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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)
Symplectic_representation
Topics referred to by the same term
algebra Symplectic integrator Symplectic manifold Symplectic matrix Symplectic representation Symplectic vector space, a vector space with a symplectic bilinear
Symplectic
Mathematical group
central role in symplectic geometry, Hamiltonian mechanics, and representation theory. A related but different family is the compact symplectic group, usually
Symplectic_group
Mathematical concept
In mathematics, a symplectic vector space is a vector space V {\displaystyle V} over a field F {\displaystyle F} (for example the real numbers R {\displaystyle
Symplectic_vector_space
Mathematical concept
In mathematics, a symplectic matrix is a 2 n × 2 n {\displaystyle 2n\times 2n} matrix M {\displaystyle M} with real entries that satisfies the condition
Symplectic_matrix
Mathematical concept
In mathematics, particularly in representation theory, a symplectic resolution is a morphism that combines symplectic geometry and resolution of singularities
Symplectic_resolution
representation Semisimple Complex representation Real representation Quaternionic representation Pseudo-real representation Symplectic representation
List of representation theory topics
List_of_representation_theory_topics
Representation of a group or algebra in terms of an algebra with quaternionic structure
If V is a unitary representation and the quaternionic structure j is a unitary operator, then V admits an invariant complex symplectic form ω, and hence
Quaternionic_representation
Group in mathematical representation theory
such as the Weil representation described below. It can be proved that if F is any local field other than C, then the symplectic group Sp2n(F) admits
Metaplectic_group
Writing Lie algebra sets as matrices
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra
Lie_algebra_representation
Branch of mathematics that studies abstract algebraic structures
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of
Representation_theory
Mathematical structure in differential geometry
Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics
Poisson_manifold
submanifolds of g ∗ {\displaystyle {\mathfrak {g}}^{*}} and carry a natural symplectic structure. On each orbit O μ {\displaystyle {\mathcal {O}}_{\mu }} , there
Coadjoint_representation
Representation theory of the symplectic group
In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David
Oscillator_representation
Mathematical term
representation of the adjoint representation. Alexandre Kirillov observed that the orbit of any vector in a co-adjoint representation is a symplectic
Adjoint_representation
Area of mathematics
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
Type of group and algebra representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation ( ρ , V ) {\displaystyle (\rho ,V)} or
Irreducible_representation
Austrian mathematician and mathematical physicist
Metaplectic representation, Conley–Zehnder index, and Weyl calculus on phase space. Rev. Math. Phys. 19 (2007), no. 10, 1149–1188. Symplectically covariant
Maurice_A._de_Gosson
Oscillator Oscillator representation orbit orbit method, an approach to representation theory that uses tools from symplectic geometry Peter–Weyl The
Glossary of representation theory
Glossary_of_representation_theory
Branch of mathematics
example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are
Differential_geometry
Random matrix with gaussian entries
three main examples are the Gaussian orthogonal (GOE), unitary (GUE), and symplectic (GSE) ensembles. These are classified by the Dyson index β, which takes
Gaussian_ensemble
via the metaplectic representation. The metaplectic representation of the metaplectic group — the two-fold covering of the symplectic group — gives rise
Symplectic_spinor_bundle
Concept in mathematics
the mapping class group is a linear group or not. Besides the symplectic representation on homology explained above there are other interesting finite-dimensional
Mapping class group of a surface
Mapping_class_group_of_a_surface
transformations are the linear transforms of the time–frequency representation that preserve the symplectic form. These include and generalize the Fourier transform
Time–frequency_representation
Representation theory of an important group in physics
In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group
Representation theory of the Poincaré group
Representation_theory_of_the_Poincaré_group
Four-dimensional number system
Clarysse, P.; Pujol, R.; Delachartre, P. (2025). "Hyperquaternionic unitary symplectic groups: A unifying tool for physics". Advances in Applied Clifford Algebras
Quaternion
Differential algebra
