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Pure-injective modules in mathematics
In mathematics, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution
Algebraically_compact_module
Direct summand of a free module (mathematics)
mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring
Projective_module
W XYZ See also References algebraically compact algebraically compact module (also called pure injective module) is a module in which all systems of equations
Glossary_of_module_theory
Branch of mathematics that studies algebraic structures
related module Algebraically compact module Reflexive module Composition series Length of a module Structure theorem for finitely generated modules over a principal
List of abstract algebra topics
List_of_abstract_algebra_topics
Writing Lie algebra sets as matrices
isomorphism. If V is an irreducible g {\displaystyle {\mathfrak {g}}} -module over an algebraically closed field and f : V → V {\displaystyle f:V\to V} is a homomorphism
Lie_algebra_representation
In mathematics, the Hecke algebra of a pair (G, K) of locally compact or reductive Lie groups is an algebra of measures under convolution. It can also
Hecke_algebra_of_a_pair
Ring that is also a vector space or a module
the structure of a module or vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over K. A standard
Associative_algebra
nonassociative algebras. An algebra is a module, wherein you can also multiply two module elements. (The multiplication in the module is compatible with
List_of_algebras
Mathematical concept
regarded algebraically. For a ring an ideal is primitive if and only if it is the annihilator of a simple module. It turns out that for a C*-algebra A, an
Spectrum_of_a_C*-algebra
Manifold or algebraic variety of dimension n in a space of dimension n+1
of p in the affine space K n , {\displaystyle K^{n},} where K is an algebraically closed extension of k. A hypersurface may have singularities, which
Hypersurface
Direct sum of simple Lie algebras
characteristic zero but is not algebraically closed, there is no general structure theory like the one for those over an algebraically closed field of characteristic
Semisimple_Lie_algebra
Mathematical objects that generalise the notion of Hilbert spaces
\mathbb {C} } -module under scalar multipliation by complex numbers and its inner product. If X {\displaystyle X} is a locally compact Hausdorff space
Hilbert_C*-module
Topological complex vector space
is a locally compact Hausdorff space. C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables
C*-algebra
Algebraic structure in ring theory
In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module M over
Flat_module
Algebra used in 2D conformal field theories and string theory
construct the moonshine module. They observed that many vertex algebras that appear 'in nature' carry an action of the Virasoro algebra, and satisfy a bounded-below
Vertex_operator_algebra
Algebraic ring without a multiplicative identity
unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields Formally, a rng is a set R with two binary operations
Rng_(algebra)
Concept in Lie algebra representation theory
of a connected compact Lie group Highest-weight category Root system In fact, given a set of commuting matrices over an algebraically closed field, they
Weight (representation theory)
Weight_(representation_theory)
*-algebra of bounded operators on a Hilbert space
some apparently topological properties in von Neumann algebras can be defined purely algebraically. von Neumann, J. (1949), "On Rings of Operators. Reduction
Von_Neumann_algebra
Construction in algebra
Hopf algebra. The axioms are partly chosen so that the category of H-modules is a rigid monoidal category. The unit H-module is the separable algebra HL
Hopf_algebra
Cohomology theory for Lie algebras
properties of the Lie algebra. It was later extended by Claude Chevalley and Samuel Eilenberg (1948) to coefficients in an arbitrary Lie module. If G {\displaystyle
Lie_algebra_cohomology
Algebraic structure used in analysis
problem of classifying the simple Lie algebras. The simple Lie algebras of finite dimension over an algebraically closed field F of characteristic zero
Lie_algebra
Algebraic construct of interest in theoretical physics
noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix
Quantum_group
Two closely related mathematical subjects
Lefschetz, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field K of characteristic
Algebraic geometry and analytic geometry
Algebraic_geometry_and_analytic_geometry
