TOPICS
Search

Symplectic Group


For every even dimension 2n, the symplectic group Sp(2n) is the group of 2n×2n matrices which preserve a nondegenerate antisymmetric bilinear form omega, 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 omega^n is a volume form, the symplectic group preserves volume and vector space orientation. Hence, Sp(2n) subset SL(2n). In fact, Sp(2) is just the group of matrices with determinant 1. The three symplectic (0,1)-matrices are therefore

 [1 0; 0 1],[1 0; 1 1],[1 1; 0 1].
(1)

The matrices

 [1 0 0 s; 0 1 s 0; 0 0 1 0; 0 0 0 1]
(2)

and

 [cosht sinht 0 sinht; sinht cosht sinht 0; 0 0 cosht -sinht; 0 0 -sinht cosht]
(3)

are in Sp(4), where

 omega=e_1 ^ e_3+e_2 ^ e_4.
(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 (2n)^2 coordinate functions, the set of matrices is identified with R^((2n)^2). The symplectic matrices are the solutions to the (2n)^2 equations

 A^(T)JA=J,
(5)

where J is defined by

 omega(x,y)=<x,Jy>.
(6)

Note that these equations are redundant, since only 2n^2-n of these are independent, leaving 2n^2+n "free" variables. In fact, the symplectic group is a smooth (2n^2+n)-dimensional submanifold of R^((2n)^2).

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 sp(2n). The symplectic group is not compact.

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


See also

Determinant, Field, General Linear Group, Group, Lie Algebra, Lie Group, Lie-Type Group, Linear Algebraic Group, Metaplectic Group, Projective Symplectic Group, Quadratic Form, Siegel's Upper Half-Space, Submanifold, Symplectic Basis, Symplectic Form, Unitary Group, Vector Space

Portions of this entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Symplectic Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SymplecticGroup.html

Subject classifications