In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the name symplectic group can refer to two different, but closely related, collections of mathematical
groups, denoted and for positive integer ''n'' and
field F (usually C or R). The latter is called the compact symplectic group and is also denoted by
. Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common
matrices which represent the groups. In
Cartan's classification of the
simple Lie algebras, the Lie algebra of the complex group is denoted , and is the
compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension .
The name "symplectic group" is
due to Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex".
The
metaplectic group is a double cover of the symplectic group over R; it has analogues over other
local fields,
finite fields, and
adele rings.
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
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
. 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 group becomes the group of
symplectic matrices, with entries in , under the operation of
matrix multiplication. This group is denoted either or . If the bilinear form is represented by the
nonsingular skew-symmetric matrix Ω, then
:
where ''M''
T is the
transpose of ''M''. Often Ω is defined to be
:
where ''I
n'' is the identity matrix. In this case, can be expressed as those block matrices
, where
, satisfying the three equations:
:
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/complex
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
of real/complex dimension . 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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of is the set
:
equipped with the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
as its Lie bracket. For the standard skew-symmetric bilinear form
, this Lie algebra is the set of all block matrices
subject to the conditions
:
The symplectic group over the field of complex numbers is a
non-compact,
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
,
simple Lie group.
is the
complexification of the real group . is a real,
non-compact,
connected,
simple Lie group. It has a
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the group of
integers under addition. As the
real form of a
simple Lie group its Lie algebra is a
splittable Lie algebra.
Some further properties of :
* The
exponential map from the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
to the group is not
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
. However, any element of the group can be represented as the product of two exponentials. In other words,
::
* For all in :
::
:The matrix is
positive-definite and
diagonal. The set of such s forms a non-compact subgroup of whereas forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition. Further
symplectic matrix In mathematics, a symplectic matrix is a 2n\times 2n matrix M with real entries that satisfies the condition
where M^\text denotes the transpose of M and \Omega is a fixed 2n\times 2n nonsingular, skew-symmetric matrix. This definition can be ext ...
properties can be found on that Wikipedia page.
* As a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
, has a manifold structure. The
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
for is
diffeomorphic to the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
of the
unitary group with a
vector space of dimension .
Infinitesimal generators
The members of the symplectic Lie algebra are the
Hamiltonian matrices.
These are matrices,
such that
where and are
symmetric matrices. See
classical group for a derivation.
Example of symplectic matrices
For , the group of matrices with determinant , the three symplectic -matrices are:
Sp(2n, R)
It turns out that
can have a fairly explicit description using generators. If we let
denote the symmetric
matrices, then
is generated by
where
are subgroups of
pg 173pg 2.
Relationship with symplectic geometry
Symplectic geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate different ...
is the study of
symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
s. 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.
The compact symplectic group is the intersection of with the unitary group:
:
It is sometimes written as . Alternatively, can be described as the subgroup of (invertible
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
ic matrices) that preserves the standard
hermitian form on :
:
That is, is just the
quaternionic unitary group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic grou ...
, . Indeed, it is sometimes called the hyperunitary group. Also Sp(1) is the group of quaternions of norm , equivalent to and topologically a
-sphere .
Note that is ''not'' a symplectic group in the sense of the previous section—it does not preserve a non-degenerate skew-symmetric -bilinear form on : there is no such form except the zero form. Rather, it is isomorphic to a subgroup of , and so does preserve a complex symplectic form in a vector space of twice the dimension. As explained below, the Lie algebra of is the compact
real form of the complex symplectic Lie algebra .
is a real Lie group with (real) dimension . It is
compact and
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
.
The Lie algebra of is given by the quaternionic
skew-Hermitian matrices, the set of quaternionic matrices that satisfy
:
where is the
conjugate transpose of (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.
Important subgroups
Some main subgroups are:
:
:
:
Conversely it is itself a subgroup of some other groups:
:
:
:
There are also the
isomorphisms of the
Lie algebras and .
Relationship between the symplectic groups
Every complex,
semisimple Lie algebra has a
split real form and a
compact real form; the former is called a
complexification of the latter two.
The Lie algebra of is
semisimple and is denoted . Its
split real form is and its
compact real form is . These correspond to the Lie groups and respectively.
The algebras, , which are the Lie algebras of , are the
indefinite signature equivalent to the compact form.
Physical significance
Classical mechanics
The compact symplectic group comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket.
Consider a system of particles, evolving under
Hamilton's equations whose position in
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
at a given time is denoted by the vector of
canonical coordinates
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of ...
,
:
The elements of the group are, in a certain sense,
canonical transformations on this vector, i.e. they preserve the form of
Hamilton's equations.
If
:
are new canonical coordinates, then, with a dot denoting time derivative,
:
where
:
for all and all in phase space.
For the special case of a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
, Hamilton's equations describe the
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s on that manifold. The coordinates
live in the
tangent bundle to the manifold, and the momenta
live in the
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
. This is the reason why these are conventionally written with upper and lower indexes; it is to distinguish their locations. The corresponding Hamiltonian consists purely of the kinetic energy: it is
where
is the inverse of the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
on the Riemannian manifold.
[ Ralph Abraham and ]Jerrold E. Marsden
Jerrold Eldon Marsden (August 17, 1942 – September 21, 2010) was a Canadian mathematician. He was the Carl F. Braun Professor of Engineering and Control & Dynamical Systems at the California Institute of Technology.. Marsden is listed as an ISI ...
, ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London In fact, the cotangent bundle of ''any'' smooth manifold can be a given a (non-trivial)
symplectic structure in a canonical way, with the symplectic form defined as the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
of the
tautological one-form.
Quantum mechanics
Consider a system of particles whose
quantum state encodes its position and momentum. These coordinates are continuous variables and hence the
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, in which the state lives, is infinite-dimensional. This often makes the analysis of this situation tricky. An alternative approach is to consider the evolution of the position and momentum operators under the
Heisenberg equation in
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
.
Construct a vector of
canonical coordinates
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of ...
,
:
The
canonical commutation relation can be expressed simply as
:
where
:
and is the identity matrix.
Many physical situations only require quadratic
Hamiltonians, i.e.
Hamiltonians of the form
:
where is a real,
symmetric matrix. This turns out to be a useful restriction and allows us to rewrite the
Heisenberg equation as
:
The solution to this equation must preserve the
canonical commutation relation. It can be shown that the time evolution of this system is equivalent to an
action of
the real symplectic group, , on the phase space.
See also
*
Orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
*
Unitary group
*
Projective unitary group
*
Symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
,
Symplectic matrix In mathematics, a symplectic matrix is a 2n\times 2n matrix M with real entries that satisfies the condition
where M^\text denotes the transpose of M and \Omega is a fixed 2n\times 2n nonsingular, skew-symmetric matrix. This definition can be ext ...
,
Symplectic vector space,
Symplectic representation In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (''V'', ''ω'') which preserves the symplectic form ''ω''. Here ''ω'' is a nondegenerate ske ...
*
Representations of classical Lie groups
*
Hamiltonian mechanics
*
Metaplectic group
*
Θ10
In representation theory, a branch of mathematics, θ10 is a cuspidal unipotent complex irreducible representation of the symplectic group Sp4 over a finite, local, or global field.
introduced θ10 for the symplectic group Sp4(F''q'') over a fini ...
Notes
References
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Lie groups
Symplectic geometry