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Topological group with compact topology
mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space. Compact groups are a natural
Compact_group
Type of topological group in mathematics
a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important
Locally_compact_group
Catalogue of 100 galaxy groups made by Paul Hickson
Hickson Compact Group (abbreviation: HCG) is a collection of galaxies designated as published by Paul Hickson in 1982. The most famous group on Hickson's
Hickson_Compact_Group
Concept in algebraic topology
mathematics, in particular algebraic topology, a p-compact group is a homotopical version of a compact Lie group, but with all the local structure concentrated
P-compact_group
Topological algebra associated to continuous groups
related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally
Group algebra of a locally compact group
Group_algebra_of_a_locally_compact_group
Abstract structure in mathematics
In mathematics, compact quantum groups are generalisations of compact groups, where the commutative C ∗ {\displaystyle \mathrm {C} ^{*}} -algebra of continuous
Compact_quantum_group
Group that is a topological space with continuous group operations
group operations that are smooth Locally compact field Locally compact group – Type of topological group in mathematics Locally compact quantum group
Topological_group
Duality for locally compact abelian groups
locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex
Pontryagin_duality
Type of topological space in mathematics
space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it
Locally_compact_space
the realm of abelian group theory, a group is said to be algebraically compact if it is a direct summand of every abelian group containing it as a pure
Algebraically_compact_group
Topological group structure arising in Fourier analysis
theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped
Locally_compact_abelian_group
Aggregation of galaxies with 50 or fewer members
Astronomer Paul Hickson created a catalogue of such groups in 1982, the Hickson Compact Groups. Compact groups of galaxies readily show the effect of dark matter
Galaxy_group
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
each non-compact simple Lie group G, one compact and one non-compact. The non-compact one is a cover of the quotient of G by a maximal compact subgroup
Simple_Lie_group
Locally compact topological group with an invariant averaging operation
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant
Amenable_group
a locally compact quantum group is a C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra
Locally_compact_quantum_group
mathematics, a compactly generated (topological) group is a topological group G which is algebraically generated by one of its compact subsets. This should
Compactly_generated_group
Representations of finite groups, particularly on vector spaces
compact, if any cover of G , {\displaystyle G,} which is open in the topology, has a finite subcover. Closed subgroups of a compact group are compact
Representation theory of finite groups
Representation_theory_of_finite_groups
locally compact or reductive Lie groups is an algebra of measures under convolution. It can also be defined for a pair (g, K) of a maximal compact subgroup
Hecke_algebra_of_a_pair
Barred Spiral or Lenticular galaxy in the constellation Ursa Major
the Hickson Compact Group #56 is located. HCG 56 is a compact group of galaxies of which 4 members are interacting with each other. This group is about 120
NGC_3718
Left-invariant (or right-invariant) measure on locally compact topological group
volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced
Haar_measure
Theorem in harmonic analysis
locally compact abelian groups. There is also a version of the Plancherel theorem which makes sense for non-commutative locally compact groups satisfying
Plancherel_theorem
Digital optical disc data storage format
The compact disc (CD) is a digital optical disc data storage format co-developed by Philips and Sony to store and play digital audio recordings. It employs
Compact_disc
Universal construction of a complex Lie group from a real Lie group
original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear. For compact Lie groups, the complexification
Complexification_(Lie_group)
Decomposition of periodic functions
to compact groups is the Peter–Weyl theorem, which proves results about representations of compact groups analogous to those about finite groups. If
Fourier_series
Theorem in representation theory
theorem classifying the irreducible representations of a connected compact Lie group K {\displaystyle K} . The theorem states that there is a bijection
Theorem_of_the_highest_weight
Mathematical concept
compact groups is however much weaker than the Tannaka–Krein duality theory for compact topological groups or Pontryagin duality for locally compact abelian
Spectrum_of_a_C*-algebra
Type of mathematical space
