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Duality for locally compact abelian groups
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups,
Pontryagin_duality
Soviet mathematician (1908–1988)
became vice president of the International Mathematical Union. Pontryagin worked on duality theory for homology while still a student. He went on to lay
Lev_Pontryagin
General concept and operation in mathematics
duality Langlands dual Linear programming#Duality List of dualities Matlis duality Petrie duality Pontryagin duality S-duality T-duality, Mirror symmetry
Duality_(mathematics)
Duality between a group and its representations
category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups
Tannaka–Krein_duality
Topological group structure arising in Fourier analysis
dual. (In fact, any finite extension of Q p {\displaystyle \mathbb {Q} _{p}} is also self-dual.) It follows that the adeles are self-dual. Pontryagin
Locally_compact_abelian_group
Area of mathematical analysis
on functions and representations on topological groups, including Pontryagin duality, the Peter–Weyl theorem, and Plancherel-type theorems. Harmonic analysis
Harmonic_analysis
Lie group of complex numbers of unit modulus; topologically a circle
character group of T {\displaystyle \mathbb {T} } , also called its Pontryagin dual, is an infinite cyclic group generated by ϕ 1 {\displaystyle \phi
Circle_group
Concept in mathematics
by Pontryagin duality. In general, the unitary equivalence classes (see below) of irreducible unitary representations of G make up its unitary dual. This
Unitary_representation
Theorem about Diophantine approximations
on P. By Pontryagin duality we have T′ contained in the kernel of χ, and therefore not equal to T. In fact a thorough use of Pontryagin duality here shows
Kronecker's_theorem
Variables that are Fourier transform duals
way that they become Fourier transform duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty
Conjugate_variables
In mathematics, Cartier duality is an analogue of Pontryagin duality for commutative group schemes. It was introduced by Pierre Cartier (1962). Given any
Cartier_duality
In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic
Tate_duality
Topics referred to by the same term
mathematics, the dual group refer to: Pontryagin dual, of a locally compact abelian group Langlands dual, of a reductive algebraic group The dual group in the
Dual_group
Fourier transform of a real-space lattice, important in solid-state physics
stated simply in terms of Pontryagin duality. The dual group V^ to V is again a real vector space, and its closed subgroup L^ dual to L turns out to be a
Reciprocal_lattice
Mathematical transform that expresses a function of time as a function of frequency
formula, or Parseval's relation, or even Parseval's theorem. See Pontryagin duality for a general formulation of this concept in the context of locally
Fourier_transform
Theorem in algebra
In algebra, Matlis duality is a duality between Artinian and Noetherian modules over a complete Noetherian local ring. In the special case when the local
Matlis_duality
In mathematics, vector space of linear forms
contravariance of vectors Dual module Dual norm Duality (mathematics) Duality (projective geometry) Pontryagin duality Reciprocal lattice – dual space basis, in
Dual_space
Type of topological space in mathematics
The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact. More precisely, Pontryagin duality defines
Locally_compact_space
Mathematical theory
In mathematics, Alexander duality refers to a duality theory initiated by a result of J. W. Alexander in 1915, and subsequently further developed, particularly
Alexander_duality
Mathematical concept
duality theory for locally compact groups is however much weaker than the Tannaka–Krein duality theory for compact topological groups or Pontryagin duality
Spectrum_of_a_C*-algebra
Type of topological group in mathematics
representation theory for locally compact abelian groups is described by Pontryagin duality. Any compact group is locally compact. In particular the circle group
Locally_compact_group
Algebraic structure used in topology
configuration space, the deleted product). In 1934, Lev Pontryagin proved the Pontryagin duality theorem; a result on topological groups. This (in rather
Cohomology
Theorem of Fourier transforms of Borel measures
compact abelian group corresponds to a finite positive measure on the Pontryagin dual group. The case of sequences was first established by Gustav Herglotz
Bochner's_theorem
Physical spaces representing position and momentum, Fourier-transform duals
mass⋅length⋅time−1. Mathematically, the duality between position and momentum is an example of Pontryagin duality. In particular, if a function is given
Position_and_momentum_spaces
Application of Fourier analysis to non-abelian topological groups
Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier
Noncommutative harmonic analysis
Noncommutative_harmonic_analysis
Theorem in mathematics
given an abelian locally compact group G with Pontryagin dual G^, Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between
