Lie algebra bundle

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

$\theta : \xi \otimes \xi \rightarrow \xi$

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.

As an example of a weak Lie algebra bundle that is not a strong Lie algebra bundle, consider the total space $\mathfrak{so}(3)\times\mathbb{R}$ over the real line $\mathbb{R}$. Let [.,.] denote the Lie bracket of $\mathfrak{so}(3)$ and deform it by the real parameter as:

$[X,Y]_x = x\cdot[X,Y]$

for $X,Y\in\mathfrak{so}(3)$ and $x\in\mathbb{R}$.

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.^[1]

References

1. A. Weinstein, A.C. da Silva: Geometric models for noncommutative algebras, 1999 Berkley LNM, online readable at , in particular chapter 16.3.

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• Kiranagi, B.S.; Ranjitha, Kumar; Prema, G. (2014), "On completely semisimple Lie algebra bundles", Journal of Algebra and its Applications 14 (2): 1–11, doi:10.1142/S0219498815500097.

See also

• Algebra bundle
• Adjoint bundle