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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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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)
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
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
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
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
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
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
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 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
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
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
*-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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
factors of cotorsion groups are algebraically compact. A right module M over a ring R is said to be a cotorsion module if Ext R 1 ( F , M ) = 0 {\displaystyle
Cotorsion_group
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
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
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
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
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
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
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)
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
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
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
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
many other algebraic structures studied by algebra. They include magmas, semigroups, monoids, abelian groups, commutative rings, modules, lattices, vector
Algebra
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
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
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
Type of Kac–Moody algebras
representations of the affine compact groups only exist when k is a natural number. More generally, if one considers a semi-simple algebra, there is a central charge
Affine_Lie_algebra
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
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)
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
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
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
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)
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
Cryptographic primitives that involve lattices
CRYSTALS-Dilithium, which is built upon module learning with errors (module-LWE) and module short integer solution (module-SIS). Dilithium was selected for standardization
Lattice-based_cryptography
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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)
{\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
right-invariant) measure on locally compact topological group Locally compact field Locally compact quantum group Locally compact group – Type of topological
Topological_abelian_group
Class of representations
g , K ) {\displaystyle ({\mathfrak {g}},K)} -module which is easier to deal with as it is an algebraic object. Two admissible representations are said
Admissible_representation
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
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
(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
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
Unitary representations of a Lie group
theory of SU(2), its compact form, essentially because their Lie algebras have the same complexification and they are "algebraically simply connected".
Representation theory of SL2(R)
Representation_theory_of_SL2(R)
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
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
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)
Mathematical structure
functions with compact support. However, the group algebra k [ G ] {\displaystyle k[G]} and k G {\displaystyle k^{G}} – the commutative algebra of functions
Group_Hopf_algebra
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