Samiksha Jaiswal (Editor)

Lie algebra bundle

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In mathematics, a weak Lie algebra bundle

ξ = ( ξ , p , X , θ )

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

θ : ξ ξ ξ

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

ϕ : U × L p 1 ( U )

such that

ϕ x : x × L p 1 ( x )

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:

[ X , Y ] x = x [ X , Y ]

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.

References

Lie algebra bundle Wikipedia


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