{\displaystyle V} (of dimension 2 n {\displaystyle 2n} ) equipped with a symplectic form ω {\displaystyle \omega } . Define the Weyl algebra W ( V ) {\displaystyle
Weyl_algebra
Type of group in mathematics
quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups, together with their indefinite analogues. In the language of linear
Classical_group
Russian mathematician (1937–2010)
systems, algebra, catastrophe theory, topology, real algebraic geometry, symplectic geometry, differential equations, classical mechanics, differential-geometric
Vladimir_Arnold
Canonical differential form
derivative of this form defines a symplectic form, giving T ∗ Q {\displaystyle T^{*}Q} the structure of a symplectic manifold. The tautological one-form
Tautological_one-form
In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose
Symplectic_cut
Key result in Hamiltonian mechanics and statistical mechanics
of the symplectic 2-form, and is just another representation of the measure on the phase space described above. On our phase space symplectic manifold
Liouville's theorem (Hamiltonian)
Liouville's_theorem_(Hamiltonian)
Group in group theory and physics
groups associated to n-dimensional systems, and most generally, to any symplectic vector space. In the three-dimensional case, the product of two Heisenberg
Heisenberg_group
Group representation
theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism
Representation_of_a_Lie_group
Group that is also a differentiable manifold with group operations that are smooth
whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich
Lie_group
special orthogonal groups and unitary symplectic groups; Littlewood (1950) from the unitary groups to the unitary symplectic groups and special orthogonal groups
Restricted_representation
Group of unitary matrices
unitary group is the 3-fold intersection of the orthogonal, complex, and symplectic groups: U ( n ) = O ( 2 n ) ∩ GL ( n , C ) ∩ Sp ( 2 n , R ) .
Unitary_group
Subalgebra of E8 containing E7
56-dimensional irreducible representation of E7. This representation has an invariant symplectic form, and this symplectic form equips (56) ⊕ R with the structure of
E7½
Representation theory of the symmetries of non-relativistic quantum space
Wigner's classification of relativistic mechanics) in terms of the representation theory of the Galilean group, which is the spacetime symmetry group
Representation theory of the Galilean group
Representation_theory_of_the_Galilean_group
Recipe for constructing a quantum analog of a classical physical theory
polarizations. Suppose ( M , ω ) {\displaystyle (M,\omega )} is a symplectic manifold with symplectic form ω {\displaystyle \omega } . Suppose at first that ω
Geometric_quantization
Direct sum of simple Lie algebras
{\displaystyle C_{n}:} s p 2 n {\displaystyle {\mathfrak {sp}}_{2n}} , the symplectic Lie algebra. D n : {\displaystyle D_{n}:} s o 2 n {\displaystyle {\mathfrak
Semisimple_Lie_algebra
Map from algebra to geometric transforms
In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism
Projective_representation
Algebraic structure used in analysis
{su}}(n)} consists of the skew-hermitian matrices with trace zero. The symplectic group S p ( 2 n , R ) {\displaystyle \mathrm {Sp} (2n,\mathbb {R} )} is
Lie_algebra
Group of unitary complex matrices with determinant of 1
absolute value 1. For completeness, there are also the orthogonal and symplectic subgroups, SU ( n ) ⊃ SO ( n ) , SU ( 2 n ) ⊃ Sp ( n ) . {\displaystyle
Special_unitary_group
Belarus) is a Belarusian-American mathematician, specializing in representation theory, symplectic geometry, algebraic geometry, and combinatorial algebra. Losev
Ivan_Losev_(mathematician)
Concept in mathematics
supersymmetry: its fundamental representation is the so-called spinor representation, while its adjoint representation generates the Lorentz group SO(3
Special_linear_Lie_algebra
Topological group with compact topology
orthogonal group SO(n) and its covering spin group Spin(n), the compact symplectic group USp(n), the unitary group U(n) and the special unitary group SU(n)
Compact_group
16-element matrix group
(\mathbf {v} )\mathbf {N} (\mathbf {u} ).} The above binary representation and symplectic algebra are especially useful in making the relation between
Pauli_group
Criterion for vector stability in algebraic geometry
to the symplectic quotient of X by a maximal compact subgroup of G. Kempf, George; Ness, Linda (1979), "The length of vectors in representation spaces"
Kempf–Ness_theorem
Group representation
\operatorname {so} (2n+1;\mathbb {C} )} (type B n {\displaystyle B_{n}} ) and the symplectic Lie algebras sp ( n ; C ) {\displaystyle \operatorname {sp} (n;\mathbb
Dual_representation
Topics referred to by the same term
