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Concept in topology
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 such
Maximal_compact_subgroup
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 K
Hecke_algebra_of_a_pair
Maximal compact connected Abelian Lie subgroup
theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group
Maximal_torus
Orthogonal group of an indefinite quadratic form
{\displaystyle \operatorname {O} (p)\times \operatorname {O} (q)} is a maximal compact subgroup of O ( p , q ) {\displaystyle \operatorname {O} (p,q)} , while
Indefinite_orthogonal_group
Group of 𝑛 × 𝑛 invertible matrices
continuous. A maximal compact subgroup is not unique, but is essentially unique, hence one often refers to “the” maximal compact subgroup. Galois, Évariste
General_linear_group
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
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 H, and
Simple_Lie_group
Topological group with compact topology
compact, simply connected Lie group. A key idea in the study of a connected compact Lie group K is the concept of a maximal torus, that is a subgroup
Compact_group
Type of mathematical space
the compact flat model spaces for Cartan geometries of parabolic type. They are homogeneous Riemannian manifolds under any maximal compact subgroup of
Generalized_flag_variety
78-dimensional exceptional simple Lie group
has maximal compact subgroup SO(2) × Spin(10)/(center), fundamental group Z and trivial outer automorphism group. EIV (or E6(-26)), which has maximal compact
E6_(mathematics)
248-dimensional exceptional simple Lie group
group of real dimension 496. This is simply connected, has maximal compact subgroup the compact form (see below) of E8, and has an outer automorphism group
E8_(mathematics)
Dynkin diagram of a real semisimple Lie algebra that indicates the maximal compact subgroup. Although they resemble Satake diagrams they are a different way
Vogan_diagram
133-dimensional exceptional simple Lie group
real dimension 266. This has fundamental group Z/2Z, has maximal compact subgroup the compact form (see below) of E7, and has an outer automorphism group
E7_(mathematics)
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
Criterion for vector stability in algebraic geometry
complex vector space is given a norm that is invariant under a maximal compact subgroup of the reductive group, then the Kempf–Ness theorem states that
Kempf–Ness_theorem
Mathematical process dealing with Lie groups
GL_{n}(F)} can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup G L n ( O F ) {\displaystyle GL_{n}(O_{F})}
Iwasawa_decomposition
Group of unitary matrices
follows: O ( 2 n ) {\displaystyle \operatorname {O} (2n)} is the maximal compact subgroup of GL ( 2 n , R ) {\displaystyle \operatorname {GL} (2n,\mathbb
Unitary_group
Type of mathematical space
and only if every real maximal ideal is of the form Mp for some p ∈ X. Consequently, X is compact if and only if every maximal ideal of C(X) is the kernel
Compact_space
Simple Lie group; the automorphism group of the octonions
simply connected. The maximal compact subgroup of its associated group is the compact form of G2. The Lie algebra of the compact form is 14-dimensional
G2_(mathematics)
Discrete subgroup in a locally compact topological group
locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn
Lattice_(discrete_subgroup)
(pseudo-)Riemannian manifold whose geodesics are reversible
groups. For non-compact type, G is such a group and K is its maximal compact subgroup. Each such example has a corresponding example of compact type, by considering
Symmetric_space
Group that is a topological space with continuous group operations
types of Lie groups reduces to the case of compact Lie groups. For example, the maximal compact subgroup of SL ( 2 , R ) {\displaystyle {\text{SL}}(2
Topological_group
In mathematics, a minimal K-type is a representation of a maximal compact subgroup K of a semisimple Lie group G that is in some sense the smallest representation
Minimal_K-type
Representation theory
be a semisimple Lie group and K a maximal compact subgroup of G. The Hecke algebra Cc(K \G/K), consisting of compactly supported K-biinvariant continuous
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
Group of real 2×2 matrices with unit determinant
The circle group SO(2) is a maximal compact subgroup of SL(2, R), and the circle SO(2) / {±1} is a maximal compact subgroup of PSL(2, R). The Schur multiplier
SL2(R)
Sporadic simple group
mod 3, so is a subgroup of the Chevalley group E8(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson
Thompson_sporadic_group
Branch of mathematics that studies abstract algebraic structures
which is a complex Lie group Gc, and this complex Lie group has a maximal compact subgroup K. The finite-dimensional representations of G closely correspond
Representation_theory
Type of group representation for locally compact groups
if it has the same rank as a maximal compact subgroup K. In other words, a maximal torus T in K must be a Cartan subgroup in G. (This result required that
