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In [[mathematics]], the name '''symplectic group''' can refer to two different, but closely related, collections of mathematical [[Group (mathematics)|groups]], denoted {{math|Sp(2''n'', ''F'')}} and {{math|Sp(''n'')}}. The latter is called the '''compact symplectic group'''.  Many authors prefer slightly different notations, usually differing by factors of {{math|2}}. The notation used here is consistent with the size of the most common [[Matrix (math)|matrices]] which represent the groups. In [[Élie Cartan|Cartan]]'s classification of the [[simple Lie algebra]]s, the Lie algebra of the complex group {{math|Sp(2''n'', '''C''')}} is denoted {{math|''C<sub>n</sub>''}}, and {{math|Sp(''n'')}} is the [[Real form (Lie theory)#Compact real form|compact real form]] of {{math|Sp(2''n'', '''C''')}}. 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 {{math|''n''}}.

The name "symplectic group" is [[Symplectic topology#Name|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".

=={{math|Sp(2''n'', ''F'')}}==
The symplectic group of degree {{math|2''n''}} over a [[field (mathematics)|field]] {{math|''F''}}, denoted {{math|Sp(2''n'', ''F'')}}, is the [[group (math)|group]] of {{math|2''n'' × 2''n''}} [[symplectic matrix|symplectic matrices]] with entries in {{math|''F''}} with the operation of [[matrix multiplication]]:<blockquote><math>\mathrm{Sp}(2n)\ = \ \{\,M\in M_{2n\times 2n}(F) \ \text{ with } \ M^T\!\Omega\, M = \Omega\,\},</math> </blockquote>where ''M''<sup>T</sup> is the [[transpose]] of ''M'', and Ω is a  [[Nonsingular matrix|nonsingular]] [[skew-symmetric matrix]], often taken to be:

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

where ''I<sub>n</sub>'' is the identity matrix''.'' Since all symplectic matrices have [[determinant]] {{math|1}}, the symplectic group is a [[subgroup]] of the [[special linear group]] {{math|SL(2''n'', ''F'')}}.

Notational warning: What is here called {{math|Sp(2''n'', ''F'')}} is often denoted {{math|Sp(''n'', ''F'')}}.

Defined more abstractly as a [[classical group]], the symplectic group is the set of [[linear transformations]] of a {{math|2''n''}}-dimensional [[vector space]] over {{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'')}}.

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/complex [[Lie group]] of real/complex dimension {{math|''n''(2''n'' + 1)}}. 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'')}}, denoted <math>\mathfrak{sp}(2n,F)</math>, is the set of {{math|2''n'' × 2''n''}} matrices ''A'' with entries in ''F'' satisfying<ref>{{harvnb|Hall|2015}} Prop. 3.25</ref> <math>\Omega A + A^\mathrm{T} \Omega = 0,</math> with the [[Commutator#Ring theory|matrix commutator]] as its Lie bracket: <math>[A,B] = AB-BA</math>.

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'')}}.

==={{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]].

==={{math|Sp(2''n'', '''R''')}}===
{{math|Sp(2''n'', '''C''')}} is the [[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 may be generated by the group multiplication of two elements.<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,\mathbb{R}) \; \exists \; X,Y \in \mathfrak{sp}(2n,\mathbb{R}) \text{ such that } 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,\mathbb{R})\cap\operatorname{SO}(2n) \cong U(n) \quad \text{and} \quad Z = \begin{pmatrix}D & 0 \\ 0 & D^{-1}\end{pmatrix}.</math>.<br/>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
:<math>  Q = \begin{pmatrix} A & B \\ C & -A^\mathrm{T} \end{pmatrix}  </math>

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>
:<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>

===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.

=={{math|Sp(''n'')}}==

The '''compact symplectic group'''<ref>{{harvnb|Hall|2015}} Section 1.2.8</ref> {{math|Sp(''n'')}} is the intersection of {{math|Sp(2''n'', '''C''')}} with the <math>2n\times 2n</math> unitary group:<math display="block">\mathrm{Sp}(n):=\mathrm{Sp}(2n;\mathbb C)\cap\mathrm{U}(2n).</math>

It is sometimes written as {{math|USp(2''n'')}}. Alternatively, {{math|Sp(''n'')}} can be described as the subgroup of {{math|GL(''n'', '''H''')}} (invertible [[quaternion]]ic matrices) that preserves the standard [[hermitian form]] on {{math|'''H'''<sup>''n''</sup>}}:
:<math>\langle x, y\rangle = \bar x_1 y_1 + \cdots + \bar x_n y_n</math>.
That is, {{math|Sp(''n'')}} is just the [[Classical group#Sp(p, q) – the quaternionic unitary group|quaternionic unitary group]], {{math|U(''n'', '''H''')}}.<ref>{{harvnb|Hall|2015}} p. 14</ref> Indeed, it is sometimes called the '''hyperunitary group'''. Also Sp(1) is the group of quaternions of norm {{math|1}}, equivalent to [[SU(2)|{{math|SU(2)}}]] and topologically a [[3-sphere|{{math|3}}-sphere]] {{math|S<sup>3</sup>}}.

