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Hamiltonian Lie algebroids Weinstein, Alan


The constraint manifold for the initial value problem of general relativity is a coistropic subset in the symplectic manifold $P$ of 1-jets of lorentzian metrics along a space-like hypersurface. Coistropic submanifolds often arise as the zero sets of momentum maps for Lie algebra actions, but there is no evident Lie algebra action on $P$ to produce all of the constraints as momenta. Perhaps the Lie algebra should be replaced by a Lie algebroid. In an attempt to find the appropriate symmetry structure, Christian Blohmann, Marco Cezar Fernandes, and the speaker constructed a Lie algebroid over a space of \underline{infinite} jets for which the brackets relations among constant sections exactly matched the bracket relations among constraints, but this was not enough to explain the coisotropic nature of the constraint set. Two unanswered questions remain. (1) What are the appropriate notions of ``hamiltonian Lie algebroid" over a symplectic (or Poisson) manifold and associated ``momentum map" which make the zero sets of momentum maps coisotropic? (2) Is there a hamiltonian Lie algebroid over some ``extended phase space" $P'$ closely related to $P$ in which the constraint functions can be understood as the components of a momentum map? This talk will be a report on ongoing work with Blohmann and Michele Schiavina, the aim of which is to resolve these problems. Our working notion of hamiltonian structure on a Lie algebroid involves a connection with conditions relating its torsion to the symplectic structure on the base. A relevant extended phase space $P'$ appears to require a version of the BV-BFV construction currently under investigation by Cattaneo, Mnev, Reshetikhin, and Schiavina. Most of the talk will be devoted to question (1), with some remarks on question (2).

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