tailieunhanh - Symplectic groupoids and generalized almost subtangent manifolds

We obtain equivalent assertions among the integrability conditions of generalized almost subtangent manifolds, the condition of compatibility of source and target maps of symplectic groupoids with symplectic form, and generalized subtangent maps. | Turkish Journal of Mathematics Research Article Turk J Math (2015) 39 ¨ ITAK ˙ c TUB ⃝ doi: Symplectic groupoids and generalized almost subtangent manifolds ˙ ∗ Fulya S ¸ AHIN ˙ onu University, Malatya, Turkey Department of Mathematics, Faculty of Science and Art, In¨ Received: • Accepted: • Published Online: • Printed: Abstract: We obtain equivalent assertions among the integrability conditions of generalized almost subtangent manifolds, the condition of compatibility of source and target maps of symplectic groupoids with symplectic form, and generalized subtangent maps. Key words: Lie groupoid, symplectic groupoid, generalized almost subtangent manifold 1. Introduction The concept of groupoid was introduced by Ehresmann [5] in the 1950s, following his work on the concept of principal bundle. A groupoid G consists of 2 sets G1 and G0 , called arrows and objects, respectively, with maps s, t : G1 → G0 called source and target. It is equipped with a composition m : G2 → G1 defined on the subset G2 = {(g, h) ∈ G1 × G1 |s(g) = t(h)} , an inclusion map of objects e : G0 → G1 , and an inversion map i : G1 → G1 . 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) . Generally, a groupoid G is denoted by the set of arrows G1 . A topological groupoid is a groupoid G1 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 structure maps s, t, e, i, m are all smooth maps and s, t are submersions. The latter condition ensures that s and t -fibers are manifolds. One can .