Discrete series representation
Discrete_series_representation
Mathematical theory
K)-modules, for g a Lie algebra of a reductive Lie group G, with maximal compact subgroup K, in terms of tempered representations of smaller groups. The
Langlands_classification
Construction in group theory
group, PO – maximal compact subgroup of PGL Projective unitary group, PU Projective special orthogonal group, PSO – maximal compact subgroup of PSL Projective
Projective_linear_group
Double cover Lie group of the special orthogonal group
fundamental group is most easily understood by considering the maximal compact subgroup of SO(p, q), which is SO(p) × SO(q), and noting that rather than
Spin_group
Mathematical group of the homotopy classes of loops in a topological space
{\displaystyle G} is a non-compact simply connected, connected Lie group (often semisimple), K {\displaystyle K} is a maximal compact subgroup of G {\displaystyle
Fundamental_group
Special group in linear algebra
conjugacy classes of maximal parahorics. When G is commutative, it has a unique maximal compact subgroup and a unique Iwahori subgroup, which is contained
Iwahori_subgroup
Concept in mathematics
manifold G/K of G by a maximal compact subgroup K is a symmetric space of non-compact type. In fact, every symmetric space of non-compact type arises this way
Reductive_group
Mathematical object
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 are equivalent:
Gelfand_pair
Topics referred to by the same term
Vogan, Togo, a town Vogan diagram, a diagram that indicates the maximal compact subgroup Vågan This disambiguation page lists articles associated with the
Vogan
of X by G is homeomorphic to the symplectic quotient of X by a maximal compact subgroup of G (Kempf–Ness theorem). Let G be a reductive group acting on
GIT_quotient
Open convex self-dual cones
maximal compact subgroups, all conjugate, and exhaust the maximal compact subgroups of Aut C. In Aut0 C the stabilizers of points are maximal compact
Symmetric_cone
Sequence of spaces in linear algebra
the maximal torus: the flag corresponds to the Borel group, and the inner product corresponds to the maximal compact subgroup. The stabilizer subgroup of
Flag_(linear_algebra)
theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal torus. It is named after the Swiss
Borel–de_Siebenthal_theory
Type of group in mathematics
q=0} this is the compact orthogonal group O ( n ) {\displaystyle \mathrm {O} (n)} , with determinant- 1 {\displaystyle 1} subgroup S O ( n ) {\displaystyle
Classical_group
forces the Lie algebra to have a Hermitian structure, i.e. the maximal compact subgroup has center isomorphic to the circle group. The invariant convex
Invariant_convex_cone
centerless group). PSO is the maximal compact subgroup in the projective special linear group PSL, while PO is maximal compact in the projective general linear
Projective_orthogonal_group
Concept in differential geometry
an irreducible semisimple complex connected Lie subgroup and let K ⊂ H be a maximal compact subgroup. If there is an irreducible hermitian symmetric space
Holonomy
Harish-Chandra indicates that if G is a reductive Lie group with maximal compact subgroup K, and X is an irreducible ( g , K ) {\displaystyle ({\mathfrak
Harish-Chandra_module
Type of subgroup of an algebraic group
algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general
Borel_subgroup
Family of groups in group theory
one of the five real forms of E6 classified by Élie Cartan. Its maximal compact subgroup is of type F4. Reading example: If q2=22 in 2E6(q2) then q=2 in
2E6_(mathematics)
maximal compact subgroup of G. Let G be a real Lie group. Let g {\displaystyle {\mathfrak {g}}} be its Lie algebra, and K a maximal compact subgroup with
(g,K)-module
semisimple group G in terms of their restricted representations to a maximal compact subgroup K (their so-called K-types). It is named after Robert James Blattner
Blattner's_conjecture
Rational function of the form (az + b)/(cz + d)
becomes identical to the upper-half-plane model as r approaches ∞. A maximal compact subgroup of the Möbius group M {\displaystyle {\mathcal {M}}} is given by
Möbius_transformation
Universal construction of a complex Lie group from a real Lie group
polar decomposition implies that G is a maximal compact subgroup of GC, since a strictly larger compact subgroup would contain all integer powers of a positive
Complexification_(Lie_group)
semisimple Lie group G, the subgroup H is constructed starting from the Iwasawa decomposition G = KAN with K a maximal compact subgroup. Then H is chosen to
Principal series representation
Principal_series_representation
Finite group
sporadic group (the full automorphism group of this lattice) as a maximal subgroup. Huppert (1967, p.124) showed that any extension of G L n ( F q ) {\displaystyle
Dempwolff_group
Type of representation of a linear semisimple Lie group
series representation. If G is a linear semisimple Lie group with a maximal compact subgroup K, an admissible representation ρ of G is tempered if the above
Tempered_representation
