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Hamiltonian Lie algebroids


Blohmann,  Christian
Max Planck Institute for Mathematics, Max Planck Society;

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Blohmann, C., & Weinstein, A. (2024). Hamiltonian Lie algebroids. Providence, RI: American Mathematical Society.

Cite as: https://hdl.handle.net/21.11116/0000-000A-1CC7-7
In previous work with M.C. Fernandes, we found a Lie algebroid symmetry for
the Einstein evolution equations of general relativity. The present work was
motivated by the effort to explain the coisotropic structure of the constraint
subset for the initial value problem by extending the notion of hamiltonian
structure from Lie algebra actions to general Lie algebroids over presymplectic
manifolds. After comparing possible compatibility conditions between the anchor
$A\to TM$ and the presymplectic structure on the base $M$, we choose the most
natural of them, given by a suitably chosen connection on $A$. We define a
notion of momentum section of $A^*$ and a condition for compatibility with the
Lie bracket. A Lie algebroid over a presymplectic manifold with compatible
anchor and momentum section is then called hamiltonian. For an action Lie
algebroid, we retrieve the conditions of a hamiltonian action. The clean zero
locus of the momentum section of a hamiltonian Lie algebroid is a coisotropic
submanifold. We show that a bracket-compatible momentum map 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 construct
groupoids by reduction of an action Lie groupoid $G\times M$ by a subgroup $H$
of $G$ which is not necessarily normal, and we find conditions which imply that
a hamiltonian structure descends to their Lie algebroids. We consider many
examples and, in particular, find that the tangent Lie algebroid over a
symplectic manifold is hamiltonian with respect to some connection if and only
if the symplectic structure has a nowhere vanishing primitive. Recent results
of Stratmann and Tang show that this is the case whenever the symplectic
structure is exact.