Место издания:Department of Mathematics University of Patras, Greece
Первая страница:84
Последняя страница:84
Аннотация:Let g be a finite dimensional Lie algebra and L be a Lie algebra bundle (LAB).
Given a coupling Ξ between LAB L and tangent bundle TM of the
manifold M generates so called the Mackenzie obstruction Obs(Ξ)∈H3(M;ZL)
for existing of transitive Lie algebroid. We present two case of calculating of
the Mackenzie obstruction. In the case of commutative algebra g the
group Aut(g)δ is isomorphic to the group of all matrices GL(g) with discrete
topology. In this case, the coupling Ξcoincides with a flat connection ∇ in a flat
bundle L, i.e. R∇≡0. This means that the form Ω can be chosen trivial,
i.e. d∇Ω=0. So the obstacle for coupling of Obs(Ξ) equals to zero.
The second case describe the Mackenzie obstruction for simply connected
manifolds. We prove that for simply connected manifolds
Obs(Ξ)=0∈H3(M;ZL;∇Z).
References: K.Mackenzie, General Theory of Lie Groupoids and Lie
Algebroids, 2005, p.279