for more general groups), and g is the Lie algebra of G. K is a maximal compact subgroup of G. A (g,K)-module is a vector space with compatible actions
Zuckerman_functor
decomposition of a knot into a sum of prime knots is essentially unique. A maximal compact subgroup of a semisimple Lie group may not be unique, but is unique up to
Essentially_unique
Nilpotent subalgebra of a Lie algebra
‘Cartan subgroup,’ especially when dealing with disconnected groups. For compact connected Lie groups, a Cartan subgroup is essentially a maximal connected
Cartan_subalgebra
General relativity in M-theory
subsection. The available spinor representations depends on k; the maximal compact subgroup of the little group of the Lorentz group that preserves the momentum
Higher-dimensional supergravity
Higher-dimensional_supergravity
Generalized matrix decomposition for Lie groups and Lie algebras
k\cdot \mathrm {exp} (X)} is a diffeomorphism. The subgroup K {\displaystyle K} is a maximal compact subgroup of G {\displaystyle G} , whenever the center of
Cartan_decomposition
discrete series representations exist when the maximal compact subgroup of the group G has a normal subgroup isomorphic to SU(2). Every complex simple Lie
Quaternionic discrete series representation
Quaternionic_discrete_series_representation
Three dimensional analogue of uniformization conjecture
X with compact stabilizers. A model geometry is called maximal if G is maximal among groups acting smoothly and transitively on X with compact stabilizers
Geometrization_conjecture
Subgroup of a root system's isometry group
always be realized as a subgroup of G. If B is a Borel subgroup of G, i.e., a maximal connected solvable subgroup and a maximal torus T = T0 is chosen
Weyl_group
Subgroup of the group of invertible n×n matrices
field k, a Borel subgroup of G means a maximal smooth connected solvable subgroup. For example, one Borel subgroup of GL(n) is the subgroup B of upper-triangular
Linear_algebraic_group
Subspace defined by a polynomial of degree 2 over a field
G=\operatorname {SO} (n+2,\mathbf {C} )} , and also for its maximal compact subgroup, the compact Lie group SO(n + 2). From the latter point of view, this
Quadric_(algebraic_geometry)
Lie group whose manifold is complex and whose group operation is holomorphic
{Lie} (K)\otimes _{\mathbb {R} }\mathbb {C} } , and (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example, GL n
Complex_Lie_group
Manifold with inversion symmetry
Irreducible compact Hermitian symmetric spaces are exactly the homogeneous spaces of simple compact Lie groups by maximal closed connected subgroups which contain
Hermitian_symmetric_space
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
at all contractible. In fact it is homotopy equivalent to its maximal compact subgroup, the unitary group U of H. The proof that the complex general linear
Kuiper's_theorem
Device in the representation theory of Lie groups
group is a complex semisimple Lie group. For any such group G and maximal compact subgroup K, and V a complex vector space of finite dimension which is a
Unitarian_trick
Type of group in mathematics
2. The component with det(A) = 1 is SO(n). A maximal torus in a compact Lie group G is a maximal subgroup among those that are isomorphic to Tk for some
Orthogonal_group
Integral polynomial
cells in a Grassmannian. The L-V case takes a real form GR of G, a maximal compact subgroup KR in that semisimple group GR, and makes the complexification
Kazhdan–Lusztig_polynomial
real group we may not even get an algebraic set. In particular, a maximal compact subgroup generally gives a semi-algebraic set. On the other hand, whenever
Character_variety
representation of a real reductive group is called admissible if (1) a maximal compact subgroup K acts as unitary operators and (2) each irreducible representation
Glossary of representation theory
Glossary_of_representation_theory
Mathematical concept
decomposition. The set of orthogonal symplectic matrices forms a (maximal) compact subgroup of the symplectic group. This set is isomorphic to the set of
Symplectic_matrix
Writing Lie algebra sets as matrices
complexification g {\displaystyle {\mathfrak {g}}} and the connected maximal compact subgroup K. The g {\displaystyle {\mathfrak {g}}} -module structure of π
Lie_algebra_representation
Subgroup of the Clifford algebra associated to a quadratic space
the set of all 32 inequivalent double covers of O(p) x O(q), the maximal compact subgroup of O(p, q) and an explicit construction of 8 double covers of the
Pin_group
Vector bundles theorem
group, admitting a compatible reduction of structure group to a maximal compact subgroup. Annamalai Ramanathan first defined the notion of a stable principal
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
Application of topology
world manifold X {\displaystyle X} is always reducible to its maximal compact subgroup S O ( 4 ) {\displaystyle SO(4)} . The corresponding global section
World_manifold
Mathematical theorem
G)} ). When G {\displaystyle G} is a semisimple Lie group with a maximal compact subgroup K {\displaystyle K} and X = G / K {\displaystyle X=G/K} is the
