Representation of Double Lie-Groupoid
Subject Areas : StatisticsM. R. Farhangdoost 1 , S. Merati 2
1 - Department of Mathematics, College of Sciences, Shiraz University, P.O. Box 71457-44776, Shiraz, Iran.
2 - Department of Mathematics, College of Sciences,
Shiraz University, P.O. Box 71457- 44776, Shiraz, Iran
Keywords: کلاف برداری دوگانه, نمایش, جبروار لی مضاعف, گروهوار لی مضاعف,
Abstract :
In this paper we introduce the bi-VB groupoid and representation of double Lie groupoids, using the concept of vector bundle object in the category of Lie groupoids or Lie groupoid object in the category of vector bundle. A bi-VB groupoid is a bi-vector bundle object in the category of Lie groupoids. By a bi-vector bundle, we mean that a manifold by two vector bundle structures over two manifolds. We study some properties of the representation of double Lie groupoid as a cochains and smooth groupoid cohomology. We can show that there exists a one to one corresponding between a representation of a double Lie groupoid and two continues degree one operator which the space of normalized cochains, satisfying graded Leibniz identity and vanished their square. We study some properties and some example of bi-VB groupoids. And then we show that any representation of double Lie groupoids induced a bi-VB groupoid structure on its action groupoid.
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