Lie groups andLie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups inphysics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
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Algebraic structure →Group theory Group theory
Basic notions Subgroup Normal subgroup Group action Quotient group (Semi-)direct product Direct sum Free product Wreath product Group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb{Z}}$) Free group Modular groups PSL(2,${\displaystyle \mathbb{Z}}$) SL(2,${\displaystyle \mathbb{Z}}$) Arithmetic group Lattice Hyperbolic group
Topological andLie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
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Inmathematics, the namesymplectic group can refer to two different, but closely related, collections of mathematicalgroups, denotedSp(2n,F) andSp(n) for positive integern andfieldF (usuallyC orR). The latter is called thecompact symplectic group and is also denoted by${\displaystyle \mathrm{U}\mathrm{S}\mathrm{p}(n)}$. Many authors prefer slightly different notations, usually differing by factors of2. The notation used here is consistent with the size of the most commonmatrices which represent the groups. InCartan's classification of thesimple Lie algebras, the Lie algebra of the complex groupSp(2n,C) is denotedC_n, andSp(n) is thecompact real form ofSp(2n,C). Note that when we refer tothe (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimensionn.

The name "symplectic group" was coined byHermann Weyl as a replacement for the previous confusing names (line)complex group andAbelian linear group, and is the Greek analog of "complex".

Themetaplectic group is a double cover of the symplectic group overR; it has analogues over otherlocal fields,finite fields, andadele rings.

The symplectic group is aclassical group defined as the set oflinear transformations of a2n-dimensionalvector space over the fieldF which preserve anon-degenerateskew-symmetricbilinear form. Such a vector space is called asymplectic vector space, and the symplectic group of an abstract symplectic vector spaceV is denotedSp(V). Upon fixing a basis forV, the symplectic group becomes the group of2n × 2nsymplectic matrices, with entries inF, under the operation ofmatrix multiplication. This group is denoted eitherSp(2n,F) orSp(n,F). If the bilinear form is represented by thenonsingularskew-symmetric matrix Ω, then

${\displaystyle \mathrm{Sp}(2n,F)=\{M\in M_{2n\times 2n}(F):M^{\mathrm{T}}\mathrm{\Omega }M=\mathrm{\Omega }\},}$

whereM^T is thetranspose ofM. Often Ω is defined to be

whereI_n is the identity matrix. In this case,Sp(2n,F) can be expressed as those block matrices, where${\displaystyle A,B,C,D\in M_{n\times n}(F)}$, satisfying the three equations:

Since all symplectic matrices havedeterminant1, the symplectic group is asubgroup of thespecial linear groupSL(2n,F). Whenn = 1, the symplectic condition on a matrix is satisfiedif and only if the determinant is one, so thatSp(2,F) = SL(2,F). Forn > 1, there are additional conditions, i.e.Sp(2n,F) is then a proper subgroup ofSL(2n,F).

Typically, the fieldF is the field ofreal numbersR orcomplex numbersC. In these casesSp(2n,F) is a real or complexLie group of real or complex dimensionn(2n + 1), respectively. These groups areconnected butnon-compact.

Thecenter ofSp(2n,F) consists of the matricesI_2n and−I_2n as long as thecharacteristic of the field is not2.^[1] Since the center ofSp(2n,F) is discrete and its quotient modulo the center is asimple group,Sp(2n,F) is considered asimple Lie group.

The real rank of the corresponding Lie algebra, and hence of the Lie groupSp(2n,F), isn.

TheLie algebra ofSp(2n,F) is the set

${\displaystyle \mathfrak{s}\mathfrak{p}(2n,F)=\{X\in M_{2n\times 2n}(F):\mathrm{\Omega }X+X^{\mathrm{T}}\mathrm{\Omega }=0\},}$

equipped with thecommutator as its Lie bracket.^[2] 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 anon-compact,simply connected,simple Lie group. The definition of this group includesno conjugates (contrary to what one might naively expect) but instead it is exactly the same as the definition bar the field change.^[3]

Sp(n,C) is thecomplexification of the real groupSp(2n,R).Sp(2n,R) is a real,non-compact,connected,simple Lie group.^[4] It has afundamental groupisomorphic to the group ofintegers under addition. As thereal form of asimple Lie group its Lie algebra is asplittable Lie algebra.

Some further properties ofSp(2n,R):

• Theexponential map from theLie algebrasp(2n,R) to the groupSp(2n,R) is notsurjective. However, any element of the group can be represented as the product of two exponentials.^[5] In other words,

${\displaystyle \mathrm{\forall }S\in \mathrm{Sp}(2n,\mathbf{R})\mathrm{\exists }X,Y\in \mathfrak{s}\mathfrak{p}(2n,\mathbf{R})S=e^X{e}^Y.}$

• For allS inSp(2n,R):

The matrixD ispositive-definite anddiagonal. The set of suchZs forms a non-compact subgroup ofSp(2n,R) whereasU(n) forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition.^[6] Furthersymplectic matrix properties can be found on that Wikipedia page.

