Symplectic Group

For every even dimension , the symplectic group is the group of matrices which preserve a nondegenerate antisymmetric bilinear form , i.e., a symplectic form.

Every symplectic form can be put into a canonical form by finding a symplectic basis. So, up to conjugation, there is only one symplectic group, in contrast to the orthogonal group which preserves a nondegenerate symmetric bilinear form. As with the orthogonal group, the columns of a symplectic matrix form a symplectic basis.

Since is a volume form, the symplectic group preserves volume and vector space orientation. Hence, . In fact, is just the group of matrices with determinant 1. The three symplectic (0,1)-matrices are therefore

(1)

The matrices

(2)

and

(3)

are in , where

(4)

In fact, both of these examples are 1-parameter subgroups.

A matrix can be tested to see if it is symplectic using the Wolfram Language code:

  SymplecticForm[n_Integer] :=
    Join[PadLeft[IdentityMatrix[n], {n, 2n}],
      PadRight[-IdentityMatrix[n], {n, 2n}]]
  SymplecticQ[a_List]:= EvenQ[Length[a]]&&
    Transpose[a] . SymplecticForm[Length[a]/2] .
       a == SymplecticForm[Length[a]/2]

Thinking of a matrix as given by coordinate functions, the set of matrices is identified with . The symplectic matrices are the solutions to the equations

(5)

where is defined by

(6)

Note that these equations are redundant, since only of these are independent, leaving "free" variables. In fact, the symplectic group is a smooth -dimensional submanifold of .

Because the symplectic group is a group and a manifold, it is a Lie group. Its submanifold tangent space at the identity is the symplectic Lie algebra . The symplectic group is not compact.

Instead of using real numbers for the coefficients, it is possible to use coefficients from any field . The symplectic group for even is the group of elements of the general linear group that preserve a given nonsingular symplectic form. Any such matrix has determinant 1.