In mathematics, especially general topology and analysis, compactness is a property of a space that makes it behave in many ways like a finite set. For
Compact_space
Type of group in mathematics
_{n}(\mathbb {R} ).} When q = 0 {\displaystyle q=0} this is the compact orthogonal group O ( n ) {\displaystyle \mathrm {O} (n)} , with determinant- 1 {\displaystyle
Classical_group
Basic result in harmonic analysis on compact topological groups
result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann
Peter–Weyl_theorem
Generalization of Lie groups
finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3)
Schur_orthogonality_relations
Topological complex vector space
important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics. Another
C*-algebra
Concept in topology
In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst
Maximal_compact_subgroup
Proposed lower bound on the Mahler measure for polynomials with integer coefficients
measure-theoretic entropy of an ergodic automorphism of a compact metrizable abelian group is known to be given by the logarithmic Mahler measure of a
Lehmer's_conjecture
Group of galaxies in the constellation Leo
Hickson Compact Group 44 (HCG 44) is a group of galaxies in the constellation Leo It is situated around 80 million light years from Earth. Also a part
Hickson_44
Function that "converges" to periodicity
There is also a notion of almost periodic functions on locally compact abelian groups, first studied by John von Neumann. Almost periodicity is a property
Almost_periodic_function
Deformation of the group algebra of a Coxeter group
very general construction in group theory. Let (G, K) be a pair consisting of a unimodular locally compact topological group G and a closed subgroup K of
Iwahori–Hecke_algebra
Coefficients coupled with angular momentum
{\displaystyle 3j} symbols for compact groups has been performed based on these principles. The Special unitary group SU(n) is the Lie group of n × n unitary matrices
3-j_symbol
Galaxy group from Hickson Compact Group
HCG 2 is one of the compact groups of galaxies in the Hickson Compact Group of galaxies. It is located in Pisces east of HD 2679 and is 192 million light-years
HCG_2
Duality between a group and its representations
interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete
Tannaka–Krein_duality
Second smallest sport utility vehicle size class
Compact crossover SUV is an automobile classification used mainly in North America to describe a segment of crossover SUV, a type of sport utility vehicle
Compact_crossover_SUV
Visual grouping of five galaxies
grouping of five galaxies of which four form the first compact galaxy group ever discovered. The group, visible in the constellation Pegasus, was discovered
Stephan's_Quintet
First case of a Lie group that is both compact and non-abelian
semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abelian group. The first condition implies the representation
Representation theory of SU(2)
Representation_theory_of_SU(2)
Principal mass of a mountain
means "massive"), where it is used to refer to a large mountain mass or compact group of connected mountains forming an independent portion of a range. The
Massif
U.S. agreement on presidential elections
The National Popular Vote Interstate Compact (NPVIC) is an agreement among a group of U.S. states and the District of Columbia to award all their electoral
National Popular Vote Interstate Compact
National_Popular_Vote_Interstate_Compact
Mathematical object
G is a reductive Lie group and K is a maximal compact subgroup. When G is a locally compact topological group and K is a compact subgroup, the following
Gelfand_pair
Type of motorcycle
high performance ECU, track-use only. Mick Withers set the 'Sports Compact' group 5 Australian record for a twin-cylinder motorcycle on a Stage 3 Yamaha
Yamaha_MT-01
Measure on group representations
defined on the set of irreducible unitary representations of a locally compact group G {\displaystyle G} , that describes how the regular representation
Plancherel_measure
Oligarchic political clique in Upper Canada (1810s to 1840s)
The Family Compact was a small closed group of men who exercised most of the political, economic and judicial power in Upper Canada (mostly analogous to
Family_Compact
Area of mathematical analysis
abstract harmonic analysis, where the emphasis is on symmetry, locally compact groups, and representation theory. A real-variable method in harmonic analysis
Harmonic_analysis
Representation theory
theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by Hermann Weyl (1925
Weyl_character_formula
Group homomorphism into the general linear group over a vector space
groups. Compact groups or locally compact groups — Many of the results of finite group representation theory are proved by averaging over the group. These
Group_representation
Branch of mathematics that studies abstract algebraic structures
provides a way to recover a compact group from its category of unitary representations. If the group is neither abelian nor compact, no general theory is known
Representation_theory
