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151 relations: Abelian group, Absolute Galois group, Admissible representation, Algebraic group, Analytic function, Andrew M. Gleason, Armand Borel, Élie Cartan, Banach algebra, Banach manifold, Banach space, Bounded operator, Cambridge University Press, Cantor set, Category of sets, Category of topological spaces, Category theory, Character theory, Circle group, Classical group, Classifying space, Closed set, Closed-subgroup theorem, Closure (topology), Cohomology, Compact group, Compact space, Complete metric space, Connected space, Continuous function, Continuous group action, Contractible space, Coset, Countable set, Covering space, Deane Montgomery, Diffeomorphism, Differentiable manifold, Direct integral, Direct sum, Discrete group, Discrete space, Euclidean distance, Euclidean geometry, Euclidean group, Euclidean space, Exterior algebra, Field (mathematics), Fourier series, Fourier transform, ..., Fundamental group, Garrett Birkhoff, Gauge theory, Gelfand–Raikov theorem, General linear group, Graded-commutative ring, Group (mathematics), Group action, Group homomorphism, Group isomorphism, Group object, H-space, Haar measure, Harmonic analysis, Hausdorff space, Heinz Hopf, Hermann Weyl, Hilbert space, Hilbert's fifth problem, Homeomorphism, Homeomorphism group, Homogeneous space, Homotopy, Hopf algebra, Hyperbolic geometry, Identity component, Integral, Inverse limit, Irreducible representation, Isometry, Isomorphism theorems, John Wiley & Sons, Kac–Moody algebra, Langlands classification, Leo Zippin, Lie algebra, Lie group, Linear algebra, Linear map, Locally compact group, Loop group, Loop space, Mathematics, Mathematische Annalen, Matrix (mathematics), Maximal compact subgroup, Measure (mathematics), Metrization theorem, N-sphere, Neighbourhood system, Normal subgroup, Number theory, Open and closed maps, Orthogonal group, P-adic number, Paratopological group, Peter–Weyl theorem, Polynomial ring, Pontryagin duality, Prime number, Principal bundle, Product topology, Profinite group, Quotient group, Quotient space (topology), Rational number, Real number, Real projective space, Representation theory, Representation theory of finite groups, Representation theory of SL2(R), Rotation group SO(3), Semisimple Lie algebra, Semitopological group, Shizuo Kakutani, Smoothness, Special unitary group, Springer Science+Business Media, Subgroup, Submanifold, Subspace topology, Symmetry, Symmetry (physics), Tensor product of algebras, Topological group, Topological manifold, Topological ring, Topological space, Topological vector space, Torus, Totally disconnected group, Tychonoff space, Uniform continuity, Uniform convergence, Uniform space, Unitary representation, University of Chicago Press, Weyl character formula, Whitehead product, Wilhelm Killing, World Scientific. Expand index (101 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Absolute Galois group
In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism.
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Admissible representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups.
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Algebraic group
In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
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Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series.
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Andrew M. Gleason
Andrew Mattei Gleason (19212008) was an American mathematician who as a young World War II naval officer broke German and Japanese military codes, then over the succeeding sixty years made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in teaching at all levels.
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Armand Borel
Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993.
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Élie Cartan
Élie Joseph Cartan, ForMemRS (9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.
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Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, i.e. a normed space and complete in the metric induced by the norm.
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Banach manifold
In mathematics, a Banach manifold is a manifold modeled on Banach spaces.
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Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
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Bounded operator
In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).
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Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.
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Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets.
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Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps or some other variant; for example, objects are often assumed to be compactly generated.
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Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
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Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix.
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Circle group
In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers.
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Classical group
In mathematics, the classical groups are defined as the special linear groups over the reals, the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces.
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Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e. a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG.
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Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
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Closed-subgroup theorem
In mathematics, the closed-subgroup theorem (sometimes referred to Cartan's theorem) is a theorem in the theory of Lie groups.
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Closure (topology)
In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
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Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.
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Compact group
In mathematics, a compact (topological) group is a topological group whose topology is compact.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Complete metric space
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).
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Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Continuous group action
In topology, a continuous group action on a topological space X is a group action of a topological group G that is continuous: i.e., is a continuous map.
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Contractible space
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map.
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Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.
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Countable set
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
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Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.
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Deane Montgomery
Deane Montgomery (September 2, 1909 – March 15, 1992) was a mathematician specializing in topology who was one of the contributors to the final resolution of Hilbert's fifth problem in the 1950s.
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Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.
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Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
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Direct integral
In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum.
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Direct sum
The direct sum is an operation from abstract algebra, a branch of mathematics.
