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92 relations: Abelian group, Abstract algebra, Adjoint functors, Affine variety, Algebra over a field, Algebraic group, Algebraic structure, Associative property, Axiom, Azumaya algebra, Banach algebra, Banach space, Bialgebra, Bilinear map, Canonical map, Category (mathematics), Category of abelian groups, Category of modules, Category of rings, Center (ring theory), Change of rings, Characteristic (algebra), Clifford algebra, Coalgebra, Combinatorics, Comma category, Commutative diagram, Commutative ring, Complex Lie group, Complex number, Continuous function, Coproduct, Cornell University, Derived algebraic geometry, Dual (category theory), Endomorphism ring, Exterior algebra, F-coalgebra, Field (mathematics), Free algebra, Free module, Free product of associative algebras, Functor, Generic matrix ring, Geometry, Group algebra, Homomorphism, Hopf algebra, Ideal (ring theory), Incidence algebra, ..., Lie algebra, Limit of a function, Linear map, Mathematics, Matrix (mathematics), Matrix multiplication, Module (mathematics), Module homomorphism, Monoid (category theory), Monoidal category, Morphism, Non-associative algebra, Noncommutative algebraic geometry, Partially ordered set, Physics, Polynomial, Polynomial ring, Quaternion, Quotient ring, Real number, Representation theory, Ring (mathematics), Ring homomorphism, Ringed space, Scalar multiplication, Semimartingale, Sheaf of algebras, Spectrum of a ring, Square matrix, Stochastic calculus, Subcategory, Subring, Symmetric algebra, Tensor algebra, Tensor product, Tensor product of algebras, Tensor product of modules, Tensor product of representations, Topology, Universal enveloping algebra, Vector space, Weyl algebra. Expand index (42 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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Abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
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Adjoint functors
In mathematics, specifically category theory, adjunction is a possible relationship between two functors.
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Affine variety
In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine ''n''-space k^n of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal.
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Algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.
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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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Algebraic structure
In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.
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Associative property
In mathematics, the associative property is a property of some binary operations.
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Axiom
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
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Azumaya algebra
In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field.
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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 space
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
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Bialgebra
In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a coalgebra.
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Bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments.
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Canonical map
In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects.
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Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.
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Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms.
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Category of modules
In algebra, given a ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules.
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Category of rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity).
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Center (ring theory)
In algebra, the center of a ring R is the subring consisting of the elements x such that xy.
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Change of rings
In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a right R-module M and a right S-module N,.
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Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
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Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra.
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Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras.
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Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
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Comma category
In mathematics, a comma category (a special case being a slice category) is a construction in category theory.
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Commutative diagram
The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result.
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Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
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Complex Lie group
In geometry, a complex Lie group is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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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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Coproduct
In category theory, the coproduct, or categorical sum, is a category-theoretic construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces.
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Cornell University
Cornell University is a private and statutory Ivy League research university located in Ithaca, New York.
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Derived algebraic geometry
Derived algebraic geometry (also called spectral algebraic geometry) is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by ring spectra in algebraic topology, whose higher homotopy accounts for the non-discreteness (e.g., Tor) of the structure sheaf.
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Dual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop.
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Endomorphism ring
In abstract algebra, the endomorphism ring of an abelian group X, denoted by End(X), is the set of all endomorphisms of X (i.e., the set of all homomorphisms of X into itself) endowed with an addition operation defined by pointwise addition of functions and a multiplication operation defined by function composition.
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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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F-coalgebra
In mathematics, specifically in category theory, an F-coalgebra is a structure defined according to a functor F. For both algebra and coalgebra, a functor is a convenient and general way of organizing a signature.
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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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Free algebra
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables.
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Free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements.
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Free product of associative algebras
In algebra, the free product (coproduct) of a family of associative algebras A_i, i \in I over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the A_i's.
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Functor
In mathematics, a functor is a map between categories.
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Generic matrix ring
In algebra, a generic matrix ring is a sort of a universal matrix ring.
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Geometry
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
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Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are related to representations of the group.
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Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
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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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Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
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Incidence algebra
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity.
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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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Limit of a function
Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1.
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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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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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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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Matrix multiplication
In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.
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Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
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Module homomorphism
In algebra, a module homomorphism is a function between modules that preserves module structures.
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Monoid (category theory)
In category theory, a monoid (or monoid object) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms.
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Monoidal category
In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism.
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Morphism
In mathematics, a morphism is a structure-preserving map from one mathematical structure to another one of the same type.
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Non-associative algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative.
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Noncommutative algebraic geometry
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients).
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Partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.
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Physics
Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.
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Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
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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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Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers.
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Quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.
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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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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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Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
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Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.
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Ringed space
In mathematics, a ringed space can be equivalently thought of as either Ringed spaces appear in analysis as well as complex algebraic geometry and scheme theory of algebraic geometry.
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Scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra).
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Semimartingale
In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process.
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Sheaf of algebras
In algebraic geometry, a sheaf of algebras on a ringed space X is a sheaf of commutative rings on X that is also a sheaf of \mathcalO_X-modules.
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Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by \operatorname(R), is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
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Square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns.
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Stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes.
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Subcategory
In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms.
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Subring
In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R).
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Symmetric algebra
In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is the free commutative unital associative algebra over K containing V. It corresponds to polynomials with indeterminates in V, without choosing coordinates.
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Tensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product.
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Tensor product
In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.
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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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Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps.
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Tensor product of representations
In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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Universal enveloping algebra
In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a Lie algebra.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Weyl algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form More precisely, let F be the underlying field, and let F be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F. ∂X is the derivative with respect to X. The algebra is generated by X and ∂X.
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Abelian algebra, Associative Algebra, Associative R-algebra, Associative algebras, Commutative R-algebra, Commutative algebra (structure), Linear associative algebra, R-algebra, Unital associative algebra.
References
[1] https://en.wikipedia.org/wiki/Associative_algebra