Parseval's_theorem
Mathematical function
Infinitesimal character Alternating character Characterization (mathematics) Pontryagin duality Base (topology) § Weight and character "character in nLab". ncatlab
Character_(mathematics)
Mathematical object
(products of a finite number of copies of the circle group), or the Pontryagin dual of a discrete torsion-free abelian group. Some examples of protori
Protorus
Left-invariant (or right-invariant) measure on locally compact topological group
analysis on locally compact groups, particularly in the theory of Pontryagin duality. To prove the existence of a Haar measure on a locally compact group
Haar_measure
Branch of mathematics
more generally, as the Plancherel theorem, and most generally via Pontryagin duality). The transforms are usually invertible. The exponential functions
Fourier_analysis
Very general problem in computer science
the HSP, and whether or not they are solvable. Hidden shift problem Pontryagin duality Mark Ettinger; Peter Høyer (1999). "A quantum observable for the graph
Hidden_subgroup_problem
Type of topological space
cases, this can be usefully applied, for example in combination with Pontryagin duality. A 0-dimensional manifold (or differentiable or analytic manifold)
Discrete_space
Poincaré duality Poitou–Tate duality Pontryagin duality S-duality (homotopy theory) Schur–Weyl duality Series-parallel duality Serre duality Spanier–Whitehead
List_of_dualities
Type of mathematical object
replaced by discs of positive radius. Cartier duality is a scheme-theoretic analogue of Pontryagin duality taking finite commutative group schemes to finite
Group_scheme
theorem (measure theory) Peter–Weyl theorem (representation theory) Pontryagin duality theorem (representation theory) Final value theorem (mathematical
List_of_theorems
Theorem in harmonic analysis
In the case of an abelian group G {\displaystyle G} , there is a Pontryagin dual group G ^ {\displaystyle {\widehat {G}}} of characters on G {\displaystyle
Plancherel_theorem
(declined both) Lev Pontryagin, blind mathematician, developed Pontryagin duality and Pontryagin classes in topology, and Pontryagin's minimum principle
List of Russian mathematicians
List_of_Russian_mathematicians
Function that "converges" to periodicity
developments and the advent of abstract methods (the Peter–Weyl theorem, Pontryagin duality and Banach algebras) a general theory became possible. The general
Almost_periodic_function
Group homomorphism into the general linear group over a vector space
The resulting theory is a central part of harmonic analysis. The Pontryagin duality describes the theory for commutative groups, as a generalised Fourier
Group_representation
Involutive change of basis in linear algebra
/ 2 Z ) n {\displaystyle r\in (\mathbb {Z} /2\mathbb {Z} )^{n}} (Pontryagin duality) and define f ^ : ( Z / 2 Z ) n → C {\displaystyle {\widehat {f}}\colon
Hadamard_transform
Topological group with compact topology
case of infinite degree. Pontryagin duality provides a large supply of examples of compact commutative groups. These are in duality with abelian discrete
Compact_group
Construction in algebra
1007/978-3-319-49834-8, ISBN 978-3-319-49833-1. Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra"
Hopf_algebra
Collection of mathematical theories
transform's basic properties, and this is carried out by means of Pontryagin duality. One can also study the spectral properties of operators on Banach
Spectral_theory
Equation in Fourier analysis
can also be proved quite conceptually using the compatibility of Pontryagin duality with short exact sequences such as 0 → Z → R → R / Z → 0. {\textstyle
Poisson_summation_formula
French mathematician (1947–2025)
reached a state of maturity Specifically, Enock co-developed a general Pontryagin duality theory for all locally compact groups. Enock completed his postgraduate
Michel_Enock
Number system extending the rational numbers
Pontryagin dual of the group of p-adic integers is the Prüfer p-group Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty })} , and the Pontryagin dual of
P-adic_number
Locally compact group – Type of topological group in mathematics Pontryagin duality – Duality for locally compact abelian groups Protorus – Mathematical object
Topological_abelian_group
Number-theoretic concept
Hausdorff abelian group, and thus its Pontryagin dual must be a discrete abelian group. In fact, the Pontryagin dual of Z ^ {\displaystyle {\widehat {\mathbb
Profinite_integer
transform Dirichlet character Amenable group Von Neumann's conjecture Pontryagin duality Kronecker's theorem on diophantine approximation Almost periodic function
List of harmonic analysis topics
List_of_harmonic_analysis_topics
Branch of number theory
The 3-adic integers, with selected corresponding characters on their Pontryagin dual group
P-adic_analysis
Topological group that is in a certain sense assembled from a system of finite groups
to being 'ind-finite'. By applying Pontryagin duality, one can see that abelian profinite groups are in duality with locally finite discrete abelian
