Lie algebra bundle

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In Mathematics, a weak Lie algebra bundle

$\xi=(\xi, p, X, \theta)\,$

is a vector bundle $\xi\,$ over a base space X together with a morphism

$\theta : \xi \oplus \xi \rightarrow \xi$

which induces a Lie algebra structure on each fibre $\xi_x\,$ .

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

$\phi:U\times L\to p^{-1}(U)\,$

such that

$\phi_x:x\times L \rightarrow p^{-1}(x)\,$

is a Lie algebra isomorphism.

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

[edit] References

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