• As aLie group,Sp(2n,R) has a manifold structure. Themanifold forSp(2n,R) isdiffeomorphic to theCartesian product of theunitary groupU(n) with avector space of dimensionn(n+1).^[7]

The members of the symplectic Lie algebrasp(2n,F) are theHamiltonian matrices.

These are matrices,${\displaystyle Q}$ such that

whereB andC aresymmetric matrices. Seeclassical group for a derivation.

ForSp(2,R), the group of2 × 2 matrices with determinant1, the three symplectic(0, 1)-matrices are:^[8]

It turns out that${\displaystyle \mathrm{Sp}(2n,\mathbf{R})}$ can have a fairly explicit description using generators. If we let${\displaystyle \mathrm{Sym}(n)}$ denote the symmetric${\displaystyle n\times n}$ matrices, then${\displaystyle \mathrm{Sp}(2n,\mathbf{R})}$ is generated by${\displaystyle D(n)\cup N(n)\cup \{\mathrm{\Omega }\},}$ where

are subgroups of${\displaystyle \mathrm{Sp}(2n,\mathbf{R})}$^{[9]pg 173[10]pg 2}.

Symplectic geometry is the study ofsymplectic manifolds. Thetangent space at any point on a symplectic manifold is asymplectic vector space.^[11] As noted earlier, structure preserving transformations of a symplectic vector space form agroup and this group isSp(2n,F), depending on the dimension of the space and thefield over which it is defined.

A symplectic vector space is itself a symplectic manifold. A transformation under anaction of the symplectic group is thus, in a sense, a linearised version of asymplectomorphism which is a more general structure preserving transformation on a symplectic manifold.

Thecompact symplectic group^[12]Sp(n) is the intersection ofSp(2n,C) with the${\displaystyle 2n\times 2n}$ unitary group:

${\displaystyle \mathrm{Sp}(n):=\mathrm{Sp}(2n;\mathbf{C})\cap \mathrm{U}(2n)=\mathrm{Sp}(2n;\mathbf{C})\cap \mathrm{SU}(2n).}$

It is sometimes written asUSp(2n). Alternatively,Sp(n) can be described as the subgroup ofGL(n,H) (invertiblequaternionic matrices) that preserves the standardhermitian form onHⁿ:

${\displaystyle ⟨x,y⟩={\overline{x}}_1{y}_1+\cdots +{\overline{x}}_n{y}_n.}$

That is,Sp(n) is just thequaternionic unitary group,U(n,H).^[13] Indeed, it is sometimes called thehyperunitary group. Also Sp(1) is the group of quaternions of norm1, equivalent toSU(2) and topologically a3-sphereS³.

Note thatSp(n) isnot a symplectic group in the sense of the previous section—it does not preserve a non-degenerate skew-symmetricH-bilinear form onHⁿ: there is no such form except the zero form. Rather, it is isomorphic to a subgroup ofSp(2n,C), and so does preserve a complex symplectic form in a vector space of twice the dimension. As explained below, the Lie algebra ofSp(n) is the compactreal form of the complex symplectic Lie algebrasp(2n,C).

Sp(n) is a real Lie group with (real) dimensionn(2n + 1). It iscompact andsimply connected.^[14]

The Lie algebra ofSp(n) is given by the quaternionicskew-Hermitian matrices, the set ofn-by-n quaternionic matrices that satisfy

${\displaystyle A+A^{†}=0}$

whereA^† is theconjugate transpose ofA (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.

Some main subgroups are:

${\displaystyle \mathrm{Sp}(n)\supset \mathrm{Sp}(n-1)}$

${\displaystyle \mathrm{Sp}(n)\supset \mathrm{U}(n)}$

${\displaystyle \mathrm{Sp}(2)\supset \mathrm{O}(4)}$

Conversely it is itself a subgroup of some other groups:

${\displaystyle \mathrm{SU}(2n)\supset \mathrm{Sp}(n)}$

${\displaystyle {\mathrm{F}}_4\supset \mathrm{Sp}(4)}$

${\displaystyle {\mathrm{G}}_2\supset \mathrm{Sp}(1)}$

There are also theisomorphisms of theLie algebrassp(2) =so(5) andsp(1) =so(3) =su(2).

Every complex,semisimple Lie algebra has asplit real form and acompact real form; the former is called acomplexification of the latter two.

The Lie algebra ofSp(2n,C) issemisimple and is denotedsp(2n,C). Itssplit real form issp(2n,R) and itscompact real form issp(n). These correspond to the Lie groupsSp(2n,R) andSp(n) respectively.

