Algebra bundle

Lie algebra bundle

In mathematics, a weak Lie algebra bundle ξ = {\displaystyle \xi =(\xi ,p,X,\theta )\,} is a vector bundle ξ {\displaystyle \xi \,} over a base space X together with a morphism θ : ξ ⊗ ξ → ξ {\displaystyle \theta :\xi \otimes \xi \rightarrow \xi } which induces a Lie algebra structure on each fibre ξ x {\displaystyle \xi _{x}\,} . A Lie algebra bundle ξ = ( ξ , p , X ) {\displaystyle \xi =(\xi ,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 {\displaystyle U} containing x, a Lie algebra L and a homeomorphism ϕ : U × L → p − 1 ( U ) {\displaystyle \phi :U\times L\to p^{-1}(U)\,} such that ϕ x : { x } × L → p − 1 ( { x } ) {\displaystyle \phi _{x}:\{x\}\times L\rightarrow p^{-1}(\{x\})\,} is a Lie algebra isomorphism.