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.
symplectic group, affine symplectic group, metaplectic group, extended affine symplectic group, Heisenberg group
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) |