Editing Symplectic group (section)

==={{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>