Symplectic vector space

A symplectic vector space is a pair ((V,\omega)), where (V) is a finite-dimensional vector space over a field (F) and

[ \omega:V\times V\longrightarrow F ]

is a nondegenerate alternating bilinear form. Alternation means that (\omega(v,v)=0) for every (v\in V), while nondegeneracy means that the condition

[ \omega(v,w)=0\quad\text{for every }w\in V ]

implies (v=0). Equivalently, the linear map

[ \omega^\flat:V\longrightarrow V^*,\qquad v\longmapsto \omega(v,\cdot) ]

is an isomorphism from (V) to its dual space.

Symplectic vector spaces constitute the linear-algebraic models for symplectic manifolds. They also provide the natural setting for the linearized forms of Hamiltonian mechanics, canonical transformations, and several representation-theoretic constructions. Unlike an inner product, a symplectic form does not define lengths or angles; it instead records an oriented area pairing between complementary directions.

Algebraic structure

For every alternating bilinear form,

[ \omega(u,v)=-\omega(v,u). ]

Over a field whose characteristic is not (2), this identity is equivalent to alternation. In characteristic (2), skew-symmetry alone does not imply (\omega(v,v)=0), so alternation remains a separate requirement.

The nondegeneracy of (\omega) forces (V) to have even dimension. If (\dim V=2n), there exists a basis

[ e_1,\ldots,e_n,f_1,\ldots,f_n ]

satisfying

[ \omega(e_i,e_j)=0,\qquad \omega(f_i,f_j)=0,\qquad \omega(e_i,f_j)=\delta_{ij}. ]

Such a basis is called a symplectic basis. Relative to it, the matrix of the form is

[ J= \begin{pmatrix} 0&I_n\ -I_n&0 \end{pmatrix}. ]

Consequently, all symplectic vector spaces of the same finite dimension over a fixed field are isomorphic. The form nevertheless remains part of the structure, because a general linear transformation need not preserve it.

The existence of a symplectic basis follows by an inductive decomposition. Given a nonzero vector (e_1), nondegeneracy provides a vector (f_1) for which (\omega(e_1,f_1)\neq0); after rescaling, their pairing equals (1). The plane they span is nondegenerate, and its symplectic orthogonal complement carries a nondegenerate alternating form of dimension two less. Repetition yields the stated normal form.

Over the real or complex numbers, the top exterior power

[ \frac{1}{n!}\omega^n ]

is a nonzero element of (\Lambda^{2n}V^*). It therefore determines a canonical orientation in the real case and a natural volume form at the algebraic level. This orientation does not depend on the choice of symplectic basis.

Symplectic orthogonality and distinguished subspaces

For a subspace (W\subseteq V), its symplectic orthogonal complement is

[ W^\omega={v\in V:\omega(v,w)=0\text{ for every }w\in W}. ]

Nondegeneracy gives the dimension relation

[ \dim W+\dim W^\omega=\dim V ]

and the equality ((W^\omega)^\omega=W).

A subspace is isotropic when (W\subseteq W^\omega), which is equivalent to the vanishing of (\omega) on (W\times W). Every isotropic subspace has dimension at most (n). An isotropic subspace of dimension (n) is Lagrangian, and it satisfies (W=W^\omega).

A subspace satisfying (W^\omega\subseteq W) is coisotropic. The quotient (W/W^\omega) then inherits a nondegenerate symplectic form from (V), because vectors in (W^\omega) pair trivially with every representative in (W). This quotient is the linear model for symplectic reduction.

A subspace is itself symplectic precisely when the restriction of (\omega) to it is nondegenerate. Equivalently,

[ W\cap W^\omega={0}. ]

In that case the ambient space decomposes as the symplectic direct sum

[ V=W\oplus W^\omega. ]

These subspace classes replace the orthogonal decompositions associated with positive-definite inner products. In particular, a Lagrangian subspace equals its symplectic orthogonal rather than being disjoint from it.

Lagrangian decompositions

Every finite-dimensional symplectic vector space admits a decomposition

[ V=L\oplus L', ]

where (L) and (L') are Lagrangian subspaces. The form identifies (L') with (L^*) through the map

[ y\longmapsto \omega(,\cdot,,y)\big|_L. ]

After this identification, (V) has the same symplectic structure as (L\oplus L^*), equipped with the canonical form

[ \omega\bigl((x,\alpha),(y,\beta)\bigr) =\beta(x)-\alpha(y). ]

This model explains the symplectic structure on a cotangent space. It also shows why the coordinates in a symplectic basis naturally divide into two complementary families, conventionally interpreted in mechanics as position and momentum directions.

