Lie algebra bundle: Difference between revisions
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In [[Mathematics]], a '''weak Lie algebra bundle''' |
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:<math> \xi=(\xi, p, X, \theta)\,</math> |
:<math> \xi=(\xi, p, X, \theta)\,</math> |
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is a [[vector bundle]] <math>\xi\,</math> over a base space ''X'' together with a morphism |
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:<math> \theta : \xi \otimes \xi \rightarrow \xi </math> |
:<math> \theta : \xi \otimes \xi \rightarrow \xi </math> |
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which induces a [[Lie algebra]] structure on each fibre <math> \xi_x\, </math>. |
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Bu her lif <math> \xi_x\, </math> üzerindeki bir [[Lie cebiri]] yapısını uyarır. |
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A '''Lie algebra bundle''' <math> \xi=(\xi, p, X)\,</math> is a [[vector bundle]] in which |
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Bir '''Lie cebiri demeti''' <math> \xi=(\xi, p, X)\,</math> içindeki bir [[vektör demeti]] bunun her lifi bir Lie cebiridir ve ''X'' içindeki her ''x'' için , burada ''x'' içeren bir [[açık küme]] <math> U </math> dur, bir Lie cebiri ''L'' ve bir homomorfizm |
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each fibre is a Lie algebra and for every ''x'' in ''X'', there is an [[open set]] <math> U </math> containing ''x'', a Lie algebra ''L'' and a homeomorphism |
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:<math> \phi:U\times L\to p^{-1}(U)\,</math> |
:<math> \phi:U\times L\to p^{-1}(U)\,</math> |
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such that |
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böylece |
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:<math> \phi_x:x\times L \rightarrow p^{-1}(x)\,</math> |
:<math> \phi_x:x\times L \rightarrow p^{-1}(x)\,</math> |
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is a Lie algebra isomorphism. |
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Any Lie algebra bundle is a weak Lie algebra bundle but the converse need not be true in general. |
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Herhangi Lie cebiri demeti bir zayıf Lie cebiri demetidir ama genel içinde tersi doğru olması gerekmez. |
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<math>\mathfrak{so}(3)\times\mathbb{R}</math> toplam uzayı düşünüldüğünde zayıf bir Lie cebiri demetinin bir örneği olarak bu bir sert Lie cebiri demeti değildir <math>\mathbb{R}</math> üzerinde gerçek hattır.Diyelimki [[So(3)#Lie algebra|<math>\mathfrak{so}(3)</math>]]ün Lie braketi [.,.] ifadesi ve gerçek parametre olarak onun deformesi: |
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As an example of a weak Lie algebra bundle that is not a strong Lie algebra bundle, consider the total space <math>\mathfrak{so}(3)\times\mathbb{R}</math> over the real line <math>\mathbb{R}</math>. Let [.,.] denote the Lie bracket of [[So(3)#Lie algebra|<math>\mathfrak{so}(3)</math>]] and deform it by the real parameter as: |
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:<math>[X,Y]_x = x\cdot[X,Y]</math> |
:<math>[X,Y]_x = x\cdot[X,Y]</math> |
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⚫ | [[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 space|Hausdorff]].<ref>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.</ref> |
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ve <math>x\in\mathbb{R}</math> dir. |
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⚫ | [[Lie' |
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==References== |
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{{Refimprove|date=February 2010}}<references/> |
{{Refimprove|date=February 2010}}<references/> |
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*A.Douady et M.Lazard, Espaces fibres en algebre de Lie et en groups, Invent. math., Vol. 1, 1966, pp. 133–151 |
*A.Douady et M.Lazard, Espaces fibres en algebre de Lie et en groups, Invent. math., Vol. 1, 1966, pp. 133–151 |
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*B.S.Kiranagi, G.Prema and C.Chidambara, Rigidity theorem for Lie algebra Bundles, Communications in Algebra 20 (6), 1992, pp. 1549 – 1556. |
*B.S.Kiranagi, G.Prema and C.Chidambara, Rigidity theorem for Lie algebra Bundles, Communications in Algebra 20 (6), 1992, pp. 1549 – 1556. |
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==See also== |
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*[[ |
*[[Algebra bundle]] |
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*[[ |
*[[Adjoint bundle]] |
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[[Category:Differential topology]] |
[[Category:Differential topology]] |
Revision as of 22:51, 26 January 2014
In Mathematics, a weak Lie algebra bundle
is a vector bundle over a base space X together with a morphism
which induces a Lie algebra structure on each fibre .
A Lie algebra bundle is a vector bundle in which each fibre is a Lie algebra and for every x in X, there is an open set containing x, a Lie algebra L and a homeomorphism
such that
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 over the real line . Let [.,.] denote the Lie bracket of and deform it by the real parameter as:
for and .
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
This article needs additional citations for verification. (February 2010) |
- 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., 2e 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.