Lie algebra bundle

In mathematics, a weak Lie algebra bundle

is a vector bundle ξ over a base space X together with a morphism

which induces a Lie algebra structure on each fibre ξ x .

A Lie algebra bundle ξ = ( ξ , p , X ) is a vector bundle in which each fibre is a Lie algebra and for every x in X, there is an open set U containing x, a Lie algebra L and a homeomorphism

such that

is a Lie algebra isomorphism.

Any Lie algebra bundle is a weak Lie algebra bundle, but the converse need not be true in general.

As an example of a weak Lie algebra bundle that is not a strong Lie algebra bundle, consider the total space s o ( 3 ) × R over the real line R . Let [.,.] denote the Lie bracket of s o ( 3 ) and deform it by the real parameter as:

for X , Y ∈ s o ( 3 ) and x ∈ R .

Lie's third theorem states that every bundle of Lie algebras can locally be integrated to a bundle of Lie groups. However globally the total space might fail to be Hausdorff.