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Monograph

#### Hamiltonian Lie algebroids

##### External Resource

https://doi.org/10.48550/arXiv.1811.11109

(Preprint)

https://doi.org/10.1090/memo/1474

(Publisher version)

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##### Citation

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

##### Abstract

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.

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.