M(K(E)) = B(E) for any Hilbert module E. The C*-algebra A is isomorphic to the compact operators on the Hilbert module A. Therefore, M(A) is the adjointable
Multiplier_algebra
Set of finitely supported functions from a group to a ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free
Group_ring
Relates the geometric vector bundles to algebraic projective modules
Jean-Pierre Serre in 1955, is more algebraic in nature, and concerns vector bundles on an algebraic variety over an algebraically closed field (of any characteristic)
Serre–Swan_theorem
Representation theory of groups
the explicit form in this case of the abstract result that over an algebraically closed field K (such as the complex numbers) the regular representation
Regular_representation
Algebraic structure with "nice" duality properties
module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly
Frobenius_algebra
Mathematical object
modules over R (a perfect module being defined as a compact object in the ∞-category of module spectra). G-spectrum J. Lurie, Lecture 19: Algebraic K-theory
Module_spectrum
Mathematical object studied in the field of algebraic geometry
be not algebraically closed. Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed
Algebraic_variety
Category whose objects are R-modules and whose morphisms are module homomorphisms
algebra, given a ring R {\displaystyle R} , the category of left modules over R {\displaystyle R} is the category whose objects are all left modules over
Category_of_modules
Mathematical property
modules over a semi-simple ring must split, i.e., M ≅ M ′ ⊕ M ″ {\displaystyle M\cong M'\oplus M''} . From the point of view of homological algebra,
Semi-simplicity
In algebra, a perfect complex of modules over a commutative ring A is an object in the derived category of A-modules that is quasi-isomorphic to a bounded
Perfect_complex
finite-dimensional module over g {\displaystyle {\mathfrak {g}}} is semisimple as a module (i.e., a direct sum of simple modules.) Weyl's theorem implies
Weyl's theorem on complete reducibility
Weyl's_theorem_on_complete_reducibility
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
(topology) Adjunction space Topological algebra Topological group Topological ring Topological vector space Topological module Topological abelian group Properly
List of general topology topics
List_of_general_topology_topics
Generalization of algebraic variety
algebraic geometry over the real numbers is simplified by working over the field of complex numbers, which has the advantage of being algebraically closed
Scheme_(mathematics)
Construction using Verma modules. This approach is purely Lie algebraic. (Generally applicable to complex semisimple Lie algebras.) The compact group approach using
Representation theory of semisimple Lie algebras
Representation_theory_of_semisimple_Lie_algebras
Nilpotent subalgebra of a Lie algebra
semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over an algebraically closed field of characteristic
Cartan_subalgebra
In algebra, completion w.r.t. powers of an ideal
abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion
Completion_of_a_ring
scheme X is a sheaf of OX-modules that is locally given by modules. quasi-compact A morphism f : Y → X is called quasi-compact, if for some (equivalently:
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Particular kind of algebraic structure
is compact. The complex conjugation being an involution, C 0 ( X ) {\displaystyle C_{0}(X)} is in fact a C*-algebra. More generally, every C*-algebra is
Banach_algebra
generated modules in algebra. (There are other notions of compactness in mathematics.) In a partially ordered set (P,≤) an element c is called compact (or finite)
Compact_element
Operation in algebra and mathematics
{\displaystyle R} -algebra on the right is considered as a module. Then, an algebra over this monad are commutative R {\displaystyle R} -algebras. There are also
Monad_(category_theory)
Group of unitary matrices
unitary group is a linear algebraic group. The unitary group of a quadratic module is a generalisation of the linear algebraic group U {\displaystyle U}
Unitary_group
introduced by Atiyah (1970). If A is an involutive algebra over the complex numbers C, then a Fredholm module over A consists of an involutive representation
Fredholm_module
Mathematical concept
coproduct. The compact objects in the category of sets are precisely the finite sets. For a ring R, the compact objects in the category of R-modules are precisely
Compact_object_(mathematics)
Sheaf cohomology on the étale site
