# momentum map

Let $(M,\omega )$ be a symplectic manifold^{}, $G$ a Lie group acting on that manifold^{}, $\U0001d524$
its Lie algebra^{}, and ${\U0001d524}^{*}$ the dual of the Lie algebra. This action induces a map
$\alpha :\U0001d524\to \U0001d51b(M)$ where $\U0001d51b(M)$ is the Lie algebra of vector fields on $M$, such that
$\mathrm{exp}(tX)(m)={\rho}_{t}(m)$ where $\rho $ is the flow of $\alpha (X)$. Then a moment map
$\mu :M\to {\U0001d524}^{*}$ for the action of $G$ is a map such that

$${H}_{\mu (X)}=\alpha (X).$$ |

Here $\mu (X)(m)=\mu (m)(X)$, that is, $\mu (m)$ is a covector, so we apply it to the vector $X$ and get a scalar function $\mu (X)$, and ${H}_{\mu (X)}$ is its Hamiltonian vector field.

Generally, the moment maps we are interested in are equivariant with respect to the coadjoint action, that is, they satisfy

$${\mathrm{Ad}}_{g}^{*}\circ \mu =\mu \circ g.$$ |

Title | momentum map |
---|---|

Canonical name | MomentumMap |

Date of creation | 2013-03-22 13:14:36 |

Last modified on | 2013-03-22 13:14:36 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 4 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 53D20 |

Synonym | moment map |