by its fans The Kirillov-Kostant-Souriau symplectic form of symplectic geometry, see Coadjoint representation. This disambiguation page lists articles
KKS
Formulation of classical mechanics using momenta
Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical
Hamiltonian_mechanics
Multi particle state space
and annihilation operators close under commutator and give a representation of the symplectic Lie algebra s p ( 2 n ) {\displaystyle {\mathfrak {sp}}(2n)}
Fock_space
Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. It is associated most classically
Maslov_index
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
unitary symplectic matrices, Sp(r) and as its associated centerless group the Lie group PSp(r) = Sp(r)/{I, −I} of projective unitary symplectic matrices
Simple_Lie_group
Group of 𝑛 × 𝑛 invertible matrices
on V {\displaystyle V} , symplectic group, Sp ( V ) {\displaystyle \operatorname {Sp} (V)} , which preserves a symplectic form on V {\displaystyle V}
General_linear_group
|} is the dual vector of | ψ ⟩ {\displaystyle |\psi \rangle } . This symplectic Hilbert space is denoted by H ( Γ ) {\displaystyle {\mathcal {H}}(\Gamma
Phase-space_wavefunctions
Russian American mathematician (born 1957)
(1997), Representation theory and complex geometry, Boston, MA: Birkhäuser, MR 1433132 Etingof, Pavel; Ginzburg, Victor (2002), "Symplectic reflection
Victor_Ginzburg
Undergraduate textbook on mathematics and mathematical physics
differential forms, pushforwards and pullbacks, symplectic manifolds, Hamiltonian energy functions, the representation of finite and infinitesimal physical symmetries
Symmetry_in_Mechanics
Group of real 2×2 matrices with unit determinant
covering group, Mp(2, R), a metaplectic group (thinking of SL(2, R) as a symplectic group). Another related group is SL±(2, R), the group of real 2 × 2 matrices
SL2(R)
Type of Riemannian manifold
holomorphically symplectic manifolds. The holonomy group of any Calabi–Yau metric on a simply connected compact holomorphically symplectic manifold of complex
Hyperkähler_manifold
Differential form
generally, the n {\displaystyle n} th exterior power of the symplectic form on a symplectic manifold is a volume form. Many classes of manifolds have canonical
Volume_form
Mathematical theorem
a symplectic space of dimension 2n. More formally, there is a unique (up to scale) non-trivial central strongly continuous unitary representation. This
Stone–von_Neumann_theorem
Lie group of Lorentz transformations
3), is isomorphic to both the special linear group SL(2, C) and to the symplectic group Sp(2, C). These isomorphisms allow the Lorentz group to act on a
Lorentz_group
Canadian mathematician
Research Chair in Equivariant Symplectic and Algebraic Geometry. Harada's research involves the symmetries of symplectic spaces and their connections to
Megumi_Harada
Type of generalization of periodic functions in Euclidean space
theory was long in coming. The Siegel modular forms, for which G is a symplectic group, arose naturally from considering moduli spaces and theta functions
Automorphic_form
Subgroup of a root system's isometry group
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
Weyl_group
Mathematics award
ingenious and surprising solutions to long standing open problems in symplectic geometry, Riemannian geometry, harmonic analysis, and combinatorial geometry
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Group of flat spacetime symmetries
Poincaré group is a group extension of the Lorentz group by a vector representation of it; it is sometimes dubbed, informally, as the inhomogeneous Lorentz
Poincaré_group
In representation theory, a branch of mathematics, θ10 is a particular unitary representation of the symplectic group Sp4, which has some exceptional properties
Θ10
Group of matrices with determinant 1
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
Special_linear_group
Lie group of complex numbers of unit modulus; topologically a circle
all 1-dimensional. Since the circle group is compact, any continuous representation ρ : T → G L ( 1 , C ) ≅ C × {\displaystyle \rho \colon \mathbb {T} \to
Circle_group
Root-finding algorithm
in comments in lower half of this code". Moler, Cleve (19 June 2012). "Symplectic Spacewar". MATLAB Central - Cleve's Corner. MATLAB. Retrieved 2014-07-21
Fast_inverse_square_root
Comprehensive physical model
spinor representation of O(16). Symplectic gauge groups could also be considered. For example, Sp(8) (which is called Sp(4) in the article symplectic group)
Grand_Unified_Theory
introduced by Eichler & Zagier (1985), is the semidirect product of the symplectic group Sp2n(R) and the Heisenberg group R1+2n. The concept is named after
Jacobi_group
Map from a Lie algebra to its Lie group
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
Exponential_map_(Lie_theory)