Selberg_trace_formula
Special types of subgroups encountered in group theory
the Weyl group of a compact Lie group G with a torus T is defined as W(G,T) = NG(T)/CG(T), and especially if the torus is maximal (i.e. CG(T) = T) it
Centralizer_and_normalizer
Class of representations
a connected reductive (real or complex) Lie group. Let K be a maximal compact subgroup. A continuous representation (π, V) of G on a complex Hilbert space
Admissible_representation
K-invariant elements of the universal enveloping algebra for a maximal compact subgroup K, the Harish-Chandra homomorphism was studied by Harish-Chandra (1958)
Harish-Chandra_homomorphism
Space of complex matrices with positive definite imaginary part
{H}}_{g}} for this action is the unitary subgroup U ( g ) {\displaystyle U(g)} , which is a maximal compact subgroup of S p ( 2 g , R ) {\displaystyle \mathrm
Siegel_upper_half-space
{\displaystyle \Xi (g)=\int _{K}a(kg)^{\rho }dk,} where K is a maximal compact subgroup of a semisimple Lie group with Iwasawa decomposition G=NAK g is
Harish-Chandra's_Ξ_function
finite-dimensional representations coincide with those of their maximal compact subgroups, respectively S U ( n ) {\displaystyle SU(n)} , S O ( n ) {\displaystyle
Representations of classical Lie groups
Representations_of_classical_Lie_groups
isometry. They turn out to be quotients of a real Lie group by a maximal compact subgroup whose Lie algebra is the fixed subalgebra of the involution obtained
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
1)/U(n)} . The stabilizer U ( n ) {\displaystyle U(n)} is the maximal compact subgroup of P U ( n , 1 ) {\displaystyle PU(n,1)} . As a consequence, the
Complex_hyperbolic_space
Mathematics of smooth surfaces
geometry. Each of the two non-compact surfaces can be identified with the quotient G / K where K is a maximal compact subgroup of G. Here K is isomorphic
Differential geometry of surfaces
Differential_geometry_of_surfaces
Theorem in abstract algebra
to κ, KG and KH are hyperspecial maximal compact subgroups of G and H, which means roughly that they are the subgroups of points with coefficients in the
Fundamental lemma (Langlands program)
Fundamental_lemma_(Langlands_program)
Representation of the symmetry group of spacetime in special relativity
principal series. The restriction of a principal series to the maximal compact subgroup K = SU(2) of G can also be realized as an induced representation
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Let G be a finite group, with O(G) being its maximal normal subgroup of odd order. If T is a Sylow 2-subgroup of G containing an involution not conjugate
Z*_theorem
K]\subseteq K\}} . maximal 1. For "maximal compact subgroup", see compact. 2. For "maximal torus", see torus. parabolic 1. Parabolic subgroup 2. Parabolic
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
Lie group of complex numbers of unit modulus; topologically a circle
compact Lie group G {\displaystyle \mathrm {G} } of positive dimension contains a non-trivial maximal torus as a subgroup, and therefore a subgroup isomorphic
Circle_group
Technique in mathematical group theory
algebraic group G defined over a finite field Fq. If B is a Borel subgroup of G and T a maximal torus of B then we write WT,B for the Weyl group (normalizer
Deligne–Lusztig_theory
Structure group sub-bundle on a tangent frame bundle
{\displaystyle G} -bundle over a group G {\displaystyle G} "comes from" a subgroup H {\displaystyle H} of G {\displaystyle G} . This is called reduction of
G-structure_on_a_manifold
Spherical triangle that can be used to tile a sphere
e to be a positive root vector in X, the stabilizer of e is a maximal compact subgroup K of G isomorphic to O(2). The homogeneous space X = G / K is a
Schwarz_triangle
Theorem about projections of coadjoint orbits of a connected compact Lie group
permutations of the coordinates of Λ. Let K be a connected compact Lie group with maximal torus T and Weyl group W = NK(T)/T. Let their Lie algebras be
Kostant's_convexity_theorem
Axiom of set theory
has a maximal ideal, every vector space has a basis, every connected graph has a spanning tree, and every product of compact spaces is compact, among
Axiom_of_choice
Isometry group of Euclidean space
of subgroups of E(n): Finite groups. They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with
Euclidean_group
subgroup of G ν is an element of the complexification of a a is the Lie algebra of A in the Langlands decomposition P = MAN. K is a maximal compact subgroup
Eisenstein_integral
Transformations induced by a mathematical group
finite-dimensional vector space, it allows one to identify many groups with subgroups of the general linear group GL ( n , K ) {\displaystyle \operatorname
Group_action
contractible. 'Compact groups of (proper) stable planes are rather small. Let Φ d {\displaystyle \Phi _{d}} denote a maximal compact subgroup of the automorphism
Topological_geometry
MAXIMAL COMPACT-SUBGROUP
MAXIMAL COMPACT-SUBGROUP
Boy/Male
Latin
Greatest.