Mathematical theory
definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition
Compact_Lie_algebra
Topics referred to by the same term
mathematics, the group algebra can mean either A group ring of an abelian group over some commutative ring. A group algebra of a locally compact group. This disambiguation
Group_algebra
compact Lie group is a torus; i.e., a Lie group isomorphic to ( S 1 ) h {\displaystyle (S^{1})^{h}} . A connected complex Lie group that is a compact
Abelian_Lie_group
Group that is also a differentiable manifold with group operations that are smooth
spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights. In 1900 David Hilbert challenged Lie
Lie_group
Algebraic construct of interest in theoretical physics
Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their
Quantum_group
Galaxy in the constellation Canes Venatici
belongs to the Hickson Compact Group 68, which consists of one spiral galaxy, and four lenticular galaxies. The other galaxies in the group are NGC 5354 [fr]
NGC_5353
Concept in measure theory
Locally compact groups which are σ-compact are σ-finite under the Haar measure. For example, all connected, locally compact groups G are σ-compact. To see
Σ-finite_measure
Algebras arising in harmonic analysis
the harmonic analysis of locally compact groups. They play an important role in the duality theories of these groups. The Fourier–Stieltjes algebra and
Fourier_algebra
Representation theory of an important group in physics
of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental
Representation theory of the Poincaré group
Representation_theory_of_the_Poincaré_group
248-dimensional exceptional simple Lie group
group, has order 214 35 52 7 = 696729600. The compact group E8 is unique among simple compact Lie groups in that its non-trivial representation of smallest
E8_(mathematics)
Topological group
that every compact group is AE(0). It follows that every zero-dimensional compact group is homeomorphic to a Cantor cube, and every compact group is a continuous
Cantor_cube
Type of topological space
said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact and (weakly)
Σ-compact_space
Problem in Lie group theory
Neumann in 1933, giving an affirmative answer for compact groups. The locally compact abelian group case was solved in 1934 by Lev Pontryagin. The final
Hilbert's_fifth_problem
flow under the group of time translations. In invariant theory, the average is often taken over a compact group or reductive algebraic group acting on a
Reynolds_operator
representation of a compact group on a complex vector space has. It can be used to classify the irreducible representations of compact groups on real vector
Frobenius–Schur_indicator
Galaxy cluster in the constellation Hydra
(Type IIb, mag. 19.97) was discovered by GOTO on 21 February 2026. Hickson Compact Group Supercluster Stephan's Quintet, a more famous HCG. Dick 2019, p. 426
Hickson_40
Set of finitely supported functions from a group to a ring
detail in the article Group algebra of a locally compact group. Categorically, the group ring construction is left adjoint to "group of units"; the following
Group_ring
locally compact group. Let F be a field of characteristic zero, G a finite group and H a subgroup of G. Let F [ G ] {\displaystyle F[G]} denote the group algebra
Hecke algebra of a finite group
Hecke_algebra_of_a_finite_group
Discrete subgroup in a locally compact topological group
Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has
Lattice_(discrete_subgroup)
Representation of the symmetry group of spacetime in special relativity
framework that applies to simply connected, compact groups. Non-compactness implies, for a connected simple Lie group, that no nontrivial finite-dimensional
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Lenticular Galaxy in the constellation Pegasus
Mpc (~406 million ly). NGC 7550 is a member of the Hickson compact group HCG 93. This group of galaxies has a total of five galaxies: HGC 93A (NGC 7550)
NGC_7550
Group of galaxies in the constellation Serpens
this group was the most compact group ever identified. Wild's Triplet Zwicky's Triplet Robert's Quartet Stephan's Quintet and NGC 7331 Group (also known
Seyfert's_Sextet
Magnetic audio tape recording format
The cassette tape, officially named the Compact Cassette, and also known as audio cassette, or simply tape or cassette, is an analog magnetic tape recording
Cassette_tape
Galaxy in the constellation Pegasus
93 imaged by SDSS NGC 7549 is a member of the Hickson Compact Group HCG 93. This five member group of galaxies also contains NGC 7547, NGC 7550, NGC 7553 [fr]
NGC_7549
Galaxy in the constellation Leo
Astronomy Magazine. Retrieved 2024-05-02. "Copeland's Septet (Hickson Compact Group 57) – Constellation Guide". www.constellation-guide.com. Retrieved 2024-05-02
NGC_3754
Framework in machine learning
arbitrarily big transformations. However, they are locally compact. For locally compact groups, invariance is achievable within certain range of transformations
