Lie algebra bundle

{{Short description|Concept in topology (mathematics)}}

In mathematics, a weak Lie algebra bundle

:$\backslash xi=(\backslash xi,\; p,\; X,\; \backslash theta)\backslash ,$

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

:$\backslash theta\; :\; \backslash xi\; \backslash otimes\; \backslash xi\; \backslash rightarrow\; \backslash xi$

which induces a Lie algebra structure on each fibre $\backslash xi\_x\backslash ,$.

A Lie algebra bundle $\backslash xi=(\backslash xi,\; p,\; X)\backslash ,$ 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

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

such that

:$\backslash phi\_x:\backslash \{x\backslash \}\backslash times\; L\; \backslash rightarrow\; p^\{-1\}(\backslash \{x\backslash \})\backslash ,$

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 $\backslash mathfrak\{so\}(3)\backslash times\backslash mathbb\{R\}$ over the real line $\backslash mathbb\{R\}$. Let [.,.] denote the Lie bracket of $\backslash mathfrak\{so\}(3)$ and deform it by the real parameter as:

:$[X,Y]\_x\; =\; x\backslash cdot[X,Y]$

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

Lie's third theorem states that every bundle of Lie algebras can locally be integrated to a bundle of Lie groups. In general globally the total space might fail to be Hausdorff.A. Weinstein, A.C. da Silva: Geometric models for noncommutative algebras, 1999 Berkley LNM, online readable at [http://math.berkeley.edu/~alanw/], in particular chapter 16.3. But if all fibres of a real Lie algebra bundle over a topological space are mutually isomorphic as Lie algebras, then it is a locally trivial Lie algebra bundle. This result was proved by proving that the real orbit of a real point under an algebraic group is open in the real part of its complex orbit. Suppose the base space is Hausdorff and fibers of total space are isomorphic as Lie algebras then there exists a Hausdorff Lie group bundle over the same base space whose Lie algebra bundle is isomorphic to the given Lie algebra bundle.B S Kiranangi: "Lie Algebra Bundles", Bull. Sc. Math., 2^{e} serie, 102, 1978, pp. 57–62 Every semi simple Lie algebra bundle is locally trivial. Hence there exist a Hausdorff Lie group bundle over the same base space whose Lie algebra bundle is isomorphic to the given Lie algebra bundle.B S Kiranangi: "Semi-Simple Lie Algebra Bundles", Bull. Math. de la Sci. Math. de la R.S. de Roumanie, 27(75),1983, p.253-257