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mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension
Separable_algebra
Algebraic field extension
contained in a separably-closed algebraic extension field. It is unique (up to isomorphism). The separable closure is the full algebraic closure if and
Algebraic_closure
Algebraic structure of set algebra
{F}}).} A separable σ {\displaystyle \sigma } -algebra (or separable σ {\displaystyle \sigma } -field) is a σ {\displaystyle \sigma } -algebra F {\displaystyle
Σ-algebra
Type of algebraic field extension
In field theory, a branch of algebra, an algebraic field extension E / F {\displaystyle E/F} is called a separable extension if for every α ∈ E {\displaystyle
Separable_extension
Algebraic structure
{\displaystyle K} is separable. Every finite extension of K {\displaystyle K} is separable. Every algebraic extension of K {\displaystyle K} is separable. Either K
Perfect_field
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
Ring that is also a vector space or a module
called the bidimension of A, measures the failure of separability. Let A be a finite-dimensional algebra over a field k. Then A is an Artinian ring. As A
Associative_algebra
Topological complex vector space
classification is possible, for separable simple nuclear C*-algebras. We begin with the abstract characterization of C*-algebras given in the 1943 paper by
C*-algebra
*-algebra of bounded operators on a Hilbert space
Neumann algebras are the direct integral of properly infinite factors. A von Neumann algebra that acts on a separable Hilbert space is called separable. Note
Von_Neumann_algebra
Topological space with a dense countable subset
In mathematics, a topological space is called separable if it contains a countable dense subset; that is, there exists a sequence ( x n ) n = 1 ∞ {\displaystyle
Separable_space
Canonical commutation or anticommutation relations
. It is well known that the CCR algebra is a simple (unless the sympletic form is degenerate) non-separable algebra and is unique up to isomorphism.
CCR_and_CAR_algebras
Algebraic structure with "nice" duality properties
example of a separable algebra extension since e = ∑ i = 1 n a i ⊗ B b i {\textstyle e=\sum _{i=1}^{n}a_{i}\otimes _{B}b_{i}} is a separability element satisfying
Frobenius_algebra
Relation algebra Relational algebra Rota–Baxter algebra Schur algebra Semisimple algebra Separable algebra Shuffle algebra Sigma-algebra Simple algebra Structurable
List_of_algebras
Mathematical expression for linear operators
associative algebra over the field K {\displaystyle K} with Jacobson radical J {\displaystyle J} . Then A / J {\displaystyle A/J} is separable if and only
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
Group of unitary matrices
field extension can be replaced by any degree 2 {\displaystyle 2} separable algebra, most notably a degree 2 {\displaystyle 2} extension of a finite field;
Unitary_group
Polynomial coprime with its derivative
mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots
Separable_polynomial
commutative algebra, an étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite separable field extensions
Étale_algebra
Type of vector space in math
Hilbert space is separable provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if
Hilbert_space
Topics referred to by the same term
Look up separable in Wiktionary, the free dictionary. Separability may refer to: Separable algebra, a generalization to associative algebras of the notion
Separability
analysis, the Calkin algebra, named after John Williams Calkin, is the quotient of B(H), the ring of bounded linear operators on a separable infinite-dimensional
Calkin_algebra
considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note
Abelian_von_Neumann_algebra
Branch of functional analysis
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with
Operator_algebra
Class of mathematical sets
complement. Then we can define the Borel σ-algebra over X {\displaystyle X} to be the smallest σ-algebra containing all open sets of X {\displaystyle
Borel_set
compactification. Multiplier algebras were introduced by Busby (1968). For example, if A is the C*-algebra of compact operators on a separable Hilbert space, M(A)
Multiplier_algebra
Mathematical concept
For a commutative C*-algebra, A ^ ≅ Prim ( A ) . {\displaystyle {\hat {A}}\cong \operatorname {Prim} (A).} Let H be a separable infinite-dimensional
Spectrum_of_a_C*-algebra
Universal C*-algebra
certain relations. These algebras were introduced as the first concrete examples of a separable infinite simple C*-algebra, meaning that as a Hilbert
Cuntz_algebra
condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra H is a Hopf algebroid over its separable subalgebra HL. The
Hopf_algebroid
Technique for solving differential equations
differential equation for the unknown f ( x ) {\displaystyle f(x)} is separable if it can be written in the form d d x f ( x ) = g ( x ) h ( f ( x ) )
Separation_of_variables
Every polynomial has a real or complex root
(hence it is a Galois extension, as every algebraic extension of a field of characteristic 0 is separable). Let G be the Galois group of this extension
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Algebraic structure with addition, multiplication, and division
exponential function exp : F → F×). For fields that are not algebraically closed (or not separably closed), the absolute Galois group Gal(F) is fundamentally
Field_(mathematics)
Dimension of the column space of a matrix
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal
Rank_(linear_algebra)
Finite dimensional algebra over a field whose central elements are that field
areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A that is simple, and for which the center
Central_simple_algebra
Mathematical representation in functional analysis
If A is a separable C*-algebra, the weak-* topology is metrizable on bounded subsets. Thus the spectrum of a separable commutative C*-algebra A can be
Gelfand_representation
Generalization of the concept of a direct sum in mathematics
von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field
Direct_integral
In mathematics, vector space of linear forms
for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace
Dual_space
Algebraic field extension
mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the
Galois_extension
Field extension that is not algebraic
transcendence basis S such that L is a separable algebraic extension over K(S). A field extension L / K is said to be separably generated if it admits a separating
Transcendental_extension
Algebraic structure modeling logical operations
logic as lattices of closed linear subspaces for separable Hilbert spaces. List of Boolean algebra topics Boolean domain Boolean function Boolean logic
Boolean_algebra_(structure)
Algebraic structure with addition and multiplication
Semisimplicity is closely related to separability. A unital associative algebra A over a field k is said to be separable if the base extension A ⊗ k F {\displaystyle
Ring_(mathematics)
C*-algebra of all bounded operators on a Hilbert space H. A C*-algebra is exact if and only if every separable sub-C*-algebra is exact. A separable C*-algebra
Exact_C*-algebra
C*-algebras it is common to restrict attention to separable C*-algebras. Since a graph C*-algebra C ∗ ( E ) {\displaystyle C^{*}(E)} is separable precisely
Graph_C*-algebra
8-dimensional real non-associative division algebra. A similar construction works for any cubic alternative separable algebra over a field containing a primitive
Okubo_algebra
Theory in mathematics
of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi
KK-theory
Neumann algebra is injective. It is amenable as a Banach algebra. (For separable algebras) It is isomorphic to a C*-subalgebra B of the Cuntz algebra 𝒪2
Nuclear_C*-algebra
Normed vector space that is complete
of a separable Banach space need not be separable, but: Theorem—Let X {\displaystyle X} be a normed space. If X ′ {\displaystyle X'} is separable, then
Banach_space
Quantum states that are not entangled
In quantum mechanics, separable states are multipartite quantum states that can be written as a convex combination of product states. Product states are
Separable_state
glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary
Glossary of commutative algebra
Glossary_of_commutative_algebra
Subject area in mathematics
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic
Algebraic_K-theory
Extension of a mathematical field with polynomial roots
In mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that
Algebraic_extension
In mathematics, element that equals its square
idempotent a of R is called a full idempotent if RaR = R. A separability idempotent; see Separable algebra. Any non-trivial idempotent a is a zero divisor (because
Idempotent_(ring_theory)
Algebraic construct of interest in theoretical physics
compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do
Quantum_group
Function in algebra
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size
Valuation_(algebra)
Branch of mathematics that studies algebraic structures
extension Algebraic extension Splitting field Algebraically closed field Algebraic element Algebraic closure Separable extension Separable polynomial
List of abstract algebra topics
List_of_abstract_algebra_topics
C*-algebra