variety is to calculate them for complete connected smooth algebraic curves X over algebraically closed fields k. The étale cohomology groups of arbitrary
Étale_cohomology
Branch of mathematics
studied through categories of sheaves or modules. In these classical examples, geometry is encoded algebraically. Addition and multiplication of functions
Noncommutative_geometry
Net in a normed algebra
any C*-algebra. This is called the canonical approximate identity of a C*-algebra. Approximate identities are not unique. For example, for compact operators
Approximate_identity
In algebra, module with a finite generating set
generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over
Finitely_generated_module
Representation of a group or algebra that is a direct sum of simple representations
can be approximated by semisimple representations. A semisimple module over an algebra over a field is an example of a semisimple representation. Conversely
Semisimple_representation
Cohomology theory for topological spaces
its closure is compact. Similar to the definition of Alexander cohomology module, one can define Alexander cohomology module with compact supports of a
Alexander–Spanier_cohomology
Algebraic structure
unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields A ring is a set R {\displaystyle R} equipped with two
Commutative_ring
Generalization of vector bundles
{\displaystyle {\mathcal {F}}} of O X {\displaystyle {\mathcal {O}}_{X}} -modules that has a local presentation, that is, every point in X {\displaystyle
Coherent_sheaf
Set of a ring's prime ideals
separate origin in commutative algebra and algebraic geometry. For an affine algebraic variety over an algebraically closed field, the Nullstellensatz
Spectrum_of_a_ring
Duality for locally compact abelian groups
{\displaystyle R} –module; in this way we can also see that discrete left R {\displaystyle R} –modules will be Pontryagin dual to compact right R {\displaystyle
Pontryagin_duality
Class of mathematical sets
Hausdorff σ-compact spaces, but can be different in more pathological spaces. In the case that X {\displaystyle X} is a metric space, the Borel algebra in the
Borel_set
Algebraic structure
an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler
Hodge_structure
Topological structure in number theory
Iwasawa algebras were introduced by Iwasawa (1959) in his study of Zp extensions in Iwasawa theory, and non-commutative Iwasawa algebras of compact p-adic
Iwasawa_algebra
see Cartan. 2. For "Generalized Kac–Moody algebra", see Kac–Moody algebra. 3. For "Generalized Verma module", see Verma. group Group analysis of differential
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
Concept in mathematics
over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups
Reductive_group
in the dual module). An equivalent characterization is that A is amenable if and only if it has a virtual diagonal. If A is a group algebra L 1 ( G ) {\displaystyle
Amenable_Banach_algebra
Tool in algebraic topology
a finitely generated R-module. Then the cohomology groups Hj(X,E) are finitely generated R-modules. For example, for a compact Hausdorff space X that
Sheaf_cohomology
Result about when a matrix can be diagonalized
descriptions of redirect targets Spectral theory of compact operators Spectral theory of normal C*-algebras Borel functional calculus Spectral theory Matrix
Spectral_theorem
Algebraic structure with addition, multiplication, and division
maximal number of elements in F that are algebraically independent over the prime field. Two algebraically closed fields E and F are isomorphic precisely
Field_(mathematics)
Concept in mathematics
Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as
Universal_enveloping_algebra
Type of topological group in mathematics
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
Branch of mathematics
many other algebraic structures studied by algebra. They include magmas, semigroups, monoids, abelian groups, commutative rings, modules, lattices, vector
Algebra
Algebra of formal sums
\mathbb {Z} } -modules, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free
Free_abelian_group
Mathematical relation in abstract algrebra
R-module. Let M be a left S-module and N a left R-module. By restriction of scalars, M is also a left R-module. If S is projective as a right R-module,
Shapiro's_lemma
Deformation of the group algebra of a Coxeter group
algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a deformation of the group algebra of a Coxeter group. The Hecke algebra can
Iwahori–Hecke_algebra
Branch of functional analysis
