symplectic vector space

A vector space $V$ over a field $k$ is **symplectic** if it is equipped with an exterior 2-form $\omega \in \Lambda^2_k V$ such that $\omega^{\wedge n}=\omega\wedge\omega\wedge\cdots\wedge\omega$ has the maximal rank.

A subspace $W\subset V$ in a symplectic vector space is **isotropic** if $\omega(v,v) = 0$ for all $v\in W$ and Lagrangean (or **lagrangian**) if it is maximal isotropic (not proper subspace in any isotropic subspace). See wikipedia.

type of subspace $W$ of inner product space | condition on orthogonal space $W^\perp$ | |
---|---|---|

isotropic subspace | $W \subset W^\perp$ | |

coisotropic subspace | $W^\perp \subset W$ | |

Lagrangian subspace | $W = W^\perp$ | (for symplectic form) |

symplectic space | $W \cap W^\perp = \{0\}$ | (for symplectic form) |

- O. T. O’Meara,
*Symplectic groups*, Math. Surveys**16**, Amer. Math. Soc. 1978. xi+122 pp.

Revised on March 18, 2013 23:48:51
by Urs Schreiber
(89.204.138.142)