Lie algebroid
Lie algebroids were first introduced and studied by J. Pradines [a11], following work by Ch. Ehresmann and P. Libermann on differentiable groupoids (later called Lie groupoids). Just as Lie algebras are the infinitesimal objects of Lie groups, Lie algebroids are the infinitesimal objects of Lie groupoids (cf. also Lie group). They are generalizations of both Lie algebras and tangent vector bundles (cf. also Lie algebra; Vector bundle; Tangent bundle). For a comprehensive treatment and lists of references, see [a8], [a9]. See also [a1], [a4], [a6], [a13], [a14].
A real Lie algebroid is a smooth real vector bundle
over a base
, with a real Lie algebra structure
on the vector space
of smooth global sections of
, and a morphism of vector bundles
, where
is the tangent bundle of
, called the anchor, such that
, for all
and
;
defines a Lie algebra homomorphism from the Lie algebra of sections of
, with Lie bracket
, into the Lie algebra of vector fields on
. Complex Lie algebroid structures [a1] on complex vector bundles over real bases can be defined similarly, replacing the tangent bundle of the base by the complexified tangent bundle.
The space of sections of a Lie algebroid is a Lie–Rinehart algebra, also called a Lie -ring or a Lie pseudo-algebra. (See [a4], [a6], [a9].) More precisely, it is a
-Lie algebra, where
is the field of real (or complex) numbers and
is the algebra of functions on the base manifold. In fact, the Lie–Rinehart algebras are the algebraic counterparts of the Lie algebroids, just as the modules over a ring are the algebraic counterparts of the vector bundles.
Examples.
1) A Lie algebroid over a one-point set, with the zero anchor, is a Lie algebra.
2) The tangent bundle of a manifold
, with as bracket the Lie bracket of vector fields and with as anchor the identity of
, is a Lie algebroid over
. Any integrable sub-bundle of
, in particular the tangent bundle along the leaves of a foliation, is also a Lie algebroid.
3) A vector bundle with a smoothly varying Lie algebra structure on the fibres (in particular, a Lie-algebra bundle [a8]) is a Lie algebroid, with pointwise bracket of sections and zero anchor.
4) If is a Poisson manifold, then the cotangent bundle
of
is, in a natural way, a Lie algebroid over
. The anchor is the mapping
defined by the Poisson bivector
. The Lie bracket
of differential
-forms satisfies
, for any functions
, where
is the Poisson bracket (cf. Poisson brackets) of functions, defined by
. When
is non-degenerate,
is a symplectic manifold (cf. also Symplectic structure) and this Lie algebra structure of
is isomorphic to that of
. For references to the early occurrences of this bracket, which seems to have first appeared in [a3], see [a4], [a6] and [a13]. It was shown in [a2] that
is a Lie algebroid bracket on
.
5) The Lie algebroid of a Lie groupoid , where
is the source mapping and
is the target mapping [a11], [a8], [a13]. It is defined as the normal bundle along the base of the groupoid, whose sections can be identified with the right-invariant,
-vertical vector fields. The bracket is induced by the Lie bracket of vector fields on the groupoid, and the anchor is
.
6) The Atiyah sequence. If is a principal bundle with structure group
, base
and projection
, the
-invariant vector fields on
are the sections of a vector bundle with base
, denoted by
, and sometimes called the Atiyah bundle of the principal bundle
. This vector bundle is a Lie algebroid, with bracket induced by the Lie bracket of vector fields on
, and with surjective anchor induced by
. The kernel of the anchor is the adjoint bundle,
. Splittings of the anchor are connections on
(cf. also Connection). The Atiyah bundle of
is the Lie algebroid of the Ehresmann gauge groupoid
. If
is the frame bundle of a vector bundle
, then the sections of the Atiyah bundle of
are the covariant differential operators on
, in the sense of [a8].
7) Other examples are: the trivial Lie algebroids ; the transformation Lie algebroids
, where the Lie algebra
acts on the manifold
; the deformation Lie algebroid
of a Lie algebroid
, where
, for
, is isomorphic to
, and
is isomorphic to the vector bundle
with the Abelian Lie algebroid structure (zero bracket and zero anchor); the prolongation Lie algebroids of a Lie algebroid, etc.
de Rham differential.
Given any Lie algebroid , a differential
is defined on the graded algebra of sections of the exterior algebra of the dual vector bundle,
, called the de Rham differential of
. Then
can be considered as the algebra of functions on a super-manifold,
being an odd vector field with square zero [a12].
If is a Lie algebra
, then
is the Chevalley–Eilenberg cohomology operator on
.
If , then
is the usual de Rham differential on forms.