Note that {{math|Sp(''n'')}} is ''not'' a symplectic group in the sense of the previous section&mdash;it does not preserve a non-degenerate skew-symmetric {{math|'''H'''}}-bilinear form on {{math|'''H'''<sup>''n''</sup>}}: there is no such form except the zero form. Rather, it is isomorphic to a subgroup of {{math|Sp(2''n'', '''C''')}}, and so does preserve a complex symplectic form in a vector space of dimension twice as high. As explained below, the Lie algebra of {{math|Sp(''n'')}} is the compact [[real form]] of the complex symplectic Lie algebra {{math|'''sp'''(2''n'', '''C''')}}.

{{math|Sp(''n'')}} is a real Lie group with (real) dimension {{math|''n''(2''n'' + 1)}}. It is [[Compact space|compact]], [[connected space|connected]], and [[simply connected]].<ref>{{harvnb|Hall|2015}} Prop. 13.12</ref>

The Lie algebra of {{math|Sp(''n'')}} is given by the quaternionic [[skew-Hermitian]] matrices, the set of {{math|''n''-by-''n''}} quaternionic matrices that satisfy
:<math>A+A^{\dagger} = 0</math>
where {{math|A<sup>†</sup>}} is the [[conjugate transpose]] of {{math|A}} (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.

===Important subgroups===
Some main subgroups are:

: <math>\operatorname{Sp}(n) \supset \operatorname{Sp}(n-1)</math>
: <math>\operatorname{Sp}(n) \supset \operatorname{U}(n) </math>
: <math>\operatorname{Sp}(2) \supset \operatorname{O}(4)</math>
Conversely it is itself a subgroup of some other groups:
: <math>\operatorname{SU}(2n) \supset \operatorname{Sp}(n)</math>
: <math>\operatorname{F}_4 \supset \operatorname{Sp}(4)</math>
: <math>\operatorname{G}_2 \supset \operatorname{Sp}(1)</math>

There are also the [[isomorphism]]s of the [[Lie algebras]] {{math|1='''sp'''(2) = '''so'''(5)}} and {{math|1='''sp'''(1) = '''so'''(3) = '''su'''(2)}}.

==Relationship between the symplectic groups==
Every complex, [[semisimple Lie algebra]] has a [[Real form (Lie theory)#Split real form|split real form]] and a [[Real form (Lie theory)#Compact real form|compact real form]]; the former is called a [[complexification]] of the latter two.

The Lie algebra of {{math|Sp(2''n'', '''C''')}} is [[Semisimple Lie algebra|semisimple]] and is denoted {{math|'''sp'''(2''n'', '''C''')}}. Its [[Real form (Lie theory)#Split real form|split real form]] is {{math|'''sp'''(2''n'', '''R''')}} and its [[Real form (Lie theory)#Compact real form|compact real form]] is {{math|'''sp'''(''n'')}}. These correspond to the Lie groups {{math|Sp(2''n'', '''R''')}} and {{math|Sp(''n'')}} respectively.

The algebras, {{math|'''sp'''(''p'', ''n'' − ''p'')}}, which are the Lie algebras of {{math|Sp(''p'', ''n'' − ''p'')}}, are the [[Metric signature|indefinite signature]] equivalent to the compact form.

==Physical significance==

===Classical mechanics===
The compact symplectic group {{math|Sp(''n'')}} comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket. 

Consider a system of {{math|''n''}} particles, evolving under [[Hamiltonian mechanics|Hamilton's equations]] whose position in [[phase space]] at a given time is denoted by the vector of [[canonical coordinates]],

:<math>\mathbf{z} = (q_1, \ldots , q_n, p_1, \ldots , p_n)^T.</math>

The elements of the group {{math|Sp(2''n'', '''R''')}} are, in a certain sense, [[canonical transformations]] on this vector, i.e. they preserve the form of [[Hamiltonian mechanics|Hamilton's equations]].<ref>{{harvnb|Arnold|1989}} gives an extensive mathematical overview of classical mechanics. See chapter 8 for [[symplectic manifold]]s.</ref> If