Neumann algebras. Commutative self-adjoint operator algebras can be regarded as the algebra of complex-valued continuous functions on a locally compact space
Operator_algebra
Concept in abstract algebra
of formal Laurent series k ( ( t ) ) {\displaystyle k((t))} over an algebraically closed field k of characteristic zero also has absolute Galois group
Cohomological_dimension
Algebraic concept in measure theory, also referred to as an algebra of sets
and compact (in which case it is described as being descriptive) The Stone representation of a Boolean algebra is always separative and compact; the
Field_of_sets
an algebraically closed field of characteristic zero, the Borel–Weil–Bott theorem realizes an irreducible representation of a reductive algebraic group
Glossary of representation theory
Glossary_of_representation_theory
Group representation
(or reductive) Lie groups, where the associated Lie algebra representation forms a (g,K)-module. Examples of unitary representations arise in quantum
Representation_of_a_Lie_group
{\displaystyle ({\mathfrak {g}},K)} -modules, where g {\displaystyle {\mathfrak {g}}} is the Lie algebra of G and K is a maximal compact subgroup of G. Let G be a
(g,K)-module
Matrices named after Élie Cartan
Lie algebras (say over some algebraically closed field of characteristic 0). The determinants of the Cartan matrices of the simple Lie algebras are given
Cartan_matrix
Subject area in mathematics
A is a finitely generated Z-algebra. (The groups Gn(A) are the K-groups of the category of finitely generated A-modules) Additive K-theory Bloch's formula
Algebraic_K-theory
Algebraic structure in linear algebra
all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a module over a ring which is a field, with the
Vector_space
Group of 𝑛 × 𝑛 invertible matrices
{\displaystyle R} -module M {\displaystyle M} of rank n {\displaystyle n} . One can also define GL(M) for any R {\displaystyle R} -module, but in general
General_linear_group
Generalization of the Cartesian product
In mathematics, the direct product of a collection of algebraic structures (such as groups, rings, or vector spaces) is a structure of the same type constructed
Direct_product
Algebra describing 2D conformal symmetry
the Virasoro algebra and of its universal enveloping algebra. Then the Shapovalov form is the symmetric bilinear form on the Verma module V c , h {\displaystyle
Virasoro_algebra
Mathematical manifold theory
theory to all complex algebraic varieties, not necessarily smooth or compact. Namely, the cohomology of any complex algebraic variety has a more general
Hodge_theory
Inputs for which a function's value is non-zero
set of values is bounded Bump function – Smooth and compactly supported function Support of a module Titchmarsh convolution theorem Folland, Gerald B. (1999)
Support_(mathematics)
General concept and operation in mathematics
algebraically), this is always an injection; see Dual space § Injection into the double-dual. This can be generalized algebraically to a dual module.
Duality_(mathematics)
right-invariant) measure on locally compact topological group Locally compact field Locally compact quantum group Locally compact group – Type of topological
Topological_abelian_group
Basic result in the representation theory of Lie groups
positive characteristic. Namely, let G be a semisimple algebraic group over an algebraically closed field of characteristic p > 0 {\displaystyle p>0}
Borel–Weil–Bott_theorem
Specific algebraic group
particular being isogenous is an equivalence relation between tori. Over any algebraically closed field k = k ¯ {\displaystyle k={\overline {k}}} there is up to
Algebraic_torus
Tool to track locally defined data attached to the open sets of a topological space
in algebraic geometry, are locally ringed spaces that are locally isomorphic to the spectrum of a ring. Given a ringed space, a sheaf of modules is a
Sheaf_(mathematics)
Theorem in functional analysis
the unit ball within, is compact. This theorem has applications in physics when one describes the set of states of an algebra of observables, namely that
Banach–Alaoglu_theorem
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
Field extension that is not algebraic
exists a maximal algebraically independent subset of L over K. It is then called a transcendence basis. By maximality, an algebraically independent subset
Transcendental_extension
maximal compact subgroup K of a semisimple Lie group G that is in some sense the smallest representation of K occurring in a Harish-Chandra module of G.
