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Lie groupoids and generalized almost paracomplex manifolds

In this paper, we show that there is a close relationship between generalized paracomplex manifolds and Lie groupoids. We obtain equivalent assertions among the integrability conditions of generalized almost paracomplex manifolds, the condition of compatibility of source and target maps of symplectic groupoids with symplectic form and generalized paraholomorphic maps. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 500 – 510 ¨ ITAK ˙ c TUB doi:10.3906/mat-1102-16 Lie groupoids and generalized almost paracomplex manifolds ˙ ∗ Mustafa Habil GURSOY, ¨ ˙ ˙ ¸ EN Fulya S ¸ AHIN, Ilhan IC ˙In¨ on¨ u University, Faculty of Science and Art, Department of Mathematics, 44280 Malatya, Turkey Received: 10.02.2011 • Accepted: 02.03.2012 • Published Online: 26.04.2013 • Printed: 27.05.2013 Abstract: In this paper, we show that there is a close relationship between generalized paracomplex manifolds and Lie groupoids. We obtain equivalent assertions among the integrability conditions of generalized almost paracomplex manifolds, the condition of compatibility of source and target maps of symplectic groupoids with symplectic form and generalized paraholomorphic maps. Key words: Lie groupoid, symplectic groupoid, generalized almost paracomplex manifold 1. Introduction A groupoid is a small category in which all morphisms are invertible. More precisely, a groupoid (G, G0 ) consists of two sets G and G0 , called arrows and objects, respectively, with maps s, t : G → G0 called source and target. It is equipped with a composition m : G2 → G defined on the subset G2 = {(g, h) ∈ G × G | s(g) = t(h)} ; an inclusion map of objects e : G0 → G and an inverse map i : G → G . For a groupoid, the following properties are satisfied: s(gh) = s(h), t(gh) = t(g), s(g−1 ) = t(g), t(g−1 ) = s(g), g(hf) = (gh)f whenever both sides are defined, g−1 g = 1s(g) , gg−1 = 1t(g) . Here we have used gh, 1x and g−1 instead of m(g, h), e(x) and i(g), respectively. Generally, a groupoid (G, G0 ) is denoted by the set of arrows G . A topological groupoid is a groupoid G whose set of arrows and set of objects are both topological spaces whose structure maps s, t, e, i, m are all continuous and s, t are open maps. A Lie groupoid is a groupoid G whose set of arrows and set of objects are both manifolds whose .