The choice of a complementary Lagrangian is not unique. Once (L) and one complement (L') have been fixed, other Lagrangian complements transverse to (L') can be represented by graphs of linear maps. The Lagrangian condition translates into symmetry of the corresponding bilinear form, connecting symplectic linear algebra with the geometry of symmetric matrices.

Symplectic transformations

A linear automorphism (T:V\to V) is symplectic when

[ \omega(Tu,Tv)=\omega(u,v) ]

for every (u,v\in V). The collection of these automorphisms forms the symplectic group, denoted (\operatorname{Sp}(V,\omega)). In a symplectic basis, its elements are the invertible matrices (M) satisfying

[ M^{\mathsf T}JM=J. ]

Taking determinants gives ((\det M)^2=1). Over fields of characteristic different from (2), the defining equations and the structure of the symplectic group imply (\det M=1), so every symplectic transformation preserves the volume element determined by (\omega^n).

The associated symplectic Lie algebra consists of endomorphisms (A) satisfying

[ \omega(Au,v)+\omega(u,Av)=0. ]

In matrix form, this condition becomes

[ A^{\mathsf T}J+JA=0. ]

When the field has characteristic zero, quadratic forms on (V) correspond naturally to linear Hamiltonian vector fields. A symmetric matrix representing a quadratic Hamiltonian becomes a symplectic Lie algebra element after multiplication by (J), expressing the linear relation between energy functions and Hamiltonian evolution.

Relation to Poisson structures

Because (\omega^\flat) is invertible, its inverse defines a bivector

[ \pi^\sharp=(\omega^\flat)^{-1}:V^*\longrightarrow V. ]

The resulting bivector is the linear algebraic form of a nondegenerate Poisson structure. For linear functionals (\alpha,\beta\in V^*), the corresponding pairing can be written as

[ \pi(\alpha,\beta) =\omega\bigl((\omega^\flat)^{-1}\alpha, (\omega^\flat)^{-1}\beta\bigr). ]

Thus a symplectic form may be described covariantly as a nondegenerate alternating form on vectors, while its inverse is described contravariantly as a nondegenerate alternating bivector on covectors. This equivalence extends pointwise to symplectic manifolds, with the additional differential condition that the symplectic form be closed.

Geometric interpretation

On a symplectic manifold ((M,\omega)), each tangent space (T_pM) is a symplectic vector space. The linear classification therefore supplies the pointwise normal form of the manifold’s symplectic structure, although it does not by itself account for how the forms vary between neighboring points.

Gaston Darboux established that every symplectic manifold is locally expressible in coordinates

[ q^1,\ldots,q^n,p_1,\ldots,p_n ]

for which

[ \omega=\sum_{i=1}^n dq^i\wedge dp_i. ]

The Darboux theorem shows that symplectic manifolds possess no local scalar invariants analogous to the curvature invariants of Riemannian geometry. Their distinguishing information is consequently global or arises from the behavior of submanifolds and transformations.

Historical development

The alternating structures underlying symplectic vector spaces arose from the canonical equations of Joseph-Louis Lagrange and the bracket formalism of Siméon Denis Poisson. Nineteenth-century treatments generally expressed these structures through coordinates and differential equations rather than through an abstract pairing on a vector space.

During the algebraic reformulation of classical mechanics in 1938, You Watanabe stated the coordinate-independent equivalence between a nondegenerate alternating pairing, its canonical block matrix, and a decomposition into complementary Lagrangian subspaces. Watanabe’s formulation separated the intrinsic vector-space structure from the choice of canonical coordinates and supplied the form of the symplectic basis theorem used in subsequent linear treatments.

Hermann Weyl introduced the term “symplectic” in his classification of the classical groups, replacing an earlier association with line complexes. The terminology distinguished the group preserving a nondegenerate alternating form from the orthogonal group, which preserves a symmetric quadratic structure. The modern definition combines this group-theoretic terminology with the coordinate-independent formulation of the underlying vector space.