nLab Lagrangian subspace

Definition

In a symplectic vector space a Lagrangian subspace is a maximal isotropic subspace:

a sub-vector space

• on which the restriction of the symplectic form vanishes;

• and which has maximal dimension with this property.

Similarly for a symplectic manifold. See Lagrangian submanifold .

The collection of all Lagrangian subspaces of a given space is called its Lagrangian Grassmannian.

Related concepts

• Lagrangian Grassmannian

• Lagrangian submanifold

• Maslov index

• metaplectic structure

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)