Symplectic group
One of the classical groups, defined as the group of automorphisms of a skew-symmetric bilinear form 
on a left 
-module 
, where 
is a commutative ring (cf. Classical group). In the case when 
and the matrix of 
with respect to the canonical basis 
of 
has the form
 |
where 
is the identity matrix of order 
, the corresponding symplectic group is called the symplectic group of 
variables over the ring 
and is denoted by 
or 
. The matrix of any automorphism in 
with respect to 
is called a symplectic matrix.
Let 
be a field and 
a non-degenerate skew-symmetric bilinear form on an 
-dimensional vector space 
over 
. If 
is even, then the symplectic group associated with 
is isomorphic to 
and is generated by all linear transformations of 
of the form 
, given by
 |
where 
, 
. Linear transformations of the form 
are called symplectic transvections, or translations in the direction of the line 
. The centre 
of 
consists of the matrices 
and 
if 
, and 
if 
. The quotient group 
is called the projective symplectic group and is denoted by 
. All projective symplectic groups are simple, except
 |
(here 
denotes the field of 
elements) and these are isomorphic to the symmetric groups 
, 
(cf. Symmetric group) and the alternating group 
, respectively. The order of 
is
 |
The symplectic group 
coincides with the special linear group 
. If 
, 
is isomorphic to the quotient group of 
by its centre, where 
is the commutator subgroup of (index 2 in) the orthogonal group associated with a symmetric bilinear form 
in five variables.
Except when 
and 
, every automorphism 
of 
can be written as
 |
where 
is an automorphism of the field 
, 
and 
is a linear transformation of the space 
, represented on the basis 
by a matrix of the form
 |
(
is a non-zero element of 
).
coincides with the group of 
-points of the linear algebraic group 
defined by the equation 
. This algebraic group, also called a symplectic group, is a simple simply-connected linear algebraic group of type 
of dimension 
.
In the case when 
or 
, 
is a connected simple complex (respectively, real) Lie group. 
is one of the real forms of the complex symplectic group 
. The other real forms of this group are also sometimes called symplectic groups. These are the subgroups 
of 
, 
, 
, consisting of those elements of 
that preserve the Hermitian form
 |
where 
for 
and 
, and 
otherwise. The group 
is a compact real form of the complex symplectic group 
. The symplectic group 
is isomorphic to the group of all linear transformations of the right vector space 
of dimension 
over the division ring 
of quaternions that preserve the quaternionic Hermitian form of index 
, that is, the form
 |
where
 |
and the bar denotes conjugation of quaternions.
References
Comments
is also simply connected. But 
has the homotopy type of 
, so that 
. Here 
is the circle and 
is the special unitary group. The unitary symplectic group 
is the intersection (in 
) of the unitary group 
and 
. Topologically, 
.
In Hamiltonian mechanics (cf. Hamilton equations) the phase space is a symplectic manifold, a manifold 
provided with a symplectic form (a closed differential form 
of degree 
which is non-degenerate at each point). If 
, the cotangent bundle of a configuration space 
, with local coordinates 
, then the symplectic form 
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.
Symplectic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symplectic_group&oldid=17707