoid
monoidal poset?2-poset
monoidal groupoid?(2,1)-category
monoidal category2-category/bicategory
2-group2-groupoid/bigroupoid

References

The concept of Lie algebroid was introduced in

In algebra a generalization of Lie algebroid, the Lie pseduoalgebra or Lie-Rinehart algebra/pair has been introduced more than a dozen of times under various names starting in early 1950-s. Atiyah’s construction of Atiyah sequence is published in 1957 and Rinehart’s paper in 1963.

Historically important is also the reference

on tangent Lie algebroids.

A bijective correspondence between Lie algebroid structures, homological vector fields of degree 1, and odd linear Poisson structures is established in the paper

Contemporary textbooks include:

For an infinite-dimensional version used in stochastic analysis see

There is also a recent “hom-version”