Editing Symplectic group (section)

=={{math|Sp(2''n'', '''F''')}}==
The symplectic group is a [[classical group]] defined as the set of [[linear transformations]] of a {{math|2''n''}}-dimensional [[vector space]] over the field {{math|'''F'''}} which preserve a [[nondegenerate form|non-degenerate]] [[skew-symmetric matrix|skew-symmetric]] [[bilinear form]]. Such a vector space is called a [[symplectic vector space]], and the symplectic group of an abstract symplectic vector space {{math|''V''}} is denoted {{math|Sp(''V'')}}.  Upon fixing a basis for {{math|''V''}}, the symplectic group becomes the group of {{math|2''n'' × 2''n''}} [[symplectic matrix|symplectic matrices]], with entries in {{math|'''F'''}}, under the operation of [[matrix multiplication]].  This group is denoted either {{math|Sp(2''n'', '''F''')}} or {{math|Sp(''n'', '''F''')}}.  If the bilinear form is represented by the [[nonsingular matrix|nonsingular]] [[skew-symmetric matrix]] Ω, then

:<math>\operatorname{Sp}(2n, F) = \{M \in M_{2n \times 2n}(F) : M^\mathrm{T} \Omega M = \Omega\},</math>

where ''M''<sup>T</sup> is the [[transpose]] of ''M''.  Often Ω is defined to be

:<math>\Omega = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \\ \end{pmatrix},</math>

where ''I<sub>n</sub>'' is the identity matrix.  In this case, {{math|Sp(2''n'', '''F''')}} can be expressed as those block matrices <math>(\begin{smallmatrix} A & B \\ C & D \end{smallmatrix})</math>, where <math>A, B, C, D \in M_{n \times n}(F)</math>, satisfying the three equations:

:<math>\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}</math>

Since all symplectic matrices have [[determinant]] {{math|1}}, the symplectic group is a [[subgroup]] of the [[special linear group]] {{math|SL(2''n'', '''F''')}}.  When {{math|1=''n'' = 1}}, the symplectic condition on a matrix is satisfied [[if and only if]] the determinant is one, so that {{math|1=Sp(2, '''F''') = SL(2, '''F''')}}. For {{math|''n'' > 1}}, there are additional conditions, i.e. {{math|Sp(2''n'', '''F''')}} is then a proper subgroup of {{math|SL(2''n'', '''F''')}}.

Typically, the field {{math|'''F'''}} is the field of [[real number]]s {{math|'''R'''}} or [[complex number]]s {{math|'''C'''}}. In these cases {{math|Sp(2''n'', '''F''')}} is a real or complex [[Lie group]] of real or complex dimension {{math|''n''(2''n'' + 1)}}, respectively. These groups are [[connected space|connected]] but [[Compact group|non-compact]].

The [[Center (group theory)|center]] of {{math|Sp(2''n'', '''F''')}} consists of the matrices {{math|''I''<sub>2''n''</sub>}} and {{math|−''I''<sub>2''n''</sub>}} as long as the [[Characteristic (algebra)|characteristic of the field]] is not {{math|2}}.<ref>[http://www.encyclopediaofmath.org/index.php/Symplectic_group "Symplectic group"], ''[[Encyclopedia of Mathematics]]'' Retrieved on 13 December 2014.</ref> Since the center of {{math|Sp(2''n'', '''F''')}} is discrete and its quotient modulo the center is a [[simple group]], {{math|Sp(2''n'', '''F''')}} is considered a [[Simple Lie group#Comments on the definition|simple Lie group]].

The real rank of the corresponding Lie algebra, and hence of the Lie group {{math|Sp(2''n'', '''F''')}}, is {{math|''n''}}.

The [[Lie algebra]] of {{math|Sp(2''n'', '''F''')}} is the set

:<math>\mathfrak{sp}(2n,F) = \{X \in M_{2n \times 2n}(F) : \Omega X + X^\mathrm{T} \Omega = 0\},</math>

equipped with the [[Commutator#Ring theory|commutator]] as its Lie bracket.<ref>{{harvnb|Hall|2015}} Prop. 3.25</ref>  For the standard skew-symmetric bilinear form <math>\Omega = (\begin{smallmatrix} 0 & I \\ -I & 0 \end{smallmatrix})</math>, this Lie algebra is the set of all block matrices <math>(\begin{smallmatrix} A & B \\ C & D \end{smallmatrix})</math> subject to the conditions

:<math>\begin{align}
A &= -D^\mathrm{T}, \\
B &= B^\mathrm{T}, \\
C &= C^\mathrm{T}.
\end{align}</math>

==={{math|Sp(2''n'', '''C''')}}===
The symplectic group over the field of complex numbers is a [[Compact group|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.{{sfn|Hall|2015|p=10}}

==={{math|Sp(2''n'', '''R''')}}===
{{math|Sp(''n'', '''C''')}} is the [[Complexification (Lie group)|complexification]] of the real group {{math|Sp(2''n'', '''R''')}}. {{math|Sp(2''n'', '''R''')}} is a real, [[Compact group|non-compact]], [[Connected space|connected]], [[simple Lie group]].<ref>[https://math.stackexchange.com/q/1051400 "Is the symplectic group Sp(2''n'', '''R''') simple?"], ''[[Stack Exchange]]'' Retrieved on 14 December 2014.</ref> It has a [[fundamental group]] [[Group isomorphism|isomorphic]] to the group of [[integers]] under addition. As the [[real form]] of a [[simple Lie group]] its Lie algebra is a [[Split Lie algebra|splittable Lie algebra]].

