The symplectic group is a
classical group defined as the set of
linear transformations of a -dimensional
vector space over the field which preserve a
non-degenerate skew-symmetric bilinear form. Such a vector space is called a
symplectic vector space, and the symplectic group of an abstract symplectic vector space is denoted . Upon fixing a basis for , the symplectic form is represented by a
nonsingular skew-symmetric matrix , and is identified with the group of matrices over satisfying :\{M \in M_{2n\times 2n}(F) : M^\mathrm{T} J M = J\}. This matrix group is denoted or , although the notation depends on the convention being used. Here denotes the
transpose of . In an arbitrary basis, the matrix need not have any particular form. However, one can choose a
symplectic basis, in which the form is represented by the standard matrix :\Omega = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}, where is the identity matrix. In such a basis, :\operatorname{Sp}(2n,F)=\{M\in M_{2n\times 2n}(F):M^\mathrm{T}\Omega M=\Omega\}. In this case, can be expressed as those block matrices (\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}), where A, B, C, D \in M_{n \times n}(F), satisfying the three equations: :\begin{align} -C^\mathrm{T}A + A^\mathrm{T}C &= 0, \\ -C^\mathrm{T}B + A^\mathrm{T}D &= I_n, \\ -D^\mathrm{T}B + B^\mathrm{T}D &= 0. \end{align} Since all symplectic matrices have
determinant , the symplectic group is a
subgroup of the
special linear group . When , the symplectic condition on a matrix is satisfied
if and only if the determinant is one, so that . For , there are additional conditions, i.e. is then a proper subgroup of . Typically, the field is the field of
real numbers or
complex numbers . In these cases is a real or complex
Lie group of real or complex dimension , respectively. These groups are
connected but
non-compact. The
center of consists of the matrices and as long as the
characteristic of the field is not . Since the center of is discrete and its quotient modulo the center is a
simple group, is considered a
simple Lie group. The real rank of the corresponding Lie algebra, and hence of the Lie group , is . The
Lie algebra of is the set :\mathfrak{sp}(2n,F) = \{X \in M_{2n \times 2n}(F) : \Omega X + X^\mathrm{T} \Omega = 0\}, equipped with the
commutator as its Lie bracket. For the standard skew-symmetric bilinear form \Omega = (\begin{smallmatrix} 0 & I \\ -I & 0 \end{smallmatrix}), this Lie algebra is the set of all block matrices (\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}) subject to the conditions :\begin{align} A &= -D^\mathrm{T}, \\ B &= B^\mathrm{T}, \\ C &= C^\mathrm{T}. \end{align} ====== The symplectic group over the field of complex numbers is a
non-compact,
simply connected,
simple Lie group. The definition of this group includes
no conjugates (contrary to what one might naively expect) but instead it is exactly the same as the definition bar the field change. ====== is a real, non-compact, connected simple Lie group of dimension n(2n+1). The exponential map from to is not surjective. For every in , one has a Cartan/polar decomposition S = OZO' with O,O' \in \operatorname{Sp}(2n,\mathbf{R})\cap\operatorname{SO}(2n)\cong U(n) and Z=\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}. As a manifold, is diffeomorphic to times a vector space of dimension .
Generation by involutions The symplectic group can be generated by linear transformations A:V\to V of the symplectic vector space such that A^2 = I and A^TJA = -J.
Example of symplectic matrices For , the group of matrices with determinant , the three symplectic -matrices are:\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\quad \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\quad \text{and} \quad \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}.
Sp(2n, R) It turns out that \operatorname{Sp}(2n,\mathbf{R}) can have a fairly explicit description using generators. If we let \operatorname{Sym}(n) denote the symmetric n\times n matrices, then \operatorname{Sp}(2n,\mathbf{R}) is generated by D(n)\cup N(n) \cup \{\Omega\}, where\begin{align} D(n) &= \left\{ \left. \begin{bmatrix} A & 0 \\ 0 & (A^T)^{-1} \end{bmatrix} \,\right| \, A \in \operatorname{GL}(n, \mathbf{R}) \right\} \\[6pt] N(n) &= \left\{ \left. \begin{bmatrix} I_n & B \\ 0 & I_n \end{bmatrix} \, \right| \, B \in \operatorname{Sym}(n)\right\} \end{align}are subgroups of \operatorname{Sp}(2n,\mathbf{R})pg 173pg 2.
Relationship with symplectic geometry Symplectic geometry is the study of
symplectic manifolds. The
tangent space at any point on a symplectic manifold is a
symplectic vector space. As noted earlier, structure preserving transformations of a symplectic vector space form a
group and this group is , depending on the dimension of the space and the
field over which it is defined. A symplectic vector space is itself a symplectic manifold. A transformation under an
action of the symplectic group is thus, in a sense, a linearised version of a
symplectomorphism which is a more general structure preserving transformation on a symplectic manifold. ====