Construction in representation theory
coadjoint orbits of a Lie group G have natural structure of symplectic manifolds whose symplectic structure is invariant under G. If an orbit is the phase
Orbit_method
Associative algebra together with a Lie bracket that satisfies Leibniz's law
Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is
Poisson_algebra
define the symplectic spinor bundle, which is the Hilbert space bundle associated to the metaplectic structure via the metaplectic representation, giving
Metaplectic_structure
Infinite family of simple groups of Lie type
Suzuki groups were the fixed points of exceptional automorphisms of some symplectic groups of dimension 4, and used this to construct two further families
Suzuki_groups
Isometry group of Euclidean space
size n + 1, as explained for the affine group. Details for the first representation are given in the next section. In the terms of Felix Klein's Erlangen
Euclidean_group
In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact
Theta_representation
Poincaré disk model of the Hyperbolic plane. Lorentz group Spinor group Symplectic group Exceptional groups G2 F4 E6 E7 E8 Affine group Euclidean group Poincaré
List_of_Lie_groups_topics
Physics-mathematics connection
There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties
Particle physics and representation theory
Particle_physics_and_representation_theory
Invariance of operations under geometric translation
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
Translational_symmetry
Topic in group theory and harmonic analysis (Niemeier lattice-mock theta connection)
moonshine starts with a theorem of Mukai, asserting that any group of symplectic automorphisms of a K3 surface embeds in the Mathieu group M23. The moonshine
Umbral_moonshine
Feature of a system that is preserved under some transformation
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
Symmetry_(physics)
Property of a differential manifold that includes complex structures
differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures were introduced by Nigel Hitchin
Generalized_complex_structure
Universal construction of a complex Lie group from a real Lie group
representations of the group. In any finite-dimensional faithful unitary representation of the compact group it can be realized concretely as a closed subgroup
Complexification_(Lie_group)
Concept in topological group theory
cover of the symplectic group Sp2n means that there are always two elements in the metaplectic group representing one element in the symplectic group. Let
Covering_group
Example of a phase-space star product in mathematics
^{2n}} , equipped with its Poisson bracket (with a generalization to symplectic manifolds, described below). It is a special case of the ★-product of
Moyal_product
Type of complex number
oscillator representation, but only up to a phase factor. The resulting operators therefore define a projective representation of the symplectic group. Passing
Phase_factor
Concept in Lie algebra mathematics
Simple Lie group Vogel plane Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics
Simple_Lie_algebra
1+10+45 Hyperovals in P2(4); three classes L4(3) PSp4(3):2 117 = 1+36+80 Symplectic polarities of P3(3); two classes G2(2)' = U3(3) PSL3(2) 36 = 1+14+21 Suzuki
Rank_3_permutation_group
Coordinate transformation that preserves the form of Hamilton's equations
all matrices M {\textstyle M} which satisfy symplectic conditions form a symplectic group. The symplectic conditions are equivalent with indirect conditions
Canonical_transformation
Differential geometry concept
semisimple Lie groups are equipped with the Kirillov-Kostant holomorphic symplectic form. This argument also explains how one can associate a unique Wolf
Quaternion-Kähler symmetric space
Quaternion-Kähler_symmetric_space
248-dimensional exceptional simple Lie group
compact Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra
E8_(mathematics)
American Jewish mathematician
involved representation theory, Lie groups, Lie algebras, homogeneous spaces, differential geometry and mathematical physics, particularly symplectic geometry
Bertram_Kostant
Generalization of Hamiltonian mechanics involving multiple Hamiltonians
mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms and hence obey Liouville's
Nambu_mechanics
American mathematician
Mathematics portal Commutation theorem for traces Metaplectic group Symplectic group Symplectic spinor bundle Shale, D. (1962). "Linear symmetries of free boson
Irving_Segal
SYMPLECTIC REPRESENTATION
SYMPLECTIC REPRESENTATION
SYMPLECTIC REPRESENTATION
Boy/Male
Tamil
Dinendra | திநேஂதà¯à®°
Lord of the day, The Sun
Boy/Male
Indian
Surname or Lastname
English
English : from a Norman form of a female personal name, Agilina, of Germanic origin.Swiss German : variant of Egli.
