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 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
[Ar] | E. Artin, "Geometric algebra", Interscience (1957) MR1529733 MR0082463 Zbl 0077.02101 |
[Bo] | N. Bourbaki, "Algebra", Elements of mathematics, 1, Addison-Wesley (1973) (Translated from French) MR0354207 Zbl 0281.00006 |
[Ch] | C. Chevalley, "Theory of Lie groups", 1, Princeton Univ. Press (1946) MR0082628 MR0015396 Zbl 0063.00842 |
[Di] | J.A. Dieudonné, "La géométrie des groups classiques", Springer (1955) Zbl 0221.20056 |
[He] | S. Helgason, "Differential geometry and symmetric spaces", Acad. Press (1962) MR0145455 Zbl 0111.18101 |
Symplectic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symplectic_group&oldid=30670