Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids
Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids
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DOI：
10.1515/crelle.2011.092
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发表时间：
2006-11
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影响因子：
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通讯作者：
K. Mackenzie
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文献类型：
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作者：
K. Mackenzie
Ehresmann用“双重”一词来表示“所有X的范畴中的一个对象X”。双范畴、双群拟和双向量束是实例，但李代数的概念不能在Ehresmann意义上轻易地被双化，因为李代数的括号不能用图解的方式定义。本文利用双向量束的对偶性定义了一个双李群的概念，并证明了该概念抽象了一个双李群的二阶无限小结构。我们进一步证明了李双代数中任一李代数的余切都具有双重李代数结构，在李双代数的情况下，这种结构可以简化为经典的Drinfel双元。因此，我们可以说李双代数群的Drinfel'd双元是Ehresmann双元，并由此得出，双李群为李双代数群的余切双元提供了全局模型。如果一个双向量束是由公共基M上的向量束A， B作为同时回拉A × M B的向量束A， B构造而成的，则称为空的双向量束。给定基M上的一对李代数集（A， B），我们证明了A × M B具有一个双李代数集结构，并且在一个空的双向量束A × M B上的任何一个双李代数集结构都以这种方式出现。特别地，Lu和Weinstein、Kosmann-Schwarzbach和Magri以及Majid意义上的双李代数具有空双李代数的结构。最后对鲁杜克数学的构造进行了推广。[j] . 86: 261-304, 1997)，将一个匹配的李代数群对与任何泊松群作用联系起来；这就产生了一个双李代数，它通常不对应于一个匹配对。这篇论文的方法完全是“经典”的，而不是使用超级技术。
Abstract The word ‘double’ was used by Ehresmann to mean ‘an object X in the category of all X’. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot be defined diagrammatically. In this paper we use the duality of double vector bundles to define a notion of double Lie algebroid, and we show that this abstracts the infinitesimal structure (at second order) of a double Lie groupoid. We further show that the cotangent of either Lie algebroid in a Lie bialgebroid has a double Lie algebroid structure, that reduces, in the case of a Lie bialgebra, to the classical Drinfel'd double. Thus one may say that the Drinfel'd double of a Lie bialgebroid is an Ehresmann double, and it follows that double Lie groupoids provide global models for the cotangent double of a Lie bialgebroid. A double vector bundle is called vacant if it is constructed from vector bundles A, B on a common base M as the simultaneous pullback A × M B. Given a matched pair (A, B) of Lie algebroids over base M, we show that A × M B has a double Lie algebroid structure, and that any double Lie algebroid structure on a vacant double vector bundle A × M B arises in this way. In particular, double Lie algebras in the sense of Lu and Weinstein, Kosmann-Schwarzbach and Magri, and Majid, have the structure of a vacant double Lie algebroid. Lastly we extend the construction of Lu (Duke Math. J. 86: 261–304, 1997) which associates a matched pair of Lie algebroids to any Poisson group action, to actions of Lie bialgebroids; this yields a double Lie algebroid which in general does not correspond to a matched pair. The methods of the paper are entirely ‘classical’ rather than utilizing super techniques.