Boy/Male
Hindu
Dignity, Power
Boy/Male
American, Australian, Chinese, French, German, Greek, Latin, Swedish
Greatest
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Tamil, Telugu, Traditional
A String of Pearls
Boy/Male
Hindu, Indian, Marathi
The Garland of Lord Vishnu
Boy/Male
Indian, Sanskrit
Fallen from Glory
Male
Russian
(МакÑим) Variant spelling of Russian Maksim, MAXIM means "the greatest." Compare with another form of Maxim.
Male
French
French form of Latin Maximus, MAXIME means "the greatest."Â
Girl/Female
Indian
Soft
Girl/Female
Hindu
Full of jewel
Girl/Female
Hindu
Greatness
Boy/Male
Hindu, Indian
Praise
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Fragrance
Boy/Male
Sikh
A king
Boy/Male
Hindu
Devoted, A promise to God
Boy/Male
Gujarati, Hindu, Indian
Rich; Maladar
Boy/Male
Celebrity, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sanskrit, Sikh, Tamil, Telugu, Traditional
King of the Earth; A King
Boy/Male
British, English
Ermine; Ferret-like Mammal; Animal Name
Boy/Male
Hindu, Indian, Punjabi, Sikh, Tamil
Great Speech
Boy/Male
Muslim
Liberal, Generous, Another name for God
MAXIMAL COMPACT-SUBGROUP
MAXIMAL COMPACT-SUBGROUP
Female
Egyptian
, wife of Nehara.
Girl/Female
Bengali, Indian
Top; Who is at the Top
Female
Italian
Feminine form of Italian Orsino, ORSINA means "bear-like."
Surname or Lastname
English
English : from one of the many Middle English pet forms of Adam, formed with the hypocoristic suffix -cok.
Girl/Female
Muslim
Love, Friendship
Male
English
Anglicized form of Latin Isaias (Hebrew Yeshayah), ISAIAH means "God is salvation." In the bible, this is the name of one of the most famous prophets. Also spelled Jesaiah and Jeshaiah.
Girl/Female
American, British, English
Dark; Variant of Darcy
Boy/Male
Tamil
Narresh | நாரà¯à®°à¯‡à®·
King
Girl/Female
Christian & English(British/American/Australian)
Dweller of the Plain
Girl/Female
Gujarati, Hindu, Indian, Kannada, Mythological
A Great Rishi; A Beauty
MAXIMAL COMPACT-SUBGROUP
MAXIMAL COMPACT-SUBGROUP
MAXIMAL COMPACT-SUBGROUP
MAXIMAL COMPACT-SUBGROUP
MAXIMAL COMPACT-SUBGROUP
adv.
In a compact manner; with close union of parts; densely; tersely.
a.
Consisting of the flesh of animals; as, animal food.
n.
Extent; reach; sweep; capacity; sphere; as, the compass of his eye; the compass of imagination.
v. t.
To compact or join anew.
n.
The crew of a ship, including the officers; as, a whole ship's company.
a.
Of or relating to animals; as, animal functions.
v. i.
To bear or endure; to put up (with); as, to comport with an injury.
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.
n.
An inclosing limit; boundary; circumference; as, within the compass of an encircling wall.
n.
One who makes a compact.
n.
Contact or impression by touch; collision; forcible contact; force communicated.
n.
Guests or visitors, in distinction from the members of a family; as, to invite company to dine.
pl.
of Maximum
imp. & p. p.
of Compact
v.
Of or pertaining to a husband; as, marital rights, duties, authority.
p. p. & a
Brief; close; pithy; not diffuse; not verbose; as, a compact discourse.
a.
Greatest in quantity or highest in degree attainable or attained; as, a maximum consumption of fuel; maximum pressure; maximum heat.
v. t.
To manure with compost.
a.
Compact; pressed close; concentrated; firmly united.