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)

Revised on November 10, 2013 11:45:14 by Urs Schreiber (89.204.135.42)