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Lie algebroid

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

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};

  • 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

  • A Lie algebra is a Lie algebroid over a point, X=ptX = pt.

  • The tangent Lie algebroid is

    1. in the vector bundle definition given by E=TXE = T X, ρ=Id\rho = \mathrm{Id};

    2. in the Chevalley-Eilenberg algebra definition: CE(TX)=(Ω (X),d deRham)\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 EXE \to X with fiber 𝔤\mathfrak{g} are Lie algebroids with ρ=0\rho = 0 and fiberwise bracket. In particular, for GG a Lie group with Lie algebra 𝔤\mathfrak{g} and PXP \to X a GG-principal bundle, the adjoint bundle adP:=P× G𝔤ad P := P \times_G \mathfrak{g} (where 𝔤\mathfrak{g} is associated using the adjoint representation of GG 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 GG a Lie group and PXP \to X a GG-principal bundle, the vector bundle At(P):=TP/GAt(P):= T P/G naturally inherits the structure of a Lie algebroid. Moreover, it fits into a short exact sequence of Lie algebroids over XX

    0adPAt(P)TX0 0 \to ad P \to At(P) \to T X \to 0

    known as the Atiyah sequence.

  • The vertical tangent Lie algebroid T vertYTYT_{vert}Y \hookrightarrow T Y of a smooth map π:YX\pi : Y \to X of manifolds is the sub-Lie algebroid of the tangent Lie algebroid TYT Y defined as follows:

    1. In the vector bundle perspective E=ker(π *)E = ker(\pi_*) is the kernel bundle of the map π *:TYTX\pi_* : T Y \to T X.

    2. In the dual picture we have CE(T vertY)=Ω vert (Y)CE(T_{vert}Y) = \Omega^\bullet_{vert}(Y), the qDGCA of vertical differential forms. This is the quotient of Ω (Y)\Omega^\bullet(Y) by the ideal of those forms which vanish when restricted in all arguments to ker(π *)ker(\pi_*).

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

  • If EXE\to X is a Lie algebroid with bracket [,][,] and anchor ρ:ETX\rho:E\to TX then it induces a Lie algebroid structure on the kk-th jet bundle j kEXj^k E\to X, called the jet Lie algebroid. More precisely, if sΓ XEs\in\Gamma_X E then call by j ksj^k s the induced section in Γ Xj kE\Gamma_X j^k E. Then there is a unique Lie algebroid structure on the bundle j kEXj^k E\to X such that the following two properties hold: [j ks,j kt]=j k[s,t][j^k s, j^k t] = j^k [s,t] and ρ(j ks)=ρ(s)\rho(j^k s) = \rho(s) for all s,tΓ XEs,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.

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.

References

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”

  • Camille Laurent-Gengoux, Joana Teles, Hom-Lie algebroids, arxiv/1211.2263

Revised on April 21, 2014 06:27:52 by Mike Shulman (108.247.151.163)