Some further properties of {{math|Sp(2''n'', '''R''')}}:

* The [[exponential map (Lie theory)|exponential map]] from the [[Lie algebra]] {{math|'''sp'''(2''n'', '''R''')}} to the group {{math|Sp(2''n'', '''R''')}} is not [[Surjective function|surjective]]. However, any element of the group can be represented as the product of two exponentials.<ref>[https://math.stackexchange.com/q/1051255 "Is the exponential map for Sp(2''n'', '''R''') surjective?"], ''[[Stack Exchange]]'' Retrieved on 5 December 2014.</ref> In other words,
::<math>\forall S \in \operatorname{Sp}(2n,\mathbf{R})\,\, \exists X,Y \in \mathfrak{sp}(2n,\mathbf{R}) \,\, S = e^Xe^Y. </math>

* For all {{math|''S''}} in {{math|Sp(2''n'', '''R''')}}:
::<math>S = OZO' \quad \text{such that} \quad O, O' \in \operatorname{Sp}(2n,\mathbf{R})\cap\operatorname{SO}(2n) \cong U(n) \quad \text{and} \quad Z = \begin{pmatrix}D & 0 \\ 0 & D^{-1}\end{pmatrix}.</math>
:The matrix {{math|''D''}} is [[Positive-definite matrix|positive-definite]] and [[Diagonal matrix|diagonal]]. The set of such {{math|''Z''}}s forms a non-compact subgroup of {{math|Sp(2''n'', '''R''')}} whereas {{math|U(''n'')}} forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition.<ref>[https://www.maths.nottingham.ac.uk/personal/ga/papers/2602.pdf "Standard forms and entanglement engineering of multimode Gaussian states under local operations – Serafini and Adesso"], Retrieved on 30 January 2015.</ref> Further [[symplectic matrix]] properties can be found on that Wikipedia page.

* As a [[Lie group]], {{math|Sp(2''n'', '''R''')}} has a manifold structure. The [[manifold]] for {{math|Sp(2''n'', '''R''')}} is [[Diffeomorphism|diffeomorphic]] to the [[Manifold#Cartesian products|Cartesian product]] of the [[unitary group]] {{math|U(''n'')}} with a [[vector space]] of dimension {{math|''n''(''n''+1)}}.<ref>[http://www.maths.ed.ac.uk/~aar/papers/arnogive.pdf "Symplectic Geometry – Arnol'd and Givental"], Retrieved on 30 January 2015.</ref>

===Infinitesimal generators===
The members of the symplectic Lie algebra {{math|'''sp'''(2''n'', '''F''')}}  are the [[Hamiltonian matrix|Hamiltonian matrices]].

These are matrices, <math>Q</math> such that<blockquote><math>Q = \begin{pmatrix} A & B \\ C & -A^\mathrm{T} \end{pmatrix}</math></blockquote>where {{math|''B''}} and {{math|''C''}} are [[Symmetric matrix|symmetric matrices]]. See [[classical group]] for a derivation.

===Example of symplectic matrices===
For {{math|Sp(2, '''R''')}}, the group of {{math|2 × 2}} matrices with determinant {{math|1}}, the three symplectic {{math|(0, 1)}}-matrices are:<ref>[http://mathworld.wolfram.com/SymplecticGroup.html Symplectic Group], (source: [[Wolfram MathWorld]]), downloaded February 14, 2012</ref><blockquote><math>\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}. </math></blockquote>

==== Sp(2n, R) ====
It turns out that <math>\operatorname{Sp}(2n,\mathbf{R})</math> can have a fairly explicit description using generators. If we let <math>\operatorname{Sym}(n)</math> denote the symmetric <math>n\times n</math> matrices, then <math>\operatorname{Sp}(2n,\mathbf{R})</math> is generated by <math>D(n)\cup N(n) \cup \{\Omega\},</math> where<blockquote><math>\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}</math></blockquote>are subgroups of <math>\operatorname{Sp}(2n,\mathbf{R})</math><ref>{{Cite book|last=Gerald B. Folland.|url=https://www.worldcat.org/oclc/945482850|title=Harmonic analysis in phase space|date=2016|publisher=Princeton Univ Press|isbn=978-1-4008-8242-7|location=Princeton|page=173|oclc=945482850}}</ref><sup>pg 173</sup><ref>{{Cite book|last=Habermann, Katharina, 1966-|url=http://worldcat.org/oclc/262692314|title=Introduction to symplectic Dirac operators|date=2006|publisher=Springer|isbn=978-3-540-33421-7|oclc=262692314}}</ref><sup>pg 2</sup>.

===Relationship with symplectic geometry===
[[Symplectic geometry]] is the study of [[symplectic manifold]]s. The [[tangent space]] at any point on a symplectic manifold is a [[symplectic vector space]].<ref>[https://empg.maths.ed.ac.uk/Activities/BRST/ "Lecture Notes – Lecture 2: Symplectic reduction"], Retrieved on 30 January 2015.</ref> As noted earlier, structure preserving transformations of a symplectic vector space form a [[Group (mathematics)|group]] and this group is {{math|Sp(2''n'', '''F''')}}, depending on the dimension of the space and the [[Field (mathematics)|field]] over which it is defined.

A symplectic vector space is itself a symplectic manifold. A transformation under an [[Group action (mathematics)|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.