Description
After comparing a number of possible compatibility conditions between an anchor map A ? TM on a vector bundle A and a presymplectic structure on the base M, we choose the most natural of them, best formulated in terms of a suitably chosen connection on A. We define a notion of momentum section of A?, and, when A is a Lie algebroid, we specify a condition for compatibility with the Lie algebroid bracket. Compatibility conditions on an anchor, a Lie algebroid bracket, a momentum section, a connection, and a presymplectic structure are then the defining properties of a hamiltonian Lie algebroid. For an action Lie algebroid with the trivial connection, the conditions reduce to those for a hamiltonian action. We show that the clean zero locus of the momentum section of a hamiltonian Lie algebroid is a coisotropic submanifold. To define morphisms of hamiltonian Lie algebroids, we express the structure in terms of a bigraded algebra generated by Lie algebroid forms and de Rham forms on its base. We give an Atiyah-Bott type characterization of a bracket-compatible momentum map; it is equivalent to a closed basic extension of the presymplectic form, within the generalization of the BRST model of equivariant cohomology to Lie algebroids. We show how to construct a groupoid by reduction of an action Lie groupoid G × M by a subgroup H of G which is not necessarily normal, and we find conditions which imply that a hamiltonian structure descends to such a reduced Lie algebroid.
Book Information
ISBN 9781470469092
Author Christian Blohmann
Format Paperback
Page Count 91
Imprint American Mathematical Society
Publisher American Mathematical Society