nLab
symplectic vector space

Definition

A vector space $V$ over a field $k$ is symplectic if it is equipped with an exterior 2-form $\omega \in {\Lambda }_k^{2}V$ 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.

Related concepts

• symplectic manifold

• symplectic 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)

References

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