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I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.

I am not comfortable with these notions and google gave http://www.pphmj.com/Images/PT11.pdf which says

In 1966, A. Douady and M. Lazard constructed a Lie group bundle $\Gamma$ (not necessarily Hausdorff) for a given Lie algebra bundle $\xi$ such that the Lie algebra bundle of a Lie group bundle $\Gamma$ is isomorphic to a given Lie algebra bundle $\xi$ in their remarkable paper.

The paper they have mentioned is

A. Douady and M. Lazard, Espaces fibres en algebre de Lie et en groupes, Invent. Math. 1 (1966), 133-151. (digizeitschriften)

I do not read that language.

Can some one give a reference in English where this result is discussed, briefly atleast.

I am guessing that Lie algebra bundle over a manifold $M$ associated to a Lie group bundle over a manifold $M$ should be something where each fibre of $x\in M$ is Lie algebra $\mathfrak{g}_x$ of Lie group $G_x$ which is fibre of $x$ of Lie group bundle.

Any comments are welcome.

Edit : I am insisting on the possibility of each fibre being different Liegroup is because there is a notion of Lie group bundle (over a manifold $M$) where there is no assumption that all fibers are isomorphic. It is in Alexandre Grothendieck's "A general theory of fibre spaces with structure sheaf". It is not clear from the discussion in page $11$ of the paper Non abelian differentiable gerbes if they assume all fibers are isomorphic.

I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.

I am not comfortable with these notions and google gave http://www.pphmj.com/Images/PT11.pdf which says

In 1966, A. Douady and M. Lazard constructed a Lie group bundle $\Gamma$ (not necessarily Hausdorff) for a given Lie algebra bundle $\xi$ such that the Lie algebra bundle of a Lie group bundle $\Gamma$ is isomorphic to a given Lie algebra bundle $\xi$ in their remarkable paper.

The paper they have mentioned is

A. Douady and M. Lazard, Espaces fibres en algebre de Lie et en groupes, Invent. Math. 1 (1966), 133-151. (digizeitschriften)

I do not read that language.

Can some one give a reference in English where this result is discussed, briefly atleast.

I am guessing that Lie algebra bundle over a manifold $M$ associated to a Lie group bundle over a manifold $M$ should be something where each fibre of $x\in M$ is Lie algebra $\mathfrak{g}_x$ of Lie group $G_x$ which is fibre of $x$ of Lie group bundle.

Any comments are welcome.

Edit : I assume the notion of group bundle is same as that of the paper Notes on 1-gerbes and 2-gerbes (because this paper is closely related to the paper Non abelian differentiable gerbes mentioned above). According to that paper the following is the definition of group bundle :

the standard notion of an $X$-group scheme $G$ will correspond in a topological context to that of a bundle of groups on a space $X$. By this we mean a total space $G$ above a space $X$ that is a group in the cartesian monoidal category of spaces over $X$. In particular, the fibers $G_x$ of $G $ at points $x \in X$ are topological groups, whose group laws vary continuously with $x$.

Thus, I consider the possibility that the fibers are not necessarily isomorphic.

I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.

I am not comfortable with these notions and google gave http://www.pphmj.com/Images/PT11.pdf which says

In 1966, A. Douady and M. Lazard constructed a Lie group bundle $\Gamma$ (not necessarily Hausdorff) for a given Lie algebra bundle $\xi$ such that the Lie algebra bundle of a Lie group bundle $\Gamma$ is isomorphic to a given Lie algebra bundle $\xi$ in their remarkable paper.

The paper they have mentioned is

A. Douady and M. Lazard, Espaces fibres en algebre de Lie et en groupes, Invent. Math. 1 (1966), 133-151. (digizeitschriften)

I do not read that language.

Can some one give a reference in English where this result is discussed, briefly atleast.

I am guessing that Lie algebra bundle over a manifold $M$ associated to a Lie group bundle over a manifold $M$ should be something where each fibre of $x\in M$ is Lie algebra $\mathfrak{g}_x$ of Lie group $G_x$ which is fibre of $x$ of Lie group bundle.

Any comments are welcome.

I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.

I am not comfortable with these notions and google gave http://www.pphmj.com/Images/PT11.pdf which says

In 1966, A. Douady and M. Lazard constructed a Lie group bundle $\Gamma$ (not necessarily Hausdorff) for a given Lie algebra bundle $\xi$ such that the Lie algebra bundle of a Lie group bundle $\Gamma$ is isomorphic to a given Lie algebra bundle $\xi$ in their remarkable paper.

The paper they have mentioned is

A. Douady and M. Lazard, Espaces fibres en algebre de Lie et en groupes, Invent. Math. 1 (1966), 133-151. (digizeitschriften)

I do not read that language.

Can some one give a reference in English where this result is discussed, briefly atleast.

I am guessing that Lie algebra bundle over a manifold $M$ associated to a Lie group bundle over a manifold $M$ should be something where each fibre of $x\in M$ is Lie algebra $\mathfrak{g}_x$ of Lie group $G_x$ which is fibre of $x$ of Lie group bundle.

Any comments are welcome.

Edit : I am insisting on the possibility of each fibre being different Liegroup is because there is a notion of Lie group bundle (over a manifold $M$) where there is no assumption that all fibers are isomorphic. It is in Alexandre Grothendieck's "A general theory of fibre spaces with structure sheaf". It is not clear from the discussion in page $11$ of the paper Non abelian differentiable gerbes if they assume all fibers are isomorphic.

I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.

I am not comfortable with these notions and google gave http://www.pphmj.com/Images/PT11.pdf which says

In 1966, A. Douady and M. Lazard constructed a Lie group bundle $\Gamma$ (not necessarily Hausdorff) for a given Lie algebra bundle $\xi$ such that the Lie algebra bundle of a Lie group bundle $\Gamma$ is isomorphic to a given Lie algebra bundle $\xi$ in their remarkable paper.

The paper they have mentioned is

A. Douady and M. Lazard, Espaces fibres en algebre de Lie et en groupes, Invent. Math. 1 (1966), 133-151.

I do not read that language.

Can some one give a reference in English where this result is discussed, briefly atleast.

I am guessing that Lie algebra bundle over a manifold $M$ associated to a Lie group bundle over a manifold $M$ should be something where each fibre of $x\in M$ is Lie algebra $\mathfrak{g}_x$ of Lie group $G_x$ which is fibre of $x$ of Lie group bundle.

Any comments are welcome.

I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.

I am not comfortable with these notions and google gave http://www.pphmj.com/Images/PT11.pdf which says

In 1966, A. Douady and M. Lazard constructed a Lie group bundle $\Gamma$ (not necessarily Hausdorff) for a given Lie algebra bundle $\xi$ such that the Lie algebra bundle of a Lie group bundle $\Gamma$ is isomorphic to a given Lie algebra bundle $\xi$ in their remarkable paper.

The paper they have mentioned is

A. Douady and M. Lazard, Espaces fibres en algebre de Lie et en groupes, Invent. Math. 1 (1966), 133-151. (digizeitschriften)

I do not read that language.

Can some one give a reference in English where this result is discussed, briefly atleast.

I am guessing that Lie algebra bundle over a manifold $M$ associated to a Lie group bundle over a manifold $M$ should be something where each fibre of $x\in M$ is Lie algebra $\mathfrak{g}_x$ of Lie group $G_x$ which is fibre of $x$ of Lie group bundle.

Any comments are welcome.