Profinite_group
Representation theory of groups
harmonic analysis. The locally compact abelian case is part of the Pontryagin duality theory. In Galois theory it is shown that for a field L, and a finite
Regular_representation
strong characters. Equivalently, a suitably defined dual set is relatively dense in the Pontryagin dual of the group. This notion was introduced by Yves
Harmonious_set
Mathematical term in group theory
discrete topology) is the Pontryagin dual of the compact group of p-adic integers, and the group of p-adic integers is the Pontryagin dual of the Prüfer p-group
Prüfer_group
Indian poet, writer and film director. Lev Pontryagin, 79, Soviet mathematician (Pontryagin duality, Pontryagin cohomology operation). Abraham Seidenberg
Deaths_in_May_1988
(declined both) Lev Pontryagin, blind mathematician, developed Pontryagin duality and Pontryagin classes in topology, and Pontryagin's minimum principle
List_of_Russian_scientists
Branch of algebraic number theory concerned with abelian extensions
Wolfgang Krull's theory of their Galois groups. This combined with Pontryagin duality to give a clearer if more abstract formulation of the central result
Class_field_theory
Group that is a topological space with continuous group operations
case, the unitary dual G ^ {\displaystyle {\hat {G}}} is a group, in fact another locally compact abelian group. Pontryagin duality states that for a
Topological_group
Commutative group (mathematics)
Grothendieck group – Abelian group extending a commutative monoid Pontryagin duality – Duality for locally compact abelian groups ^ Among mathematical adjectives
Abelian_group
Basic result in harmonic analysis on compact topological groups
not itself be a Lie group: it may for example be a profinite group. Pontryagin duality Peter, F.; Weyl, H. (1927), "Die Vollständigkeit der primitiven Darstellungen
Peter–Weyl_theorem
Austrian mathematician (1899–1982)
transforms appeared in a 1932 book. His techniques came into their own as Pontryagin duality and then the representation theory of locally compact groups developed
Salomon_Bochner
Fraction with denominator a power of two
structure of an additive abelian group. Pontryagin duality is a method for understanding abelian groups by constructing dual groups, whose elements are characters
Dyadic_rational
finite degree extension of local fields. It plays a basic role in Pontryagin duality for p-adic fields. The relative different δL / K is defined in a similar
Different_ideal
French mathematician (1906-1998)
carried out from the late 1960s. Other significant results were on Pontryagin duality and differential geometry. He introduced the concept of a uniform
André_Weil
German mathematician (born 1958)
Artin–Verdier duality. Broadly speaking, Artin–Verdier duality, a consequence of class field theory, is an arithmetic analogue of Poincaré duality, a duality for
Christopher_Deninger
Group representation
(x)=[e^{iax}]} , for some real number a {\displaystyle a} . See also Pontryagin duality for this case. Representation theory of connected compact groups Lie
Representation_of_a_Lie_group
Topological space
where F ⊂ Z {\displaystyle F\subset \mathbb {Z} } is finite. Thus the Pontryagin dual is naturally identified with the group of finite subsets of Z {\displaystyle
Cantor_space
Mathematical Society. ISBN 0-8218-0780-3. Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra"
Refinement_(category_theory)
Concept in number theory (mathematics)
rather favourable terms – the harmonic analysis required is all of the Pontryagin duality type, rather than needing more general automorphic representations
Arithmetic of abelian varieties
Arithmetic_of_abelian_varieties
Measure for Baire sets in mathematics
coincide and we say the Haar measure is translation invariant. See also Pontryagin duality. Leonard Gillman and Meyer Jerison, Rings of Continuous Functions
Baire_measure
Mathematical category formed by reversing morphisms
equivalent to the opposite of the category of commutative rings. The Pontryagin duality restricts to an equivalence between the category of compact Hausdorff
Opposite_category
the complex conjugate of f k ( g i ) {\displaystyle f_{k}(g_{i})} . Pontryagin duality Birkenhake, Christina; H. Lange (2004). Complex Abelian varieties
Character_group
Amsterdam: North Holland. ISBN 9780080871356. Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra"
Topological_algebra
Retired Australian mathematics professor
York: W. De Gruyter. ISBN 3-11-015268-1. Morris, Sidney A. (2009). Pontryagin duality and the structure of locally compact abelian groups. Cambridge: Cambridge
Sidney_Morris
American mathematician
interesting results. In Lawson, Harvey and Zweck established a Poincaré-Pontryagin Duality for the differential characters of Cheeger and Simons. They also gave
H._Blaine_Lawson
Class of compact connected topological spaces
Protorus, a class of topological groups that includes the solenoids Pontryagin duality p-adic solenoid Profinite integer Hewitt, Edwin; Ross, Kenneth A.