theorem for AF-algebras serves as a prototype for classification results for larger classes of separable simple amenable stably finite C*-algebras. Its proof
Approximately finite-dimensional C*-algebra
Approximately_finite-dimensional_C*-algebra
K0 group of A. One example of a UHF C*-algebra is the CAR algebra. It is defined as follows: let H be a separable complex Hilbert space H with orthonormal
Uniformly_hyperfinite_algebra
Local ring in which Hensel's lemma holds
depends on the choice of a separable algebraic closure of the residue field of A, and automorphisms of this separable algebraic closure correspond to automorphisms
Henselian_ring
Specific algebraic group
enough r {\displaystyle r} . In general one has to use separable closures instead of algebraic closures. If F {\displaystyle F} is a field then the multiplicative
Algebraic_torus
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Mathematical theorem in the study of analysis
Weierstrass approximation theorem, one can show that the space C[a, b] is separable: the polynomial functions are dense, and each polynomial function can
Stone–Weierstrass_theorem
we have that The residue field k ( x ) {\displaystyle k(x)} is a separable algebraic extension of k ( y ) {\displaystyle k(y)} . f # ( m y ) O x , X =
Unramified_morphism
Algebraic geometry
formally étale. Finite separable field extensions are formally étale. More generally, any (commutative) flat separable A-algebra B is formally étale. Formally
Formally_étale_morphism
dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree. If ( X , Σ
Weakly_measurable_function
of A as an (Aop ⊗R A)-module. For example, an algebra has bidimension zero if and only if it is separable. boolean A boolean ring is a ring in which every
Glossary_of_ring_theory
Unique von Neumann algebra
the Clifford algebra of an infinite separable Hilbert space. If p is any non-zero finite projection in a hyperfinite von Neumann algebra A of type II
Hyperfinite_type_II_factor
Concept in ring theory
the opposite algebra of A {\displaystyle A} . The center of A {\displaystyle A} is R {\displaystyle R} , and A {\displaystyle A} is separable. A {\displaystyle
Azumaya_algebra
Construction of a larger algebraic field by "adding elements" to a smaller field
property. An algebraic extension L / K {\displaystyle L/K} is called separable if the minimal polynomial of every element of L over K is separable, i.e., has
Field_extension
Net in a normed algebra
approximate identity and a C*-algebra with a sequential approximate identity is called σ-unital. Every separable C*-algebra is σ-unital, though the converse
Approximate_identity
Mathematical ring whose elements are matrices
the algebra B(H) of continuous operators; this identifies Mn(A) with a subalgebra of B(H⊕n). For simplicity, if we further suppose that H is separable and
Matrix_ring
Mathematical problem in von Neumann algebra theory
conjecture in C*-algebra theory Tsirelson's problem in quantum information theory The predual of any (separable) von Neumann algebra is finitely representable
Connes_embedding_problem
Classification in abstract algebra
) {\displaystyle Z(q)} is either a separable quadratic extension field of F {\displaystyle F} or the split algebra F ⊕ F {\displaystyle F\oplus F} . If
Classification of Clifford algebras
Classification_of_Clifford_algebras
German mathematician (1882–1935)
German mathematician who made many important contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental
Emmy_Noether
contains neither 2413 nor 3142 as a pattern. The separable permutations also have a characterization from algebraic geometry: if a collection of distinct real
Separable_permutation
Type of feedforward neural network
layers, notable for being able to distinguish data that is not linearly separable. Modern neural networks are trained using backpropagation and are colloquially
Multilayer_perceptron
Type of field extension
^ {\displaystyle {\hat {k}}} is the set of elements in L algebraic over k) and L is separable over k, or equivalently, L ⊗ k k ¯ {\displaystyle L\otimes
Regular_extension
Mathematical function
x_{m})=0.} In one variable, algebraic functions are closely related to algebraic curves and their function fields; in the separable case, they may also be
Algebraic_function
On invertibility of polynomial maps (mathematics)
Stuart Sui-Sheng (August 1980), "A Jacobian criterion for separability", Journal of Algebra, 65 (2): 453–494, doi:10.1016/0021-8693(80)90233-1 Bass, Hyman;
Jacobian_conjecture
Counterintuitive mathematical object
Separable field extensions are better-behaved than non-separable ones. Normed division algebras are better-behaved than general composition algebras.
