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Farhangdoost, M. R., Merati, S.. (1401). Representation of Double Lie-Groupoid. , 8(36), 149-156.
M. R. Farhangdoost; S. Merati. "Representation of Double Lie-Groupoid". , 8, 36, 1401, 149-156.
Farhangdoost, M. R., Merati, S.. (1401). 'Representation of Double Lie-Groupoid', , 8(36), pp. 149-156.
Farhangdoost, M. R., Merati, S.. Representation of Double Lie-Groupoid. , 1401; 8(36): 149-156.

Representation of Double Lie-Groupoid

Journal of New Researches in Mathematics
مقاله 12، دوره 8، شماره 36، آذر و دی 2022، صفحه 149-156  XML اصل مقاله (546.62 K)
نوع مقاله: research paper
نویسندگان
M. R. Farhangdoost email 1؛ S. Merati2
1Department of Mathematics, College of Sciences, Shiraz University, P.O. Box 71457-44776, Shiraz, Iran.
2Department of Mathematics, College of Sciences, Shiraz University, P.O. Box 71457- 44776, Shiraz, Iran
چکیده
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.
کلیدواژه‌ها
double Lie groupoid؛ double Lie algebroid؛ representation؛ bi-vector bundle
مراجع
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[2] A. Garcia-Saz and R. A. Mehta., VB-groupoids and representation theory of Lie groupoids., J. Symplect. Geom., vol. 15, no. 3, pp. 741-783, 2017.
 

[3] S. Merati and M. R. Farhangdoost, Representation up to homotopy of hom-Lie algebroids, Int. J. Geom. Methods Mod. Phys., vol. 15, no. 5, pp. 1850074, 2018.
 

[4] S. Merati and M. R. Farhangdoost, Representation and central extension of hom-Lie algebroids, J. Algebra Appl., vol. 17, no. 11, pp. 185219, 2018.
 

[5] C. A. Abad and M. Crainic., Representation up to homotopy and Bott's sequnce for Lie groupoids, Adv. Math., vol. 248, pp. 416-452, 2013.
 

[6] K. C. H. Mackenzie., Double Lie algebroids and second-order geometry. I, Adv. Math., vol. 94, no. 2, pp. 180-239, 1992.
 

[7] L. Stefanini., On the integration of LA-groupoids and duality for Poisson groupoids, Travaux mathe'matiques. Fascicule XVII, Universite' du Luxembourg, pp. 39-59, 2007.
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