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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 \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

^ A. Weinstein, A.C. da Silva: Geometric models for noncommutative algebras, 1999 Berkley LNM, online readable at [1], 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.

@@ Line 1: / Line 1: @@
-[[Matematik]]te, bir '''zayıf Lie cebri demeti'''
+In [[Mathematics]], a '''weak Lie algebra bundle'''
 :<math> \xi=(\xi, p, X, \theta)\,</math>
-bir [[vektör demeti]] <math>\xi\,</math> bir  ''X'' uzay tabanı  üzerinde bir morfizmle beraber
+is a [[vector bundle]] <math>\xi\,</math> over a base space ''X'' together with a morphism
 :<math> \theta : \xi \otimes \xi \rightarrow \xi </math>
+which induces a [[Lie algebra]] structure on each fibre <math> \xi_x\, </math>.
-Bu her lif <math> \xi_x\, </math> üzerindeki bir [[Lie cebiri]]  yapısını uyarır.
+A '''Lie algebra bundle''' <math> \xi=(\xi, p, X)\,</math> is a [[vector bundle]] in which
-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
+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
 :<math> \phi:U\times L\to p^{-1}(U)\,</math>
+such that
-böylece
 :<math> \phi_x:x\times L \rightarrow p^{-1}(x)\,</math>
-bir Lie cebiri izomorfizmidir.
+is a Lie algebra isomorphism.
+Any Lie algebra bundle is a weak Lie algebra bundle but the converse need not be true in general.
-Herhangi Lie cebiri demeti bir zayıf Lie cebiri demetidir ama  genel içinde tersi doğru olması gerekmez.
-<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:
-:<math>X,Y\in\mathfrak{so}(3)</math> için
+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:
 :<math>[X,Y]_x = x\cdot[X,Y]</math>
+for <math>X,Y\in\mathfrak{so}(3)</math> and <math>x\in\mathbb{R}</math>.
+[[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>
-ve <math>x\in\mathbb{R}</math> dir.
-[[Lie'nin üçüncü teoremi]] durumunda bu Lie cebirinin her demeti Lie gruplarının bir demetine yerel entegre olabilir.Ancak küresel toplam uzay [[Hausdorff space|Hausdorff]] için başarısız olabilir.<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>
-==Kaynakça==
+==References==
 {{Refimprove|date=February 2010}}<references/>
 *A.Douady et M.Lazard, Espaces fibres en algebre de Lie et en groups, Invent. math., Vol. 1, 1966, pp.&nbsp;133–151
@@ Line 43: / Line 40: @@
 *B.S.Kiranagi, G.Prema and C.Chidambara, Rigidity theorem for Lie algebra Bundles, Communications in Algebra 20 (6), 1992, pp.&nbsp;1549 – 1556.
-==Ayrıca bakınız==
+==See also==
-*[[Cebir demeti]]
+*[[Algebra bundle]]
-*[[Eşlenik demeti]]
+*[[Adjoint bundle]]
 [[Category:Differential topology]]