2015/08/23 by Pedro Frejlich, Frejlich, Pedro, Ioan Mărcuț +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1508.05670
openalex publication_date 2015/08/23 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Poisson transversals are those submanifolds in a Poisson manifold which\nintersect all symplectic leaves transversally and symplectically. In a previous\nnote we proved a normal form theorem around such submanifolds. In this\ncommunication, we promote that result to a normal form theorem for Poisson maps\naround Poisson transversals. A Poisson map pulls a Poisson transversal back to\na Poisson transversal, and our first main result states that simultaneous\nnormal forms exist around such transversals, for which the Poisson map becomes\ntransversally linear, and intertwines the normal form data of the transversals.\n Our second main result concerns symplectic integrations. We prove that a\nneighborhood of a Poisson transversal is integrable exactly when the Poisson\ntransversal itself is integrable, and in that case we prove a normal form\ntheorem for the symplectic groupoid around its restriction to the Poisson\ntransversal, which puts all its structure maps in normal form.\n We conclude the paper by illustrating our results with examples arising from\nLie algebras.\n