nLab
Lagrangian subspace

Contents

Definition

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

a sub-vector space

Similarly for a symplectic manifold. See Lagrangian submanifold .

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

type of subspace WW of inner product spacecondition on orthogonal space W W^\perp
isotropic subspaceWW W \subset W^\perp
coisotropic subspaceW WW^\perp \subset W
Lagrangian subspaceW=W W = W^\perp(for symplectic form)
symplectic spaceWW ={0}W \cap W^\perp = \{0\}(for symplectic form)

Last revised on January 2, 2015 at 19:47:20. See the history of this page for a list of all contributions to it.