Pathological_(mathematics)
Concept in topology
the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic
Polish_space
Field theory is the branch of algebra that studies fields
F is also in E. Separable extension An algebraic extension in which the minimal polynomial of every element of E over F is a separable polynomial, that
Glossary_of_field_theory
{\overline {k}}} denotes an algebraic closure of k. X × k k s {\displaystyle X\times _{k}k_{s}} is irreducible for a separable closure k s {\displaystyle
Geometrically (algebraic geometry)
Geometrically_(algebraic_geometry)
Subspace preserved by a linear mapping
ideal in A. The invariant subspace problem concerns the case where V is a separable Hilbert space over the complex numbers, of dimension > 1, and T is a bounded
Invariant_subspace
Field of knowledge
including number theory (the study of integers and their properties), algebra (the study of operations and the structures they form), geometry (the study
Mathematics
and the Borel algebra on B) if and only if it is both weakly measurable and almost surely separably valued. In the case that B is separable, since any subset
Bochner_measurable_function
A finitely generated k-algebra A is 0-smooth over k if and only if Spec A is a smooth scheme over k. A separable algebraic field extension L of k is
Smooth_algebra
Subject in mathematics
sets. For non-separable spaces it can happen that the vector addition is no longer measurable to the product algebra of borel σ-algebras because in general
Measure theory in topological vector spaces
Measure_theory_in_topological_vector_spaces
Field theory theorem
every finite separable field extension is simple, i.e. generated by a single element. This theorem implies in particular that all algebraic number fields
Primitive_element_theorem
necessarily separable) Hilbert space. The problem has been solved in the affirmative for special cases (specifically for separable and Type-I C*-algebras). Akemann
Naimark's_problem
Study of Galois symmetry groups of differential fields
{\displaystyle e^{-x^{2}}} . For a differential field F, if G is a separable algebraic extension of F, the derivation of F uniquely extends to a derivation
Differential_Galois_theory
Structure in algebraic geometry
In algebraic geometry, a motive (or sometimes motif, following French usage) is an abstract object introduced by Alexander Grothendieck in the 1960s as
Motive_(algebraic_geometry)
Equivalence relation on rings
ISBN 0-387-97845-3. Zbl 0765.16001. DeMeyer, F.; Ingraham, E. (1971). Separable algebras over commutative rings. Lecture Notes in Mathematics. Vol. 181.
Morita_equivalence
Mathematical group
In Galois theory, a branch of abstract algebra, the Galois group of a certain type of field extension is a symmetry group characterizing how it extends
Galois_group
Roots of an algebraic element's minimal polynomial
list of each element is the separable degree [L:K(α)]sep. A theorem of Kronecker states that if α is a nonzero algebraic integer such that α and all of
Conjugate element (field theory)
Conjugate_element_(field_theory)
Term in mathematics
and take T to be a maximal torus containing S defined over the separable algebraic closure K of k. Then G(K) has a Dynkin diagram with respect to some
Satake_diagram
In algebraic geometry, a morphism f : X → S {\displaystyle f:X\to S} between schemes is said to be smooth if (i) it is locally of finite presentation
Smooth_morphism
Algebraic structure
whose minimal polynomial is separable. To use a piece of jargon, finite fields are perfect. Finite fields are quasi-algebraically closed: every degree d homogeneous
Finite_field
group is not special because there exist Azumaya algebras, which are trivial over a finite separable extension, but not over the base field. Special groups
Special group (algebraic group theory)
Special_group_(algebraic_group_theory)
Abelian group related to division algebras
viewed as an algebraic group over K. More concretely, the cohomology group indicated means H2(Gal(Ks/K), Ks*), where Ks denotes a separable closure of K
Brauer_group
Relates the topology of a complete non-archimedean field to its algebraic extensions
complete non-archimedean field to its algebraic extensions. Let K be a complete non-archimedean field and let K be a separable closure of K. Given an element
Krasner's_lemma
American mathematician
under the direction of Daniel Zelinsky with thesis Separable Subalgebras of Commutative Algebras and Other Applications of the Boolean Spectrum. From
Andy_Magid
Collection of random variables
the separability conditions, so discrete-time stochastic processes are always separable. A theorem by Doob, sometimes known as Doob’s separability theorem
Stochastic_process
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
Surname or Lastname
English
English : variant spelling of Rimer 1.German : variant of Riemer.German : habitational name for someone from Riem (now a suburb of Munich; formerly a separate town).
