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arXiv:submit/0064778 [math.AT] 24 Jun 2010

Transitive Lie algebroids - categorical point of view
A.S.Mishchenko Moscow State University

Intro duction
Transitive Lie algebroids have specific properties that allow to lo ok at the transitive Lie algebroid as an element of the ob ject of a homotopy functor. Roughly speaking each transitive Lie algebroids can be described as a vector bundle over the tangent bundle of the manifold which is endowed with additional structures. Therefore transitive Lie algebroids admits a construction of inverse image generated by a smo oth mapping of smo oth manifolds. The construction can be managed as a homotopy functor from the category of smo oth manifolds to the transitive Lie algebroids. The intention of this article is to make a classification of transitive Lie algebroids and on this basis to construct a classifying space. The realization of the intention allows to describe characteristic classes of transitive Lie algebroids form the point of view a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles.

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Definitions and formulation of the problem
E -T M -M
a pT

Given smo oth manifold M let

be a vector bundle over T M with fiber g , pE = pT · a. So we have a commutative diagram of two vector bundles E
pE a

/ TM
pT

M


/M

2000 Mathematics Sub ject Classification: 57R20, 57R22 Key words: Lie algebroid, flat bundle, transition function, characteristic classes, classifying space

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The diagram is endowed with additional structure (commutator braces) and then is called ([1], definition 3.3.1, [2], definition 1.1.1) transitive Lie algebroid a/ E TM A = pE ; {·, ·} . pT /M M Let f : M -M be a smo oth map. Then one can define an inverse image (pullback) of the Lie algebroid ([1], page 156, [2], definition 1.1.4), f !! (A). This means that given the finite dimensional Lie algebra g there is the functor A such that with any manifold M it assigns the family A(M ) of all transitive Lie algebroids with fixed Lie algebra g . In the dissertation [3] the following statement was proved: Each transitive Lie algebroid is trivial, that is there is a trivialization of vector bundles E , T M , ker a = g such that ¯ E T M g, ¯ and the Lie bracket is defined by the formula: [(X, u), (Y , v )] = ([X, Y ], [u, v ] + X (v ) - Y (u)). Then using the construction of pullback and the idea by Allen Hatcher [4] one can prove that the functor A is homotopic functor. More exactly for two homotopic smo oth maps f0 , f1 : M1 -M2 and for the transitive Lie algebroid (E -T M2 -M2 ; {·, ·})
! ! two inverse images f0! (E ) and f1! (A) are isomorphic. Hence there is a final classifying space Bg such that the family of all transitive Lie algebroids with fixed Lie algebra g over the manifold M has one-to-one correspondence with the family of homotopy classes of continuous maps [M , Bg ]: a

A(M ) [M , Bg ]. Using this observation one can describe the family of all characteristic classes of a transitive Lie algebroids in terms of cohomologies of the classifying space Bg . Really, from the point of view of category theory a characteristic class is a natural transformation from the functor A to the cohomology functor H . a This means that for the transitive Lie algebroid E = (E -T M -M ; {·, ·}) the value of the characteristic class (E ) is a cohomology class (E ) H (M ), such that for smo oth map f : M1 -M we have
! (f0! (E )) = f ((E )) H (M1 ).

Hence the family of all characteristic classes {} for transitive Lie algebroids with fixed Lie algebra g has a one-to-one correspondence with the cohomology group H (Bg ). 2


On the base of these abstract considerations a formulated. Problem. Given finite dimensional Lie algebra space Bg for transitive Lie algebroids in more or less Below we suggest a way of solution the problem examples.

natural problem can be g describe the classifying understandable terms. and consider some trivial

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Description of transitive Lie algebroids using transition functions
E T M g, ¯ g M â g, ¯

Consider the trivial transitive Lie algebroids

and the Lie bracket is defined by the formula: [(X, u), (Y , v )] = ([X, Y ], [u, v ] + X (v ) - Y (u)), where X, Y (T M ) are smo oth vector fields, u, v g are smo oth sections ¯ which are represented as smo oth vector functions with values in the Lie algebra g . Consider a fiberwise isomorphism A : E -E that is identical on the second summands and generates the Lie algebra homomorphism A : (E )- (E ). The isomorphism A can be written by formula: (v , Y ) = A(u, X ); (v , X ) = ((x)(u(x)) + (X ), X ), where (x) : g -g is a fiberwise map of the bundle g , and is a differential ¯ form with values in g . The isomorphism A can be expressed as a matrix v (x) Y = (x) 0 1 · u(x) X