Male
Dutch
, spear sport.
Girl/Female
Hindu, Indian
Goddess Parvati
Surname or Lastname
English
English : unexplained.Possibly an Americanized form of German Eiche ‘oak’ (see Eich).
Female
Arthurian
, bright, clear, or, famous.
Boy/Male
Muslim/Islamic
Lion
Boy/Male
Indian, Punjabi, Sikh
One in whom Truth is Predominant
Girl/Female
Greek
Bee. Famous bearer: Melissa, Mythological princess of Crete transformed to a bee after learning...
SYMPLECTIC REPRESENTATION
SYMPLECTIC REPRESENTATION
SYMPLECTIC REPRESENTATION
SYMPLECTIC REPRESENTATION
SYMPLECTIC REPRESENTATION
n.
The art or act of representing a body on a perspective plane; also, a representation or description of a body, in all its dimensions, as it appears to the eye.
n.
The pictorial representation of a scene; a sketch, /ither drawn or painted; as, a fine view of Lake George.
n.
A vessel similar to that described in the first definition above, or the representation of one in a solid block of stone, or the like, used for an ornament, as on a terrace or in a garden. See Illust. of Niche.
n.
The symplectic bone.
n.
A portrait or representation of the face of our Savior on the alleged handkerchief of Saint Veronica, preserved at Rome; hence, a representation of this portrait, or any similar representation of the face of the Savior. Formerly called also Vernacle, and Vernicle.
n.
A figure or representation of something to come; a token; a sign; a symbol; -- correlative to antitype.
n.
The representation of such a memorial, as on a medal; esp. (Arch.), an ornament representing a group of arms and military weapons, offensive and defensive.
a.
Plaiting or joining together; -- said of a bone next above the quadrate in the mandibular suspensorium of many fishes, which unites together the other bones of the suspensorium.
n.
A dramatic performance; as, a theatrical representation; a representation of Hamlet.
n.
Assemblage of scenes; the paintings and hangings representing the scenes of a play; the disposition and arrangement of the scenes in which the action of a play, poem, etc., is laid; representation of place of action or occurence.
n.
A representation of the world.
a.
Implying representation; representative.
n.
A representation of the aspects of the celestial bodies for any moment or at a given event.
n.
A description or statement; as, the representation of an historian, of a witness, or an advocate.
n.
In dramatic composition, one of the principles by which a uniform tenor of story and propriety of representation are preserved; conformity in a composition to these; in oratory, discourse, etc., the due subordination and reference of every part to the development of the leading idea or the eastablishment of the main proposition.
n.
The act or art of expressing by means of types or symbols; emblematical or hieroglyphic representation.
n.
The body of those who act as representatives of a community or society; as, the representation of a State in Congress.
n.
A perspective representation or general view of an object.
n.
A general form or structure common to a number of individuals; hence, the ideal representation of a species, genus, or other group, combining the essential characteristics; an animal or plant possessing or exemplifying the essential characteristics of a species, genus, or other group. Also, a group or division of animals having a certain typical or characteristic structure of body maintained within the group.
n.
A likeness, a picture, or a model; as, a representation of the human face, or figure, and the like.