∞-Lie theory (higher geometry)
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:
A Lie algebroid over a manifold $X$ is
a vector bundle $E \to X$;
equipped with a Lie brackets $[\cdot,\cdot] : \Gamma(E)\otimes \Gamma(E) \to \Gamma(E)$ (over the ground field) on its space of sections;
a morphisms of vector bundles $\rho : E \to TX$, whose tangent map preserves the bracket: $(d\rho)([\xi,\zeta]_{\Gamma E}) = [d\rho(\xi),d\rho(\zeta)]_{\Gamma TX}$; (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 $X, Y \in \Gamma(E)$ and all $f \in C^\infty(X)$ we have
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 $\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*$ of smooth sections of the dual bundle by the formula
for all $\omega \in \wedge^n_{C^\infty(X)} \Gamma(E)^*$ and $(e_i \in \Gamma(E))$, where $Shuff(p,q)$ denotes the set of $(p,q)$-shuffles $\sigma$ and $sgn(\sigma)$ the signature $\in \{\pm 1\}$ 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 $C^\infty(X)$ arises this way, so that one may turn this around:
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 $C^\infty(X)$) graded-commutative algebra $\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*$ (where the dual is over $C^\infty$), 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
is the Chevalley-Eilenberg algebra of the Lie algebroid (in that for $X = pt$ 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.
A Lie algebroid over the manifold $X$ is
a Lie algebra $\mathfrak{g}$;
the structure of a Lie module over $\mathfrak{g}$ on $C^\infty(X)$ (i.e. an action of $\mathfrak{g}$ on $X$);
the structure of a $C^\infty(X)$-module on $\mathfrak{g}$ (in fact: such that $\mathfrak{g}$ 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 $\mathfrak{g} = \Gamma(T X)$.
This is the special case of a Lie-Rinehart pair $(A,\mathfrak{g})$ where the associative algebra $A$ is of the form $C^\infty(X)$.
A Lie algebra is a Lie algebroid over a point, $X = pt$.
The tangent Lie algebroid is
in the vector bundle definition given by $E = T X$, $\rho = \mathrm{Id}$;
in the Chevalley-Eilenberg algebra definition: $\mathrm{CE}(T X) = (\Omega^\bullet(X), d_{deRham})$;
An action Lie algebroid is the Lie version of an action groupoid.
Bundles of Lie algebras $E \to X$ with fiber $\mathfrak{g}$ are Lie algebroids with $\rho = 0$ and fiberwise bracket. In particular, for $G$ a Lie group with Lie algebra $\mathfrak{g}$ and $P \to X$ a $G$-principal bundle, the adjoint bundle $ad P := P \times_G \mathfrak{g}$ (where $\mathfrak{g}$ is associated using the adjoint representation of $G$ on its Lie algebra) is a bundle of Lie algebras.
Lie algebroids with injective anchor maps are equivalently integrable distributions in the tangent bundle of their base manifold and hence are equivalently foliations of their base manifold.
The Atiyah Lie algebroid is the Lie algebroid of the Atiyah Lie groupoid of a principal bundle: for $G$ a Lie group and $P \to X$ a $G$-principal bundle, the vector bundle $At(P):= T P/G$ naturally inherits the structure of a Lie algebroid. Moreover, it fits into a short exact sequence of Lie algebroids over $X$
known as the Atiyah sequence.
The vertical tangent Lie algebroid $T_{vert}Y \hookrightarrow T Y$ of a smooth map $\pi : Y \to X$ of manifolds is the sub-Lie algebroid of the tangent Lie algebroid $T Y$ defined as follows:
In the vector bundle perspective $E = ker(\pi_*)$ is the kernel bundle of the map $\pi_* : T Y \to T X$.
In the dual picture we have $CE(T_{vert}Y) = \Omega^\bullet_{vert}(Y)$, the qDGCA of vertical differential forms. This is the quotient of $\Omega^\bullet(Y)$ 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 $T^* X \stackrel{\pi}{\to} t X$. This is the degree-1 example of a more general structure described at n-symplectic manifold.
If $E\to X$ is a Lie algebroid with bracket $[,]$ and anchor $\rho:E\to TX$ then it induces a Lie algebroid structure on the $k$-th jet bundle $j^k E\to X$, called the jet Lie algebroid. More precisely, if $s\in\Gamma_X E$ then call by $j^k s$ the induced section in $\Gamma_X j^k E$. Then there is a unique Lie algebroid structure on the bundle $j^k E\to X$ such that the following two properties hold: $[j^k s, j^k t] = j^k [s,t]$ and $\rho(j^k s) = \rho(s)$ for all $s,t\in\Gamma_X E$ (see pdf).
A BRST complex is a Chevalley-Eilenberg algebra of a Lie algebroid which corresponds to the action groupoid of a Lie group acting on a space.
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.
The fiberwiese linear dual of a Lie algebroid (regarded as a vector bundle) is naturally a Poisson manifold: the Lie-Poisson structure.
Lie algebra, Lie algebroid
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
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:
K. C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids, Cambridge University Press, 2005, xxxviii + 501 pages (website)
K. C. H. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp (MathSciNet)
Janez Mrčun, Ieke Moerdijk, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press 2003. x+173 pp. ISBN: 0-521-83197-0
For an infinite-dimensional version used in stochastic analysis see
There is also a recent “hom-version”