From the property of that A is a Lie algebra homomorphism: A([(X, u), (Y , v )]) = [A(X, u), A(Y , v )] one has that (x)([u1 (x), u2 (x)]) = [(x)(u1 (x)), (x)(u2 (x))]), d (X1 , X2 ) + [ (X1 ), (X2 )] = 0, d(X )(u) = [(u), (X )]. Consider an atlas of charts on the manifold M , {U },


(1)

U = M , and the


trivializations E T U (U â g ) of the Lie algebroid E over each chart U with the Lie brackets defined by the formula [(X, u), (Y , v )] = ([X, Y ], [u, v ] + X (v ) - Y (u)), 3


for X, Y (T U ), u, v (U â g ). On the intersection of two charts U = U U we have the transition function


=

-1

: TU



(U



â g )-T U



(U



â g)

which have the matrix form v (x) Y


=



u(x) X

=

(x) 0

1



·

u(x) X

.

For another choice of trivializations the correspondent transition functions satisfy the homology condition: (x) 0 = or (x) 0 = or 1


= H ·



· H-

1

1



= (x) 0 1
- 1 (x) 0 - - 1 µ 1

(x) 0

µ 1

·

·

=
- - (x) (x) 1 (x)µ + (x) 1

- (x) (x) 1 (x) 0

+µ



,

- (x) = (x) (x) 1 (x),





- = - (x) (x) 1 (x)µ + (x)



+ µ .

The elements and µ satisfy similar (1) conditions: (x)([u1 (x), u2 (x)]) = [ (x)(u1 (x)), (x)(u2 (x))]), dµ (X1 , X2 ) + [µ (X1 ), µ (X2 )] = 0, d (X )(u) = [ (u), µ (X )].

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Case of commutative Lie algebra g
(x)([u1 (x), u2 (x)]) = [ (x)(u1 (x)), (x)(u2 (x))]), d


In commutative case the conditions (1) have for simple form:

(X1 , X2 ) = 0,

(2)

d (X )(u) = 0. 4


Hence (x) = const . This means that the vector bundle g is flat and the family = { ¯ a Cech co chain C 1 (U, 1 (g )) ¯ in the bigraded Cech complex C
,

} defines

=

C i (U, j (g ); d = d + d ¯

where U = {U } is the atlas of charts. One has d ( ) = 0; Hence defines cohomology class

d ( ) = 0.

[ ] H 2 (M ; g). ¯ Therefore we have the following heorem 1 The classification of al l transitive Lie algebroids with fixed commutative Lie algebra g over the manifold M is determined by a flat Lie algebra bund le g ¯ over M and a 2-dimensional cohomology class [ ] H 2 (M ; g). ¯

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Some general prop erties

In common case we can say that a little bit about the transition functions on the level of homology groups H (g ) of the Lie algebra g . Since each transition function (x) is the homomorphism of the Lie algebra g , that is (x) Aut(g ), the co cycle { (x)} generate asso ciated bundles with fibers H (g ), say, H (g ), and bundles with fibers H (g ),H (g ). The properties (1) imply that all bundles H (g ) and H (g ) are flat. In particular the differential forms 1 (U ; g) generates the co cycle ¯ = { that is d ( ) = 0, d ( ) = 0. This means that the co cycle induces a cohomology class [ ] H
2

} C 1 (U, H1 (g )) =


1 (U



; H1 (g )),

M ; H1 (g ) .

The foregoing consideration creates a conjecture that classification of the transitive Lie algebroid E induces by two things: the Lie algebra bundle with 5


structural group Aut(g ) with special topology and the cohomology class [] H 2 M ; H1 (g ) . The special topology in the group Aut(g ) is defined as a minimal topology, which is more fine topology than the classical topology in Aut(g ) and such that all homomorphisms Aut(g )-Aut(Hk (g ))d
iscr ete

are continuous. This paper is partly supported by grants RBRF No 05-01-00923-a,07-0191555-NNIO-a, 10-01-92601-KO-a and pro ject No. RNP.2.1.1.5055


[1] Mackenzie, K.C.H., General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press,(2005) [2] Kubarski, J., The Chern-Weil homomorphism of regular Lie algebroids, Publications du Department de Mathematiques, Universite Claude Bernard - Lyon-1, (1991) pp.4­63 [3] W.Walas Algebry Liego-Rineharta i pierwsze klasy charakterystyczne. PhD manuscript, Lo dz, Poland, 2007. [4] Allen Hatcher, Vector bundles and K-theory, Available at http://www.math.cornel l.edu/hatcher/VBKT/VBpage.html , page 7, 2003.

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