Minimal_K-type
Open convex self-dual cones
invariant under automorphisms of the Jordan algebra, which is thus a closed subgroup of O(E) and thus a compact Lie group. In practical examples, however
Symmetric_cone
generally one can consider algebraically closed fields of prime characteristic. In this generality, character varieties are only algebraic sets and are not actual
Character_variety
Mathematical parametrization of vector spaces by another space
real-valued functions on X, then F becomes a sheaf of OX-modules. Not every sheaf of OX-modules arises in this fashion from a vector bundle: only the locally
Vector_bundle
Theory in mathematics
C*-algebras. This notion was introduced by the Russian mathematician Gennadi Kasparov in 1980. It was influenced by Atiyah's concept of Fredholm modules
KK-theory
Construction in category theory
A_{i}} 's are sets, semigroups, topological spaces, rings, modules (over a fixed ring), algebras (over a fixed ring), etc., and the homomorphisms are morphisms
Inverse_limit
ALGEBRAICALLY COMPACT-MODULE
ALGEBRAICALLY COMPACT-MODULE
Boy/Male
Hindu, Indian
Compact; Firm; Solid
Girl/Female
Indian, Telugu
Good 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
Lord's Company
Boy/Male
Indian, Punjabi, Sikh
Company of Guru
Girl/Female
Hindu, Indian, Marathi, Tamil
Compact; Promise
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.
Boy/Male
Indian, Sanskrit
Fallen from Glory
Girl/Female
Arabic, Muslim
Beauty of Company
Boy/Male
Indian, Punjabi, Sikh
Good Company
Boy/Male
Indian, Tamil
No Compare
Girl/Female
Tamil
Compare
Girl/Female
Muslim
Beauty of company
Boy/Male
Hindu, Indian, Sanskrit
In the Company
Girl/Female
Arabic
Sensible Contact
Boy/Male
Indian, Punjabi, Sikh
Liberation through Company
Girl/Female
Hindu, Indian
Compare
Girl/Female
Arabic, Muslim
Beauty of Company
Boy/Male
Hindu, Indian, Sanskrit
Company
Boy/Male
Hindu, Indian
Compact; Safe; Secure
ALGEBRAICALLY COMPACT-MODULE
ALGEBRAICALLY COMPACT-MODULE
Girl/Female
Indian
Unique
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Moon; Lovely Person
Boy/Male
German
Coal Town
Girl/Female
American, Australian, Chinese, French, German, Jamaican, Latin
A Nobleman; Patrician
Male
English
English surname transferred to forename use, from a byname for someone having gray hair or a beard, from Old English græg, GRAY means "grey."
Girl/Female
Indian, Punjabi, Sikh
Divine Blessing
Boy/Male
French, German, Greek, Hindu, Indian, Latin
Pain; Lipless; Who Embodies the Grief of the People
Girl/Female
Australian, Danish, Dutch, Netherlands, Swedish
God is Gracious; God has Shown Favor
Boy/Male
English American
Forest; cup bearer.
Boy/Male
Hindu, Indian
Lord of the Earth
ALGEBRAICALLY COMPACT-MODULE
ALGEBRAICALLY COMPACT-MODULE
ALGEBRAICALLY COMPACT-MODULE
ALGEBRAICALLY COMPACT-MODULE
ALGEBRAICALLY COMPACT-MODULE
adv.
By algebraic process.
adv.
In a compact manner; with close union of parts; densely; tersely.
n.
The crew of a ship, including the officers; as, a whole ship's company.
imp. & p. p.
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.
a.
Compact; pressed close; concentrated; firmly united.
n.
Extent; reach; sweep; capacity; sphere; as, the compass of his eye; the compass of imagination.
n.
An inclosing limit; boundary; circumference; as, within the compass of an encircling wall.
n.
Guests or visitors, in distinction from the members of a family; as, to invite company to dine.
p. p. & a
Brief; close; pithy; not diffuse; not verbose; as, a compact discourse.
v. t.
To manure with compost.
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.
a.
Alt. of Algebraical
n.
One who makes a compact.
a.
Of or pertaining to algebra; containing an operation of algebra, or deduced from such operation; as, algebraic characters; algebraical writings.
v. t.
To compact or join anew.
v. t.
To mingle, as different fertilizing substances, in a mass where they will decompose and form into a compost.
n.
Contact or impression by touch; collision; forcible contact; force communicated.