Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp(2n, F) and Sp(n). The latter is sometimes called the compact symplectic group to distinguish it from the former. 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. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted C_n, and Sp(n) is the compact real form of Sp(2n, C).

The name "symplectic group" is due to Hermann Weyl (details) as a replacement for the previous confusing names of (line) complex group and Abelian group, and is the Greek analog of "complex".

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 determinant 1, 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 if and only if the determinant is one, so that Sp(2, F) = SL(2, F). For n > 1, there are additional conditions, i.e. Sp(2n, F) is then a proper subgroup of SL(2n, F).

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

$\Omega A+A^{T}\Omega =0$

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

$\Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}}.$

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:

$\langle x,y\rangle ={\bar x}_{1}y_{1}+\cdots +{\bar x}_{n}y_{n}$

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 norm 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

$A+A^{{\dagger }}=0$

where $A^{{\dagger }}$ 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, C), Sp(2n, R) and Sp(n) is most evident at the level of their Lie algebras. It turns out that the first of these Lie algebras is a complexification of the Lie algebras of either of the latter two groups.

Stated slightly differently, the complex Lie algebra sp(2n, C) of the complex Lie group Sp(2n, C) has several different real forms:

the compact form, sp(n), which is the Lie algebra of Sp(n),
the algebras, sp(p, n − p), which are the Lie algebras of Sp(p, n − p), the indefinite signature equivalent to the compact form,
the normal form (or split form), sp(2n, R), which is the Lie algebra of Sp(2n, R).

Important subgroups

The symplectic group Sp(n) is sometimes written as USp(2n) which is convenient for the following equations. The symplectic group comes up in quantum physics as a symmetry on poisson brackets so it is important to understand its subgroups. Some main subgroups are:

${\mathit {USp}}(2n)\supset {\mathit {USp}}(2n-2)$

${\mathit {USp}}(2n)\supset {\mathit {U}}(n)$

${\mathit {USp}}(4)\supset O(4)$

The symplectic groups are also subgroups of various Lie groups:

${\mathit {SU}}(n)\supset {\mathit {USp}}(n)$

$F_{4}\supset {\mathit {USp}}(8)$

$G_{2}\supset {\mathit {USp}}(2)$

There are also the isomorphisms of the Lie algebras usp(4) = so(5) and usp(2) = so(3) = su(2).

Example of symplectic matrices

For Sp(2), the group of 2 x 2 matrices with determinant 1, the three symplectic (0, 1)-matrices are:^[1]

${\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad {\begin{pmatrix}1&0\\1&1\end{pmatrix}},\quad {\begin{pmatrix}1&1\\0&1\end{pmatrix}}$

Infinitesimal generators

The symplectic matrices, M, can be written using the matrix exponential as:

$M=\exp(a_{n}\lambda _{m})\,$

where λ are the infinitesimal generators. The infinitesimal generators of the symplectic matrices are the Hamiltonian matrices.

$\lambda \subset {\begin{pmatrix}A&B\\C&-A^{T}\end{pmatrix}}$

where B and C are symmetric matrices.

References

↑ Symplectic Group, (source: Wolfram MathWorld), downloaded February 14, 2012

Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6 .

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