The algebras,sp(p,n −p), which are the Lie algebras ofSp(p,n −p), are theindefinite signature equivalent to the compact form.

The non-compact symplectic groupSp(2n,R) comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket.

Consider a system ofn particles, evolving underHamilton's equations whose position inphase space at a given time is denoted by the vector ofcanonical coordinates,

${\displaystyle \mathbf{z}=(q^1,\dots ,q^n,p_1,\dots ,p_n{)}^{\mathrm{T}}.}$

The elements of the groupSp(2n,R) are, in a certain sense,canonical transformations on this vector, i.e. they preserve the form ofHamilton's equations.^[15][16] If

${\displaystyle \mathbf{Z}=\mathbf{Z}(\mathbf{z},t)=(Q^1,\dots ,Q^n,P_1,\dots ,P_n{)}^{\mathrm{T}}}$

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

${\displaystyle \dot{\mathbf{Z}}=M(\mathbf{z},t)\dot{\mathbf{z}},}$

where

${\displaystyle M(\mathbf{z},t)\in \mathrm{Sp}(2n,\mathbf{R})}$

for allt and allz in phase space.^[17]

For the special case of aRiemannian manifold, Hamilton's equations describe thegeodesics on that manifold. The coordinates${\displaystyle q^i}$ live on the underlying manifold, and the momenta${\displaystyle p_i}$ live in thecotangent bundle. 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${\displaystyle H=\frac{1}{2}{g}^{ij}(q)p_i{p}_j}$ where${\displaystyle g^{ij}}$ is the inverse of themetric tensor${\displaystyle g_{ij}}$ on the Riemannian manifold.^[18][16] In fact, the cotangent bundle ofany smooth manifold can be a given asymplectic structure in a canonical way, with the symplectic form defined as theexterior derivative of thetautological one-form.^[19]

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Consider a system ofn particles whosequantum state encodes its position and momentum. These coordinates are continuous variables and hence theHilbert 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 theHeisenberg equation inphase space.

Construct a vector ofcanonical coordinates,

${\displaystyle \hat{\mathbf{z}}=({\hat{q}}^1,\dots ,{\hat{q}}^n,{\hat{p}}_1,\dots ,{\hat{p}}_n{)}^{\mathrm{T}}.}$

Thecanonical commutation relation can be expressed simply as

${\displaystyle [\hat{\mathbf{z}},{\hat{\mathbf{z}}}^{\mathrm{T}}]=i\hslash \mathrm{\Omega }}$

where

andI_n is then ×n identity matrix.

Many physical situations only require quadraticHamiltonians, i.e.Hamiltonians of the form

${\displaystyle \hat{H}=\frac{1}{2}{\hat{\mathbf{z}}}^{\mathrm{T}}K\hat{\mathbf{z}}}$

whereK is a2n × 2n real,symmetric matrix. This turns out to be a useful restriction and allows us to rewrite theHeisenberg equation as

${\displaystyle \frac{d\hat{\mathbf{z}}}{dt}=\mathrm{\Omega }K\hat{\mathbf{z}}}$

The solution to this equation must preserve thecanonical commutation relation. It can be shown that the time evolution of this system is equivalent to anaction ofthe real symplectic group,Sp(2n,R), on the phase space.

• Hamiltonian mechanics
• Metaplectic group
• Orthogonal group
• Paramodular group
• Projective unitary group
• Representations of classical Lie groups
• Symplectic manifold,Symplectic matrix,Symplectic vector space,Symplectic representation
• Unitary group
• Θ10

• Arnold, V. I. (1989),Mathematical Methods of Classical Mechanics,Graduate Texts in Mathematics, vol. 60 (second ed.),Springer-Verlag,ISBN 0-387-96890-3
• Hall, Brian C. (2015),Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,ISBN 978-3319134666
• Fulton, W.;Harris, J. (1991),Representation Theory, A first Course,Graduate Texts in Mathematics, vol. 129,Springer-Verlag,ISBN 978-0-387-97495-8.
• Goldstein, H. (1980) [1950]. "Chapter 7".Classical Mechanics (2nd ed.). Reading MA:Addison-Wesley.ISBN 0-201-02918-9.
• Lee, J. M. (2003),Introduction to Smooth manifolds,Graduate Texts in Mathematics, vol. 218,Springer-Verlag,ISBN 0-387-95448-1
• Rossmann, Wulf (2002),Lie Groups – An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications,ISBN 0-19-859683-9
• Ferraro, Alessandro; Olivares, Stefano; Paris, Matteo G. A. (March 2005), "Gaussian states in continuous variable quantum information",arXiv:quant-ph/0503237.

• Lie groups
• Symplectic geometry

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