Solenoid_(mathematics)
Mathematical theorem
and momentum. Let G be a locally compact abelian group and G^ be the Pontryagin dual of G. The Fourier–Plancherel transform defined by f ↦ f ^ ( γ ) = ∫
Stone–von_Neumann_theorem
uniqueness is a mathematical result that relies on the Pontryagin duality theorem, the Tannaka–Krein duality theorem, and related results of Iwahori-Sugiura
Triple_correlation
original one. This result gives a far-reaching generalization of Pontryagin duality for locally compact Hausdorff abelian groups. The theory has an equivalent
Locally_compact_quantum_group
Japanese mathematician
is known for developing the theory of Tannaka–Krein duality, which generalizes Pontryagin duality to noncommutative compact groups and led to the development
Tadao_Tannaka
{\displaystyle I} to be the circle group we obtain the theory of Pontryagin duality. In the category of abelian groups, the group of integers Z {\displaystyle
Generator_(category_theory)
Concept in mathematics
{\displaystyle \coprod _{n=0}^{\infty }\mathbb {Z} /2\mathbb {Z} } , the Pontryagin dual of the Cantor group ∏ n = 0 ∞ Z / 2 Z {\displaystyle \prod _{n=0}^{\infty
Walsh_function
Type of character in number theory
dissertation, written under the supervision of Emil Artin, applied Pontryagin duality systematically, to remove the need for any special functions. A similar
Hecke_character
Branch of mathematics that studies dynamical systems
normalized Haar measure, and T a group automorphism of G. Let G* be the Pontryagin dual group, consisting of the continuous characters of G, and T* be the
Ergodic_theory
Type of operator in Fourier analysis
its Fourier transform (where G ^ {\displaystyle {\hat {G}}} is the Pontryagin dual of G). Let m : G ^ → C {\displaystyle m:{\hat {G}}\to \mathbb {C} }
Multiplier_(Fourier_analysis)
Overview of and topical guide to category theory
Representable functor Functor category Adjoint functors Galois connection Pontryagin duality Affine scheme Monad (category theory) Comonad Combinatorial species
Outline_of_category_theory
Ukrainian and French mathematician
with George I. Kac (Georgii Isaakovich Kac) on generalizations of Pontryagin duality to non-commutative groups and developed the concept now known as Kac
Leonid_I._Vainerman
Theory in abstract algebra
multiplication). This group (with discrete topology) can also be viewed as Pontryagin dual of Gal ( L / K ) {\displaystyle \operatorname {Gal} (L/K)} , assuming
Kummer_theory
(declined both) Lev Pontryagin, blind mathematician, developed Pontryagin duality and Pontryagin classes in topology, and Pontryagin's minimum principle
List_of_Russian_people
Generalization of the discrete Fourier transform
{\displaystyle S^{1}=\{z\in \mathbb {C} ,|z|=1\}} . This group is known as the Pontryagin dual of G. The Fourier transform of a function f : G → C {\displaystyle
Fourier transform on finite groups
Fourier_transform_on_finite_groups
Group in mathematical representation theory
{\displaystyle \mathbb {R} } by any locally compact abelian group G whose Pontryagin dual (the group of characters) is isomorphic to G. The Hilbert space H is
Metaplectic_group
Topological space associated to a vector bundle
mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential
Thom_space
measure Hardy space Sobolev space Topological group Set of uniqueness Pontryagin duality Plancherel theorem Peter–Weyl theorem Fourier integral operator Oscillatory
List of Fourier analysis topics
List_of_Fourier_analysis_topics
notion of envelope plays a key role in the generalizations of the Pontryagin duality theory to the classes of non-commutative groups: the holomorphic,
Envelope_(category_theory)
Soviet mathematician (1896–1982)
death. Among the students of P. S. Alexandrov, the most famous are Lev Pontryagin, Andrey Tychonoff and Aleksandr Kurosh. The older generation of his students
Pavel_Alexandrov
Pontrjagin duality theorem in linear spaces". Annals of Mathematics. 56 (2): 248–253. doi:10.2307/1969798. JSTOR 1969798. Akbarov, S.S. (2003). "Pontryagin duality
Smith_space
845–855. doi:10.1215/S0012-7094-73-04078-7. Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra"
Brauner_space
Algebraic topology uses abstract algebra to study topological spaces
Obstruction theory Characteristic class Chern class Chern–Simons form Pontryagin class Pontryagin number Stiefel–Whitney class Poincaré conjecture Cohomology operation
List of algebraic topology topics
List_of_algebraic_topology_topics
Solitons in Euclidean spacetime
"sector" of the true vacuum) is labelled by an unaltered transform, the Pontryagin index. As the third homotopy group of S 3 {\displaystyle S^{3}} has been
Instanton
Space homeomorphic to some ring spectrum
the lattice K ∘ {\displaystyle \circ } (X). A.V. Arkhangel'skii, L.S. Pontryagin (Eds.) General Topology I (1990) Springer-Verlag ISBN 3-540-18178-4 (See
Spectral_space
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