Boy/Male
Muslim/Islamic
Inseparable friend
Girl/Female
Arabic, Muslim, Sindhi
Inseparable Friend
Girl/Female
Muslim
Inseparable friend
Boy/Male
Arabic, Australian, Muslim
Considerate; Inseparable Friend
Boy/Male
Sikh
Triumph for gods name, Triumph of the inseparable creator
Girl/Female
Arabic
Separate
Girl/Female
Indian, Punjabi, Sikh
Love of the Inseparable Creator
Boy/Male
Muslim
Considerate, Inseparable friend
Girl/Female
Arabic, Muslim
Inseparable Friend
Boy/Male
Sikh
Triumph for gods name, Triumph of the inseparable creator
Boy/Male
Indian, Marathi
Separate
Girl/Female
Biblical
A parable, governing.
Surname or Lastname
English
English : occupational name for a maker of arms and armor, from Anglo-Norman French armer ‘arms-maker’ (Old French armier). Originally this was a separate name from Armour, but in due course the two became inextricably confused.
Girl/Female
Indian, Punjabi, Sikh
Triumph of the Inseparable Creator
Girl/Female
Indian
Inseparable
Biblical
a parable; governing
Girl/Female
Arabic, Muslim
Example; Allegory; Parable
Girl/Female
Muslim
Example, Allegory, Parable
Girl/Female
Muslim/Islamic
Inseparable friend
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
Boy/Male
Hindu
Air circulating in the body
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil
God of Beauty; Husband of a Beautiful Wife
Boy/Male
American, British, Christian, English, Hindu, Indian
Hidden; Obvious; Whether Hidden or Obvious; Favour; Grace
Girl/Female
Muslim/Islamic
Pearls
Boy/Male
Arabic, Muslim
Abundant; Copious
Boy/Male
Hindu, Indian, Kannada, Punjabi, Sikh
Immortal Love of God
Boy/Male
Hindu, Indian, Tamil
Leader; Born to Win as a Leader; Lord Ayyapa's Alternative Name
Boy/Male
Arabic
One who Serves the God
Boy/Male
Indian, Punjabi, Sanskrit, Sikh
Sun
Boy/Male
Irish
From dubh “â€blackâ€â€ and lan “â€blade, swordâ€â€ means “â€black sword.â€â€ Dubhlainn loved the fairy queen and legendary harpist Aoibhell who gave him her cloak of invisibility to wear in battle.
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
a.
Capable of being repaired, restored to a sound or good state, or made good; restorable; as, a reparable injury.
a.
Capable of being prepared.
v. t.
To represent by parable.
a.
Reparable.
a.
Invariably attached to some word, stem, or root; as, the inseparable particle un-.
a.
Separable.
n.
See Sperable.
p. a.
Disunited from the body; disembodied; as, a separate spirit; the separate state of souls.
a.
Capable of being severed.
n.
A kind of small nail used by shoemakers.
adv.
In a reparable manner.
a.
Capable of being overcome or conquered; surmountable.
a.
Capable of being separated, disjoined, disunited, or divided; as, the separable parts of plants; qualities not separable from the substance in which they exist.
a.
Inseparable.
a.
Not separable; incapable of being separated or disjoined.
a.
That may be secured.
a.
Capable of being, or proper to be , repaid; due; as, a loan repayable in ten days; services repayable in kind.
adv.
In an inseparable manner or condition; so as not to be separable.
a.
Able to speak.
a.
Capable of being spoken; fit to be spoken.