Symplectic group

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In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. In this article, we shall denote these two groups Sp(2n, F) and Sp(n). The latter is sometimes called the compact symplectic group to distinguish it from the former. Note that many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the matrices used to represent the groups.

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Sp(2n, F)

The symplectic group of degree 2n over a field F, denoted Sp(2n, F), is the group of 2n by 2n symplectic matrices with entries in F, and with the group operation that of matrix multiplication. Since all symplectic matrices have unit determinant, the symplectic group is a subgroup of the special linear group SL(2n, F).

More abstractly, the symplectic group can be defined as the set of linear transformations of a 2n-dimensional vector space over F that preserve a nondegenerate, skew-symmetric, bilinear form. Such a vector space is called a symplectic vector space. The symplectic group of an abstract symplectic vector space V is also denoted Sp(V).

When n = 1, the symplectic condition on a matrix is satisfied iff the determinant is one so that Sp(2, F) = SL(2, F). For n > 1, there are additional conditions.

Typically, the field F is the field of real numbers, R, or complex numbers, C. In this case Sp(2n, F) is a real/complex Lie group of real/complex dimension n(2n + 1). These groups are connected but noncompact. Sp(2n, C) is simply connected while Sp(2n, R) has a fundamental group isomorphic to Z.

The Lie algebra of Sp(2n, F) is given by the set of 2n×2n matrices A (with entries in F) that satisfy

<math>\Omega A + A^T \Omega = 0<math>

where <math>A^T<math> is the transpose of A and Ω is the skew-symmetric matrix

<math>\Omega =

\begin{pmatrix} 0 & I_n \\ -I_n & 0 \\ \end{pmatrix} <math>

Sp(n)

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

<math>\langle x, y\rangle = \bar x_1 y_1 + \cdots + \bar x_n y_n<math>

That is, Sp(n) is just the quaternionic unitary group, U(n, H). Indeed, it is sometimes called the hyperunitary group. Note that Sp(n) is not a symplectic group in the sense of the previous section—it does not preserve a skew-symmetric form on Hn (in fact, there is no such form). The justification for calling this group symplectic is explained in the next section.

Sp(n) is a real Lie group of dimension n(2n + 1). It is compact, connected, and simply connected. The Lie algebra of Sp(n) is given by the set of n by n quaternionic matrices that satisfy

<math>A+A^{\dagger} = 0<math>

where <math>A^{\dagger}<math> is the conjugate transpose of A (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.

Relationships between the symplectic groups

The relationship between the groups Sp(2n, R), Sp(2n, C), and Sp(n) is most evident at the level of their Lie algebras. It turns out the Lie algebras of these three groups, when considered as real Lie groups, all share the same complexification. In Cartan's classification of the simple Lie algebras, this algebra is denoted Cn.

Stated slightly differently, the complex Lie algebra Cn is just the algebra sp(2n, C) of the complex Lie group Sp(2n, C). This algebra has two different real forms:

  1. the compact form, sp(n), which is the Lie algebra of Sp(n), and
  2. the normal form, sp(2n, R), which is the Lie algebra of Sp(2n, R).
Comparison of the symplectic groups
  matrices Lie group dim/R dim/C compact π1
Sp(2n, R) R real n(2n + 1) no Z
Sp(2n, C) C complex 2n(2n + 1) n(2n + 1) no 1
Sp(n) H real n(2n + 1) yes 1

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