:<math>\mathbf{Z} = \mathbf Z(\mathbf z, t) = (Q_1, \ldots , Q_n, P_1, \ldots , P_n)^T</math>

are new canonical coordinates, then, with a dot denoting time derivative,

:<math>\dot {\mathbf Z} = M({\mathbf z}, t) \dot {\mathbf z},</math>

where

:<math>M(\mathbf z, t) \in \mathrm{Sp}(2n, \mathbb R)</math>

for all {{mvar|t}} and all {{math|'''z'''}} in phase space.<ref>{{harvnb|Goldstein|1980|loc=Section 9.3}}</ref>

===Quantum mechanics===
{{More citations needed section|date=October 2019}}
Consider a system of {{math|''n''}} particles whose [[quantum state]] encodes its position and momentum. These coordinates are continuous variables and hence the [[Hilbert space]], 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 picture|Heisenberg equation]] in [[phase space]].

Construct a vector of [[canonical coordinates]], 
:<math>\mathbf{\hat{z}} = (\hat{q}_1, \ldots , \hat{q}_n, \hat{p}_1, \ldots , \hat{p}_n)^T. </math>

The [[canonical commutation relation]] can be expressed simply as

:<math> [\mathbf{\hat{z}},\mathbf{\hat{z}}^T] = i\hbar\Omega </math>

where

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

and {{math|''I''<sub>''n''</sub>}} is the {{math|''n'' × ''n''}} identity matrix.

Many physical situations only require quadratic [[Hamiltonian (quantum mechanics)|Hamiltonians]], i.e. [[Hamiltonian (quantum mechanics)|Hamiltonians]] of the form

:<math>
\hat{H} = \frac{1}{2}\mathbf{\hat{z}}^TK\mathbf{\hat{z}}
</math>

where {{math|''K''}} is a {{math|2''n'' × 2''n''}} real, [[symmetric matrix]]. This turns out to be a useful restriction and allows us to rewrite the [[Heisenberg picture|Heisenberg equation]] as

:<math>
\frac{d\mathbf{\hat{z}}}{dt} = \Omega K \mathbf{\hat{z}}
</math>

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 [[Group action (mathematics)|action]] of [[Symplectic group#Sp.282n.2C R.29|the real symplectic group, {{math|Sp(2''n'', '''R''')}}]], on the phase space.

==See also==
* [[Orthogonal group]]
* [[Unitary group]]
* [[Projective unitary group]]
* [[Symplectic manifold]], [[Symplectic matrix]], [[Symplectic vector space]], [[Symplectic representation]] 
* [[Hamiltonian mechanics]]
* [[Metaplectic group]]
* [[Θ10]]

==Notes==
{{reflist}}

==References==
*{{citation|last=Arnold|first=V. I.|title=Mathematical Methods of Classical Mechanics|year=1989|edition=second|publisher=[[Springer-Verlag]]|series=[[Graduate Texts in Mathematics]]|volume=60|isbn=0-387-96890-3|authorlink=Vladimir Arnold}} 
*{{Citation| last=Hall|first=Brian C.|title=Lie groups, Lie algebras, and representations: An elementary introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn=978-3319134666}}
*{{citation|last=Fulton|first=W.|last2=Harris|first2=J.|authorlink=William Fulton (mathematician)|authorlink2=Joe Harris (mathematician)|title=Representation Theory, A first Course|year=1991|publisher=[[Springer-Verlag]]|series=[[Graduate Texts in Mathematics]]|volume=129|isbn=978-0-387-97495-8}}.
*{{cite book|ref=harv|last=Goldstein|first=H.|authorlink=Herbert Goldstein|title=Classical Mechanics|edition=2nd|publisher=[[Addison-Wesley Publishing Company|Addison-Wesley]]|location=Reading MA|isbn=0-201-02918-9|year=1980|orig-year=1950|chapter=Chapter 7}}
*{{citation|last = Lee|first = J. M.|title=Introduction to Smooth manifolds|year=2003|publisher=[[Springer-Verlag]]|series=[[Graduate Texts in Mathematics]]|isbn=0-387-95448-1|volume=218}}
*{{citation|last=Rossmann|first= Wulf|title=Lie Groups – An Introduction Through Linear Groups|publisher=Oxford Science Publications|year=2002|series=Oxford Graduate Texts in Mathematics|isbn=0 19 859683 9}}
*{{citation|last=Ferraro|first=Alessandro|last2=Olivares|first2=Stefano|last3=Paris|first3=Matteo G. A.|title=Gaussian states in continuous variable quantum information| date=March 2005 |publisher=[[arXiv|arXiv.org]]|arxiv=quant-ph/0503237|bibcode=2005quant.ph..3237F}}.

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[[Category:Lie groups]]
[[Category:Symplectic geometry]]