M-theory_(learning_framework)
algebra Griess algebra Group algebra Group algebra of a locally compact group Hall algebra Hecke algebra of a locally compact group Heyting algebra Hopf
List_of_algebras
of 14.89. It is the brightest member of HCG 34, which is a Hickson Compact Group of galaxies identified on Palamar Observatory Sky Survey red prints
NGC_1875
Mathematic theory
In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta function twisted by a Hecke character, i
Tate's_thesis
Formal agreement between two or more U.S. states
In the United States, an interstate compact is a legally binding pact or agreement between two or more states, or between states and any foreign sub-national
Interstate_compact
Group of galaxies in the constellation Leo
The Copeland Septet (also Copeland's Septet, Hickson Compact Group 57) is a group of galaxies in the constellation Leo that includes NGC 3748, NGC 3754
Copeland_Septet
Funnel-web spider from Spain and surrounds
of this spider. The carapace is low and flat and the eyes are in a compact group. The female resembles the male but has a larger abdomen. The male can
Macrothele_calpeiana
Galaxy in the constellation Leo
Astronomy Magazine. Retrieved 2024-05-02. "Copeland's Septet (Hickson Compact Group 57) – Constellation Guide". www.constellation-guide.com. Retrieved 2024-05-02
NGC_3748
Galaxy cluster in the constellation Capricornus
HCG 87 is a compact group of galaxies listed in the Hickson Compact Group Catalogue. This group is about 400 million light-years away in the constellation
HCG_87
Gaussian distribution on a locally compact Abelian group is a generalization of the classical normal distribution. Roughly speaking, it extends the concept
Gaussian distribution on a locally compact Abelian group
Gaussian_distribution_on_a_locally_compact_Abelian_group
Integral expressing the amount of overlap of one function as it is shifted over another
analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften
Convolution
Complement of an open subset
representations decompose into discrete pieces. A subgroup of a compact group need not be compact, but a closed subgroup is. In totally disconnected spaces
Closed_set
spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient
Zonal_spherical_function
Theorem about projections of coadjoint orbits of a connected compact Lie group
states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a special
Kostant's_convexity_theorem
Process of extending a representation of a subgroup to the parent group
}}k\in H\\0&{\text{otherwise}}\end{cases}}} If G is a locally compact topological group (possibly infinite) and H is a closed subgroup then there is a
Induced_representation
Galaxy in Copeland Septet
Astronomy Magazine. Retrieved 2024-05-01. "Copeland's Septet (Hickson Compact Group 57) – Constellation Guide". www.constellation-guide.com. Retrieved 2024-05-01
NGC_3753
disconnected group is a topological group that is totally disconnected. Such topological groups are necessarily Hausdorff. Interest centres on locally compact totally
Totally_disconnected_group
Galaxy cluster Galaxy group Illustris project List of galaxies Catalogue of Galaxies and of Clusters of Galaxies Hickson Compact Group List of Abell clusters
List of galaxy groups and clusters
List_of_galaxy_groups_and_clusters
Mountain range in northern Colombia
(18,947 ft) elevation that is often quoted. The Sierra Nevada is a compact group, relatively small in area, and completely surrounded by lands with elevations
Sierra_Nevada_de_Santa_Marta
Elliptical galaxy in the constellation Virgo
member of a subcluster of four galaxies, the M60 group, which is the closest-known isolated compact group of galaxies. It has several satellite galaxies
Messier_60
Representation of a group or algebra that is a direct sum of simple representations
π : G → G L ( V ) {\displaystyle \pi :G\to GL(V)} of a finite group or a compact group G, by the averaging argument, one can define an inner product ⟨
Semisimple_representation
known that under the Lie correspondence, compact Lie algebras correspond to compact Lie groups. The compact form corresponds to the Satake diagram with
Real_form_(Lie_theory)
Role of coherent states
considering a particle whose configuration space is the group manifold of a compact Lie group K. Hall introduced coherent states in which the usual Gaussian
Coherent states in mathematical physics
Coherent_states_in_mathematical_physics
COMPACT GROUP
COMPACT GROUP
Boy/Male
Indian, Punjabi, Sikh
Lord's Company
Girl/Female
Hindu, Indian, Marathi, Tamil
Compact; Promise
Boy/Male
Indian, Punjabi, Sikh
Good Company
Girl/Female
Muslim
Beauty of company
Girl/Female
Hindu, Indian
Compare
Boy/Male
Hindu, Indian, Sanskrit
Company
Girl/Female
Arabic
Sensible Contact
Boy/Male
Indian, Punjabi, Sikh
Liberation through Company
Surname or Lastname
Americanized spelling of German Kahle. Compare Kahley or Köhler (see Kohler).English and Manx
Americanized spelling of German Kahle. Compare Kahley or Köhler (see Kohler).English and Manx : variant spelling of Caley.
