Symplectic group

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

One of the classical groups, defined as the group of automorphisms of a skew-symmetric bilinear form $\Phi$ on a left $K$-module $E$, where $K$ is a commutative ring (cf. Classical group). In the case when $E=K^{2m}$ and the matrix of $\Phi$ with respect to the canonical basis $\{e_i\}$ of $E$ has the form

$$J_m = \begin{pmatrix}0 & I_m\\ -I_m & 0\end{pmatrix},$$ where $I_m$ is the identity matrix of order $m$, the corresponding symplectic group is called the symplectic group of $2m$ variables over the ring $K$ and is denoted by $\def\Sp{ {\rm Sp}}\Sp(m,K)$ or $\Sp_{2m}(K)$. The matrix of any automorphism in $\Sp_{2m}(K)$ with respect to $\{e_i\}$ is called a symplectic matrix.

Let $K$ be a field and $\Phi$ a non-degenerate skew-symmetric bilinear form on an $n$-dimensional vector space $E$ over $K$. If $n$ is even, then the symplectic group associated with $\Phi$ is isomorphic to $\Sp_{n}(K)$ and is generated by all linear transformations of $E$ of the form $\def\a{\alpha}\def\s{\sigma}\s_{e,\a}$, given by

$$x\mapsto \s_{e,\a}(x) = x+\a\Phi(e,x)e,$$ where $e\in E$, $\a\in K$. Linear transformations of the form $\s_{e,\a}$ are called symplectic transvections, or translations in the direction of the line $Ke$. The centre $Z$ of $\Sp_{n}(K)$ consists of the matrices $I_n$ and $-I_n$ if ${\rm char}\; K \ne 2$, and $Z=\{I_n\}$ if ${\rm char}\; K = 2$. The quotient group $\Sp_{n}(K)/Z$ is called the projective symplectic group and is denoted by $\def\PSp{ {\rm PSp}}\PSp_{n}(K)$. All projective symplectic groups are simple, except

$$\PSp_2(\F_2) = \Sp_2(\F_2),\quad \PSp_4(\F_2) = \Sp_4(\F_2) \textrm{ and }\PSp_2(\F_3)$$ (here $\F_q$ denotes the field of $q$ elements) and these are isomorphic to the symmetric groups $S_3$, $S_6$ (cf. Symmetric group) and the alternating group $A_4$, respectively. The order of $\Sp_{2m}(\F_q)$ is

$$q^{m^2}(q^2-1)\cdots(q^{2m-2}-1)(q^{2m}-1).$$ The symplectic group $\Sp_2(K)$ coincides with the special linear group ${\rm SL}_2(K)$. If ${\rm char}\; K \ne 2$, $\PSp_4(K)$ is isomorphic to the quotient group of $\def\Om{\Omega}\Om_5(K,f)$ by its centre, where $\Om_5(K,f)$ is the commutator subgroup of (index 2 in) the orthogonal group associated with a symmetric bilinear form $f$ in five variables.

Except when $m=2$ and ${\rm char}\; K = 2$, every automorphism $\def\phi{\varphi}\Phi$ of $\Sp_{2m}(K)$ can be written as

$$\phi(g)=h_1h_2g^\tau h_2^{-1}h_1^{-1},$$ where $\tau$ is an automorphism of the field $K$, $h_1\in\Sp_{2m}(K)$ and $h_2$ is a linear transformation of the space $E$, represented on the basis $\{e_i\}$ by a matrix of the form

$$\begin{pmatrix}I_m & 0 \\ 0 & \beta I_m\end{pmatrix}$$ ($\beta$ is a non-zero element of $K$).

$\Sp_{2m}(K)$ coincides with the group of $K$-points of the linear algebraic group $\Sp_{2m}$ defined by the equation $A^tJ_m A = J_m$. This algebraic group, also called a symplectic group, is a simple simply-connected linear algebraic group of type $C_m$ of dimension $2m^2+m$.

In the case when $K=\C$ or $\R$, $\Sp_{2m}(K)$ is a connected simple complex (respectively, real) Lie group. $\Sp_{2m}(\R)$ is one of the real forms of the complex symplectic group $\Sp_{2m}(\C)$. The other real forms of this group are also sometimes called symplectic groups. These are the subgroups $\Sp(p,q)$ of $\Sp_{2m}(\C)$, $p,q\ge 0$, $p+q=m$, consisting of those elements of $\Sp_{2m}(\C)$ that preserve the Hermitian form

$$\def\e{\epsilon}\sum_{i=1}^{2m} \e_i z_i\bar z_i,$$ where $\e_i=1$ for $1\le i\le p$ and $m+1\le i \le m+p$, and $\e_i=-1$ otherwise. The group $\Sp(0,m)$ is a compact real form of the complex symplectic group $\Sp_{2m}(\C)$. The symplectic group $\Sp(p,q)$ is isomorphic to the group of all linear transformations of the right vector space $\def\H{ {\mathbb H}}\H^m$ of dimension $m=p+q$ over the division ring $\H$ of quaternions that preserve the quaternionic Hermitian form of index $\min(p,q)$, that is, the form

$$(x,y) = \sum_{i=1}^p x_i\bar y_i - \sum_{i=p+1}^m x_i\bar y_i,$$ where

$$x=(x_1,\dots,x_m,\ y = (y_1,\dots,y_m) \in \H^m,$$ and the bar denotes conjugation of quaternions.

Comments

$\Sp_{2m}(\C)$ is also simply connected. But $\Sp_{2m}(\R)$ has the homotopy type of $S^1\times {\rm SU}_n$, so that $\pi_1(\Sp_{2m}(\R)) = \Z$. Here $S^1$ is the circle and ${\rm SU}_n$ is the special unitary group. The unitary symplectic group ${\rm USp}_{2m}(\C)$ is the intersection (in ${\rm GL}_{2m}(\C)$) of the unitary group ${\rm U}_{2m}$ and $\Sp_{2m}(\C)$. Topologically, $\Sp_{2m}(\C) \simeq {\rm USp}_{2m}(\C)\times \R^{2n^2+n}.$.

In Hamiltonian mechanics (cf. Hamilton equations) the phase space is a symplectic manifold, a manifold $M$ provided with a symplectic form (a closed differential form $\omega$ of degree $2$ which is non-degenerate at each point). If $M=T^* Q$, the cotangent bundle of a configuration space $Q$, with local coordinates $(q_1,\dots,q_n;p1,\dots,p_n)$, then the symplectic form $\sum_{j=1}^n dp_j\wedge dq_j$ is called canonical. The flow of a Hamiltonian system leaves the symplectic form invariant. As a consequence, its tangent mapping at a fixed point belongs to the symplectic group of the tangent space.

Cf. Symplectic homogeneous space; Symplectic structure.

References

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