If is the cotangent bundle of a Poisson manifold, then
is the Lichnerowicz–Poisson differential
on fields of multi-vectors on
.
Schouten algebra.
Given any Lie algebroid , there is a Gerstenhaber algebra structure (see Poisson algebra), denoted by
, on the graded algebra of sections of the exterior algebra of the vector bundle
,
. With this graded Lie bracket,
is called the Schouten algebra of
.
If is a Lie algebra
, then
is the algebraic Schouten bracket on
.
If , then
is the usual Schouten bracket of fields of multi-vectors on
.
If is the cotangent bundle of a Poisson manifold, then
is the Koszul bracket [a7], [a13], [a5] of differential forms.
Morphisms of Lie algebroids and the linear Poisson structure on the dual.
A base-preserving morphism from a Lie algebroid to a Lie algebroid
, over the same base
, is a base-preserving vector-bundle morphism,
, such that
, inducing a Lie-algebra morphism from
to
.
If is a Lie algebroid, the dual vector bundle
is a Poisson vector bundle. This means that the total space of
has a Poisson structure such that the Poisson brackets of two functions which are linear on the fibres is linear on the fibres. A base-preserving morphism from a vector bundle
to a vector bundle
is a morphism of Lie algebroids if and only if its transpose is a Poisson morphism.
Lie bi-algebroids.
These are pairs of Lie algebroids in duality satisfying the compatibility condition that
be a derivation of the graded Lie bracket
[a10], [a5]. They generalize the Lie bi-algebras in the sense of V.G. Drinfel'd (see Quantum groups and Poisson Lie group) and also the pair
, where
is a Poisson manifold.
There is no analogue to Lie's third theorem (cf. also Lie theorem) in the case of Lie algebroids, since not every Lie algebroid can be integrated to a global Lie groupoid, although there are local versions of this result. (See [a8], [a1].)
References
[a1] | A. Cannas da Silva, A. Weinstein, "Geometric models for noncommutative algebras" , Berkeley Math. Lecture Notes , 10 , Amer. Math. Soc. (1999) Zbl 1135.58300 |
[a2] | A. Coste, P. Dazord, A. Weinstein, "Groupoïdes symplectiques" Publ. Dép. Math. Univ. Claude Bernard, Lyon I , 2A (1987) pp. 1–62 MR0996653 Zbl 0668.58017 |
[a3] | B. Fuchssteiner, "The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems" Prog. Theor. Phys. , 68 (1982) pp. 1082–1104 MR0688120 Zbl 1098.37540 |
[a4] | J. Huebschmann, "Poisson cohomology and quantization" J. Reine Angew. Math. , 408 (1990) pp. 57–113 MR1058984 Zbl 0699.53037 |
[a5] | Y. Kosmann-Schwarzbach, "Exact Gerstenhaber algebras and Lie bialgebroids" Acta Applic. Math. , 41 (1995) pp. 153–165 Zbl 0837.17014 |
[a6] | Y. Kosmann-Schwarzbach, F. Magri, "Poisson–Nijenhuis structures" Ann. Inst. H. Poincaré Phys. Theor. , 53 (1990) pp. 35–81 MR1077465 Zbl 0707.58048 |
[a7] | J.-L. Koszul, "Crochet de Schouten–Nijenhuis et cohomologie" Astérisque, Hors Sér. (1985) pp. 257–271 MR0837203 Zbl 0615.58029 |
[a8] | K. Mackenzie, "Lie groupoids and Lie algebroids in differential geometry" , Cambridge Univ. Press (1987) MR0896907 Zbl 0683.53029 |
[a9] | K. Mackenzie, "Lie algebroids and Lie pseudoalgebras" Bull. London Math. Soc. , 27 (1995) pp. 97–147 MR1325261 Zbl 0829.22001 |
[a10] | K. Mackenzie, P. Xu, "Lie bialgebroids and Poisson groupoids" Duke Math. J. , 73 (1994) pp. 415–452 MR1262213 Zbl 0844.22005 |
[a11] | J. Pradines, "Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux" C.R. Acad. Sci. Paris , 264 A (1967) pp. 245–248 MR0216409 Zbl 0154.21704 |
[a12] | A. Vaintrob, "Lie algebroids and homological vector fields" Russian Math. Surveys , 52 (1997) pp. 428–429 MR1480150 Zbl 0955.58017 |
[a13] | I. Vaisman, "Lectures on the geometry of Poisson manifolds" , Birkhäuser (1994) MR1269545 Zbl 0810.53019 |
[a14] | A. Weinstein, "Poisson geometry" Diff. Geom. Appl. , 9 (1998) pp. 213–238 MR1636305 Zbl 0930.37032 |
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