Isotropic subspace - Imported Knowlpedia Content

Definition

Let $(V,\omega)$ be a finite-dimensional symplectic vector space, and let $W\subseteq V$ be a linear subspace. The subspace $W$ is isotropic if

\omega|_{W\times W}=0.

Equivalently, every vector in $W$ is symplectically orthogonal to every other vector in $W$ , or

W\subseteq W^\omega,

where $W^\omega$ is the symplectic orthogonal complement. Isotropy is a condition on the whole subspace, not merely the automatic identities $\omega(w,w)=0$ for individual vectors.

Dimension bound

If $\dim V=2n$ , then

\dim W+\dim W^\omega=2n.

The inclusion $W\subseteq W^\omega$ therefore implies $\dim W\leq n$ . An isotropic subspace of the maximal possible dimension $n$ satisfies $W=W^\omega$ and is called Lagrangian.

Examples and contrasts

Every one-dimensional subspace is isotropic because an alternating form vanishes on pairs of proportional vectors. In a symplectic basis, $\operatorname{span}(e_1,\ldots,e_k)$ is isotropic for $k\leq n$ . By contrast, $\operatorname{span}(e_1,f_1)$ is a symplectic subspace, not an isotropic one.

Quotients and geometry

The restriction of $\omega$ to $W^\omega$ has kernel $W$ when $W$ is isotropic. Hence it descends to a nondegenerate alternating form on $W^\omega/W$ . Pointwise isotropic tangent spaces similarly define isotropic submanifolds; the present knowl concerns the underlying linear notion.

References

Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: Chapter 2, isotropic, coisotropic, and Lagrangian subspaces.
Maurice A. de Gosson, Symplectic Geometry and Quantum Mechanics, Birkhäuser, 2006. DOI record. Relevant: Chapter 1, symplectic spaces and Lagrangian planes.