Symplectic Group - Sp(n)

Sp(n)

The compact symplectic group, Sp(n), is the subgroup of GL(n, H) (invertible quaternionic matrices) which preserves the standard hermitian form on Hn:

That is, Sp(n) is just the quaternionic unitary group, U(n, H). Indeed, it is sometimes called the hyperunitary group. Also Sp(1) is the group of quaternions of unit 1, equivalent to SU(2) and topologically a 3-sphere S3.

Note that Sp(n) is not a symplectic group in the sense of the previous section—it does not preserve a non-degenerate skew-symmetric (H-bilinear) form on Hn (in fact, the only skew-symmetric form is the zero form). Rather, it is isomorphic to a subgroup of Sp(2n,C), and so does preserve a complex symplectic form in a vector space of dimension twice as high. As explained below, the Lie algebra of Sp(n) is a real form of the complex symplectic Lie algebra sp(2n, C).

Sp(n) is a real Lie group with (real) dimension n(2n + 1). It is compact, connected, and simply connected. It can be defined by the intersection Sp(n)=U(2n) ∩ Sp(2n, C), where U(2n) stands for the unitary group.

The Lie algebra of Sp(n) is given by the quaternionic skew-Hermitian matrices, the set of n by n quaternionic matrices that satisfy

where is the conjugate transpose of A (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.