Место издания:Department of Mathematics University of Patras, Greece
Первая страница:124
Последняя страница:126
Аннотация:We call by the Lie algebroid a finite dimensional vector bundle
$E\rightarrow M$ over a smooth manifold $ M $ with a homomorphism
$a:E\rightarrow TM$ to the tangent bundle $TM$, called anchor,
and the space $\Gamma^{\infty}(E)$ of
smooth sections is provided with an additional structure, the commutator bracket
$\{\bullet,\bullet\}$, which satisfies the natural properties of the structure
infinite-dimensional Lie algebra, as well as the Newton-Leibniz identity with respect
to the operation of multiplication of
section by a smooth function. The anchor thus induces a homomorphism of the Lie algebra
$\Gamma^{\infty}(E)$ into the Lie algebra $\Gamma^{\infty}(TM)$ of vector fields on
the manifold $M$.
Examples of Lie algebroids are the tangent bundle $TM$, the bundle
${\cal D}(L)$ of covariant differentiations of all smooth sections
$\Gamma^{\infty}(L)$ of
any finite-dimen\-sional vector bundle $L$ over a smooth manifold $M$,
as well as the tangent bundle of an arbitrary smooth foliation ${\cal F}$ on
the manifold $M$ without singular points.
In the case where $a$ anchor is surjective,
the Lie algebroid is called
transitive.
Transitive Lie algebroids have specific properties that allow to look at the transitive
Lie algebroid as an element of the object 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.
Transitive Lie algebroids were detaily studied in the
book by K.Mackenzie (\cite {Mck-2005}) .
In particular, it was shown that the smooth map of
manifolds generate the inverse image (pullback) of
transitive Lie algebroids, which depends only on the homotopy class of the map.
Hence the construction can be managed as a homotopy functor $TLA_{\mathfrak g}$ from category of
smooth manifolds to the transitive Lie algebroids. Hence one can construct
a classifying space ${\cal B}_{\mathfrak g}$ such that
the family of all transitive Lie algebroids with fixed Lie algebra $\mathfrak g$ over the manifold $M$
has one-to-one correspondence with the family of homotopy classes of continuous maps $[M,{\cal B}_{\mathfrak g}]$: $TLA_{\mathfrak g}(M)\approx [M,{\cal B}_{\mathfrak g}].$
From this observation it follows that the classification of transitive Lie algebroids
can be reduced to the construction of the final objects for each fixed finite-dimensional Lie algebra
$g$, associated to the
transitive Lie algebroid and the classification is invariant up to homotopy. In spite of the evidence of the observation the construction of the final object still was not
conducted.
We prove (\cite{Mi-Li-2013}, \cite{Mi-Li-2013a}), that the homotopy classification is reduced
to the construction of the final space in the form of the classifying space $BG$, where $G$
is the group ${\bf Aut}(\mathfrak g)^{\delta}$ of automorphisms of the adjoint Lie algebra $\mathfrak g$ with new topology thinner
than the classical topology.
The description of the classifying space ${\cal B}_{\mathfrak g}$ is reduced to classification of coupling between Lie algebra bundle (LAB) and the tangent bundle. We show that tangent bundle $TM$ can be coupled with the Lie algebra bundle $L$ if and only if the Lie algebra bundle L admits a local trivial structure with structural group endowed with such new topology.
More of that we prove that there is a one-to-one correspondence between the family
$Coup(L)$
of all coupling of the Lie algebra bundle $L$
with fixed finite dimensional Lie algebra ${\mathfrak g}$ as the fiber and the structural group ${\bf Aut}(\mathfrak g)$ of all automorphisms of Lie algebra $\mathfrak g$ and the tangent bundle $TM$ and the family $LAB^{\delta}(L)$ of equivalent classes of local trivial structures with structural group ${\bf Aut}(\mathfrak g)$ endowed with new topology ${\bf Aut}(\mathfrak g)^{\delta}$.
This construction, in particular, allows to calculate the cohomology
Mackenzie obstacle for existence of a transitive Lie algebroid, which turns
trivial in many cases. For example, Mackenzie obstacle is trivial for any simply connected manifold as proved by Li Xiaoyu (China), Gasimov V.A-M. (Azerbaijan) and author.