Lie algebra bundle

In mathematics, a weak Lie algebra bundle

${\displaystyle \xi =(\xi ,p,X,\theta )\,}$

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

${\displaystyle \theta :\xi \otimes \xi \rightarrow \xi }$

which induces a Lie algebra structure on each fibre ${\displaystyle \xi _{x}\,}$.

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

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

such that

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

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

${\displaystyle [X,Y]_{x}=x\cdot [X,Y]}$

for ${\displaystyle X,Y\in {\mathfrak {so}}(3)}$ 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.

• A.Douady et M.Lazard, Espaces fibres en algebre de Lie et en groups, Invent. math., Vol. 1, 1966, pp. 133–151
• B.S.Kiranagi, Lie Algebra bundles, Bull. Sci. Math., 2^e serie, 102(1978), 57-62.
• B.S.Kiranagi, Semi simple Lie algebra bundles, Bull. Math de la Sci. Math de la R.S.de Roumaine, 27 (75), 1983, 253-257.
• B.S.Kiranagi and G.Prema, On complete reducibility of Module Bundles, Bull. Austral. Math Soc., 28 (1983), 401-409.
• B.S.Kiranagi and G.Prema, Cohomology of Lie algebra bundles and its applications, Ind. J. Pure and Appli. Math. 16(7): 1985, 731/735.
• B.S.Kiranagi and G.Prema, Lie algebra bundles defined by Jordan algebra bundles, Bull. Math. Soc.Sci.Math.Rep.Soc. Roum., Noun. Ser. 33 (81), 1989, 255-264.
• B.S.Kiranagi and G.Prema, On complete reducibility of Bimodule bundles, Bull. Math. Soc. Sci.Math. Repose; Roum, Nouv.Ser. 33 (81), 1989, 249-255.
• B.S.Kiranagi and G.Prema, A decomposition theorem of Lie algebra Bundles, Communications in Algebra 18 (6), 1990, 1869-1877 .
• B.S.Kiranagi, G.Prema and C.Chidambara, Rigidity theorem for Lie algebra Bundles, Communications in Algebra 20 (6), 1992, pp. 1549 – 1556.
• 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