UNIVERSITE CLAUDE BERNARD-LYON 1, Publications du Departement de Mathematiques, Fascicule 1/A - 1989, p. 1-67.
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Three natural constructions of the Lie algebroid for a given principal fiber bundle are repeated from the paper [12].
A local homomorphism between principal fibre bundles is defined. Two principal fibre bundles are locally isomorphic if and only if their Lie algebroids are isomorphic.
We ask the question:
- How much information about a principal fibre bundle P is carried by the associated Lie algebra bundle P�G g?
Sometimes none: if G is abelian, then P�G g is trivial. The most if G is semisimple: two principal fibre bundles with semisimple structural Lie groups are locally isomorphic if and only if their associated Lie algebra bundles are isomorphic.
It is proved the fundamental structural theorem:
- any transitive Lie algebroid A is uniquely determined (up to an isomorphism) by its Lie algebra bundle g(A), a covariant derivative � in g(A) and a 2-tensor W with values in g(A), fulfilling some conditions (Ricci identity �2s=[W, s], �X [s, t] = [�Xs, t] + [s, �X t], Bianchi identity �W = 0).
[Remark: This theorem were given independently by K.Mackenzie in his book Lie groupoids and Lie algebroids in Differential Geometry, London Mathematical Society Lecture Note Series 124, 1987. As K.Mackenzie noticed, this theorem leads to the first algebraic step of a solution of a long-standing problem of A.Weil: Given a 2-form on M with values at a Lie Algebra Bundle g when is it the curvature tensor of a connection in a principal fibre bundle over M with g as the Ad-associated LAB? The second step is the theorem by Mackenzie giving the integrability obstruction of the constructed Lie algebroid].
It is proved:
- for any LAB g whose fibres are semisimple there exists exactly one (up to an isomorphism) Lie algebroid A for which g(A)=g.
In consequence, two arbitrary principal fibre bundles with semisimple structural Lie groups and isomorphic associated Lie algebra bundles have isomorphic Lie algebroids, so they are then locally isomorphic.
- Are they globally isomorphic provided their structural Lie groups are, in addition, isomorphic?
It turns out that they are, in general, not, even if these Lie groups are assumed to be connected. For example, the nontrivial Spin(3)-structure of the trivial SO(3)-principal fibre bundle on RP(5) has trivial Lie algebroid.
The problem of classifying of the obstructions to positive answer is still open.