Girl/Female
Arabic, Muslim
Beauty of Company
Boy/Male
Hindu, Indian, Sanskrit
In the Company
Surname or Lastname
Americanized form of German Eisele. Compare Isley.English
Americanized form of German Eisele. Compare Isley.English : unexplained. This name is quite widespread in Britain.
Boy/Male
Indian, Punjabi, Sikh
Company of Guru
Boy/Male
Hindu, Indian
Compact; Safe; Secure
Girl/Female
Arabic, Muslim
Beauty of Company
Girl/Female
Tamil
Compare
Boy/Male
Indian, Sanskrit
Fallen from Glory
Girl/Female
Indian, Telugu
Good Company
Boy/Male
Indian, Tamil
No Compare
Boy/Male
Hindu, Indian
Compact; Firm; Solid
COMPACT GROUP
COMPACT GROUP
Boy/Male
Arabic, Muslim
Servant of the Owner (Allah)
Boy/Male
Hindu, Indian, Telugu
Shine
Boy/Male
Irish
Champion.
Boy/Male
Gujarati, Hindu, Indian
Sea
Male
Hebrew
(יָקְטָן) Hebrew name YOQTAN means "small." In the bible, this is the name of a descendant of Shem. Joktan is the Anglicized form.
Girl/Female
Biblical
South country,.
Girl/Female
Gujarati, Hindu, Indian
Touchstone; Gold Maker; Stone that Turns Iron into Gold
Girl/Female
Muslim
A name of the prophet (Pbuh), Ever ambitious
Boy/Male
German, Norse, Swedish
Dark
Boy/Male
Hindu, Indian
Master of the Right Path
COMPACT GROUP
COMPACT GROUP
COMPACT GROUP
COMPACT GROUP
COMPACT GROUP
a.
Compact; pressed close; concentrated; firmly united.
n.
Guests or visitors, in distinction from the members of a family; as, to invite company to dine.
n.
An association of persons for the purpose of carrying on some enterprise or business; a corporation; a firm; as, the East India Company; an insurance company; a joint-stock company.
a.
Strong; firm; compact.
n.
One who makes a compact.
v. t.
To manure with compost.
imp. & p. p.
of Compact
adv.
In a compact manner; with close union of parts; densely; tersely.
p. p. & a
Brief; close; pithy; not diffuse; not verbose; as, a compact discourse.
n.
Extent; reach; sweep; capacity; sphere; as, the compass of his eye; the compass of imagination.
n.
The crew of a ship, including the officers; as, a whole ship's company.
n.
Contact or impression by touch; collision; forcible contact; force communicated.
p. pr. & vb. n.
of Compact
v. i.
To be like or equal; to admit, or be worthy of, comparison; as, his later work does not compare with his earlier.
v. i.
To bear or endure; to put up (with); as, to comport with an injury.
n.
An inclosing limit; boundary; circumference; as, within the compass of an encircling wall.
n.
A mixture for fertilizing land; esp., a composition of various substances (as muck, mold, lime, and stable manure) thoroughly mingled and decomposed, as in a compost heap.
v. t.
To mingle, as different fertilizing substances, in a mass where they will decompose and form into a compost.
v. t.
To compact or join anew.