nLab
Lie algebroid

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 $X$ is

• a vector bundle $E\to X$;

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

• a morphisms of vector bundles $\rho :E\to \mathrm{TX}$;

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

  $$[X,f\cdot Y]=f\cdot [X,Y]+\rho (X)(f)\cdot Y.$$[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 $E\to X$ with anchor map $\rho$ as above, one obtains the structure of a dg-algebra on the exterior algebra ${\wedge }_{C^{\mathrm{\infty }}(X)}^{•}\Gamma (E{)}^{*}$ of smooth sections of the dual bundle by generalizing the familiar formula for the deRham differential:

for $\omega \in {\wedge }^{n-1}\Gamma (E{)}^{*}$ for all $v_i\in \Gamma (E)$ the differential is given by

$$(d\omega )(v_1,\cdots ,v_n)=\mathrm{const}\sum _{\sigma }\pm \rho (v_{{\sigma }_1})(\omega (v_{{\sigma }_2},\cdots ,v_{{\sigma }_n}))\pm \mathrm{const}\sum _{\sigma }\omega ([v_{{\sigma }_1},v_{{\sigma }_2}],v_{{\sigma }_3},\cdots ,v_{{\sigma }_n}),$$(d \omega)(v_1, \cdots,v_n) = const \sum_{\sigma} \pm \rho(v_{\sigma_1})(\omega(v_{\sigma_2}, \cdots, v_{\sigma_n})) \pm const \sum_{\sigma} \omega([v_{\sigma_1}, v_{\sigma_2}], v_{\sigma_3}, \cdots, v_{\sigma_n}) \,,

where the sums are over all permutations $\sigma$ of $\{1,\cdots ,n\}$.

Conversely, one finds that every semi-free dga finitely generated in degree 1 over $C^{\mathrm{\infty }}(X)$ 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 $X$ is a vector bundle $E\to X$ equipped with a degree +1 derivation $d$ on the free (over $C^{\mathrm{\infty }}(X)$) graded-commutative algebra ${\wedge }_{C^{\mathrm{\infty }}(X)}^{•}\Gamma (E{)}^{*}$ (where the dual is over $C^{\mathrm{\infty }}$), such that $d^2=0$.

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

The differential graded-commutative algebra

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

is the Chevalley–Eilenberg algebra of the Lie algebroid (in that for $X=\mathrm{pt}$ 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 $X$ is

• a Lie algebra $𝔤$;

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

• the structure of a $C^{\mathrm{\infty }}(X)$-module on $𝔤$ (in fact: such that $𝔤$ 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 $𝔤=\Gamma (TX)$.

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

Examples

• A Lie algebra is a Lie algebroid over a point, $X=\mathrm{pt}$.

• The tangent Lie algebroid is

  1. in the vector bundle definition given by $E=TX$, $\rho =\mathrm{Id}$;

  2. in the Chevalley–Eilenberg algebra definition: $\mathrm{CE}(TX)=({\Omega }^{•}(X),d_{\mathrm{deRham}})$;

• Bundles of Lie algebras $E\to X$ with fiber $𝔤$ are Lie algebroids with $\rho =0$ and fiberwise bracket. In particular, for $G$ a Lie group with Lie algebra $𝔤$ and $P\to X$ a $G$-principal bundle, the adjoint bundle $\mathrm{ad}P:=P{\times }_G𝔤$ (where $𝔤$ is associated using the adjoint representation? of $G$ on its Lie algebra) is a bundle of Lie algebras.

• The Atiyah Lie algebroid: for $G$ a Lie group and $P\to X$ a $G$-principal bundle, the vector bundle $\mathrm{At}(P):=TP/G$ naturally inherits the structure of a Lie algebroid. Moreover, it fits into a short exact sequence of Lie algebroids over $X$

  $$0\to \mathrm{ad}P\to \mathrm{At}(P)\to TX\to 0$$0 \to ad P \to At(P) \to T X \to 0

  known as the Atiyah sequence. For some $n$-Café blog discussion of this see n-Transport and Higher Schreier theory.

• The vertical tangent Lie algebroid $T_{\mathrm{vert}}Y\hookrightarrow TY$ of a smooth map $\pi :Y\to X$ of manifolds is the sub-Lie algebroid of the tangent Lie algebroid $TY$ defined as follows:

  1. In the vector bundle perspective $E=\ker ({\pi }_{*})$ is the kernel bundle of the map ${\pi }_{*}:TY\to TX$.

  2. In the dual picture we have $\mathrm{CE}(T_{\mathrm{vert}}Y)={\Omega }_{\mathrm{vert}}^{•}(Y)$, the qDGCA of vertical differential forms. This is the quotient of ${\Omega }^{•}(Y)$ by the ideal of those forms which vanish when restricted in all arguments to $\ker ({\pi }_{*})$.

• Each Poisson manifold $(X,\pi )$ defines and is defined by a Poisson Lie algebroid $T^{*}X\stackrel{\pi }{\to }tX$. This is the degree-1 example of a more general structure described at n-symplectic manifold.

Remarks

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.

Literature

One of the earliest reference seems to be

• Ted Courant?, Tangent Lie algebroid (pdf)