nLab symplectic vector space

Definition

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

Related concepts

• symplectic manifold

• quadratic Hamiltonian

• 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)

References

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