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Discrete group
In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of G in which every open subset contains exactly one element of H; in other words, the subspace topology of H in G is the discrete topology.
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Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense.
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Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.
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Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.
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Euclidean group
In mathematics, the Euclidean group E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space.
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Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
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Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Fourier series
In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.
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Fourier transform
The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.
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Fundamental group
In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.
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Garrett Birkhoff
Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician.
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Gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.
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Gelfand–Raikov theorem
The Gel'fand–Raikov (Гельфанд–Райков) theorem is a theorem in the theory of locally compact topological groups.
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General linear group
In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.
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Graded-commutative ring
In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements x, y satisfy where |x|, |y| denote the degrees of x, y. A commutative (non-graded) ring, with trivial grading, is a basic example.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Group homomorphism
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".
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Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
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Group object
In category theory, a branch of mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets.
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H-space
In mathematics, an H-space, or a topological unital magma, is a topological space X (generally assumed to be connected) together with a continuous map μ: X × X → X with an identity element e so that μ(e, x).
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Haar measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
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Harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).
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Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.
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Heinz Hopf
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry.
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Hermann Weyl
Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Hilbert's fifth problem
Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups.
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Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
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Homeomorphism group
In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation.
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Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively.
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Homotopy
In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
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Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.
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Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.
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Identity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group.
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Integral
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.
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Inverse limit
In mathematics, the inverse limit (also called the projective limit or limit) is a construction that allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects.
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Irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper subrepresentation (\rho|_W,W), W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hermitian vector space V is the direct sum of irreducible representations.
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Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
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Isomorphism theorems
In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects.
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John Wiley & Sons
John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.
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Kac–Moody algebra
In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently discovered them) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix.
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Langlands classification
In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group G, suggested by Robert Langlands (1973).
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Leo Zippin
Leo Zippin (1905 – May 11, 1995) was an American mathematician.
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Lie algebra
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
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Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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Linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.
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Linear map
In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
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Locally compact group
In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff.
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Loop group
In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise.
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Loop space
In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, maps from the circle S1 to X, equipped with the compact-open topology.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Mathematische Annalen
Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Maximal compact subgroup
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 such subgroups.
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Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.
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Metrization theorem
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.
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N-sphere
In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension.
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Neighbourhood system
In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x is the collection of all neighbourhoods for the point x.
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Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.
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Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
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Open and closed maps
In topology, an open map is a function between two topological spaces which maps open sets to open sets.
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Orthogonal group
In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.
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P-adic number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.
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Paratopological group
In mathematics, a paratopological group is a topological semigroup that is algebraically a group.
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Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian.
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Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.
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Pontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as \R, the circle, or finite cyclic groups.
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Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
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Principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group.
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Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.
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Profinite group
In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups.
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Quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.
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Quotient space (topology)
In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Real projective space
In mathematics, real projective space, or RPn or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in Rn+1.
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Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.
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Representation theory of finite groups
The representation theory of groups is a part of mathematics which examines how groups act on given structures.
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Representation theory of SL2(R)
In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2,'''R''') are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).
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Rotation group SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.
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Semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras \mathfrak g whose only ideals are and \mathfrak g itself.
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Semitopological group
In mathematics, a semitopological group is a topological space with a group action that is continuous with reference to a each variable considered separately.
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Shizuo Kakutani
was a Japanese-American mathematician, best known for his eponymous fixed-point theorem.
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Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.
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Special unitary group
In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Subgroup
In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
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Submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties.
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Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).
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Symmetry
Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance.
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Symmetry (physics)
In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.
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Tensor product of algebras
In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra.
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Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.
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Topological manifold
In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below.
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Topological ring
In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps where R × R carries the product topology.
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Topological vector space
In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.
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Torus
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
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Totally disconnected group
In mathematics, a totally disconnected group is a topological group that is totally disconnected.
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Tychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.
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Uniform continuity
In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between f(x) and f(y) cannot depend on x and y themselves.
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Uniform convergence
In the mathematical field of analysis, uniform convergence is a type of convergence of functions stronger than pointwise convergence.
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Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure.
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Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.
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University of Chicago Press
The University of Chicago Press is the largest and one of the oldest university presses in the United States.
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Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights.
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Whitehead product
In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space.
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Wilhelm Killing
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.
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World Scientific
World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore.
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Birkhoff-Kakutani theorem, Birkhoff–Kakutani theorem, Closed subgroup, Continuous group, Quasitopological group, Topological Group, Topological groups.
References
[1] https://en.wikipedia.org/wiki/Topological_group
