Lagrangian subspace - Imported Knowlpedia Content

Definition

Let $(V,\omega)$ be a finite-dimensional symplectic vector space. A linear subspace $L\subseteq V$ is Lagrangian if

L=L^\omega,

where $L^\omega$ is its symplectic orthogonal complement. Equivalently, if $\dim V=2n$ , then $L$ is an isotropic subspace of dimension $n$ . Thus $\omega$ vanishes on $L\times L$ , and $L$ has the largest dimension permitted by that condition. A Lagrangian subspace is simultaneously isotropic and coisotropic. No complementary subspace, basis, or inner product is included in the data.

Equivalent characterizations

For a subspace $L\subseteq V$ , the following are equivalent:

$L=L^\omega$ ;
$L$ is isotropic and $\dim L=\frac12\dim V$ ;
$L$ is maximal among isotropic subspaces under inclusion; and
$L$ is coisotropic and $\dim L=\frac12\dim V$ .

The finite-dimensional hypothesis matters: it supplies $\dim L+\dim L^\omega=\dim V$ , which turns the relevant inclusion and half-dimension condition into equality Cannas da Silva, Chapter 1, Homework 1.

Normal form and complements

Every Lagrangian subspace has a Lagrangian complement $L'$ such that $V=L\oplus L'$ . Choosing dual bases $e_1,\ldots,e_n$ of $L$ and $f_1,\ldots,f_n$ of $L'$ with $\omega(e_i,f_j)=\delta_{ij}$ produces a symplectic basis of $V$ . The complement is generally not unique.

The symplectic group acts transitively on the set of Lagrangian subspaces. After choosing one symplectic basis, every Lagrangian subspace is therefore equivalent to the span of $e_1,\ldots,e_n$ .

Examples and near-misses

In $\mathbb R^{2n}$ with $\omega=\sum_i dq_i\wedge dp_i$ , the $q$ -coordinate subspace $\{p=0\}$ and the $p$ -coordinate subspace $\{q=0\}$ are complementary Lagrangian subspaces. In $T^*Q$ at a point, the vertical tangent space is the standard geometric example.

The line $\operatorname{span}(e_1)$ in a symplectic vector space of dimension at least four is isotropic but not Lagrangian: it fails the half-dimension, maximality, and coisotropy conditions.

Conventions and scope

This knowl uses finite-dimensional real symplectic linear algebra. Over any field of characteristic different from two, the same definition works for a nondegenerate alternating form. In infinite-dimensional symplectic spaces, maximal isotropic, self-orthogonal, and complemented formulations may diverge, so the ambient topology and chosen definition must be stated separately.

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: Chapter 1 and Homework 1, isotropic, coisotropic, and Lagrangian subspaces.
Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: §2.3, Lagrangian subspaces.