∞-Lie theory (higher geometry)
Formal Lie groupoids
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 is
a vector bundle ;
equipped with a Lie brackets (over the ground field) on its space of sections;
a morphisms of vector bundles , whose tangent map preserves the bracket: ; (but this property of preserving brackets is implied by the next property, see Y. Kosmann-Shwarzbah and F. Magri. Poisson-Nijenhuis strutures. Ann. Inst. H. Poinaré Phys. Théor., 53(1):3581, 1990.)
such that the Leibniz rule holds: for all and all we have
The CE-algebra of a vector bundle with anchor
Given this data of a vector bundle with anchor map as above, one obtains the structure of a dg-algebra on the exterior algebra of smooth sections of the dual bundle by the formula
for all and , where denotes the set of -shuffles and the signature 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 arises this way, so that one may turn this around:
Definition in terms of Chevalley–Eilenberg algebra
A Lie algebroid over a manifold is a vector bundle equipped with a degree +1 derivation on the free (over ) graded-commutative algebra (where the dual is over ), such that .
This is for satisfying suitable finiteness conditions. In general, as the masters well knew, the correct definition is the algebra of alternating multilinear functions from to the ground field, assumed of characteristic 0. This can also be phrased in terms of linear maps from the corresponding coalgebra cogenerated by , 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 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.
Definition in terms of commutative Lie–Rinehart pairs
A Lie algebroid over the manifold is
a Lie algebra ;
the structure of a Lie module over on (i.e. an action of on );
the structure of a -module on (in fact: such that 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 .
This is the special case of a Lie-Rinehart pair where the associative algebra is of the form .
A Lie algebra is a Lie algebroid over a point, .
The tangent Lie algebroid is
in the vector bundle definition given by , ;
in the Chevalley-Eilenberg algebra definition: ;
An action Lie algebroid is the Lie version of an action groupoid.
Bundles of Lie algebras with fiber are Lie algebroids with and fiberwise bracket. In particular, for a Lie group with Lie algebra and a -principal bundle, the adjoint bundle (where is associated using the adjoint representation of 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 a Lie group and a -principal bundle, the vector bundle naturally inherits the structure of a Lie algebroid. Moreover, it fits into a short exact sequence of Lie algebroids over
known as the Atiyah sequence.
The vertical tangent Lie algebroid of a smooth map of manifolds is the sub-Lie algebroid of the tangent Lie algebroid defined as follows:
In the vector bundle perspective is the kernel bundle of the map .
In the dual picture we have , the qDGCA of vertical differential forms. This is the quotient of by the ideal of those forms which vanish when restricted in all arguments to .
Each Poisson manifold defines and is defined by a Poisson Lie algebroid . This is the degree-1 example of a more general structure described at n-symplectic manifold.
If is a Lie algebroid with bracket and anchor then it induces a Lie algebroid structure on the -th jet bundle , called the jet Lie algebroid. More precisely, if then call by the induced section in . Then there is a unique Lie algebroid structure on the bundle such that the following two properties hold: and for all (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.
The concept of Lie algebroid was introduced in
- Jean Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B 264 1967 A245–A248, MR0216409
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.
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
- Rémi Léandre, A Lie algebroid on the Wiener space, Adv. Math. Phys. 2010, Art. ID 146719, 17 pp. MR2011j:58064
There is also a recent “hom-version”