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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 order) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of…

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MathematicsVector spaceAlgebraTensorsTensor product
23 related topics

Construction

FieldIntegerOrderGround fieldDirect sum
10 related topics

Adjunction and universal property

FunctorLinear mapCategoryFree constructionsAlgebra homomorphism
9 related topics

Non-commutative polynomials

Basis vectorsIndeterminatesAssociativityDistributive lawCovectors
5 related topics

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Construction

Adjunction and universal property

Non-commutative polynomials

Coalgebra

Cofree cocomplete coalgebra

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Tensor algebra

Nodes62
Edges61
Triples28
Avg. degree1.97
Density0.032258
Components1

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Tensor algebra

Top relations

related to Coproduct · 7
Tensor algebra → Here, In, It, Multiplication, The, This, TV
related to Adjunction and universal property · 5
Tensor algebra → Explicitly, K-algebra, K-vector, Similarly, The
related to Coalgebra · 5
Tensor algebra → Hopf, One, That, The, This
related to Quotients · 4
Tensor algebra → Because, Clifford, Examples, Weyl
related to Cofree cocomplete coalgebra · 3
Tensor algebra → Here, It, One
related to Non-commutative polynomials · 3
Tensor algebra → If, K-linearity, Note
is a · 1
Tensor algebra → free algebra

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Important terminology

algebra displaystyle tensor one product coalgebra otimes bialgebra multiplication vector algebras hopf tv given also coproduct space structure universal definition

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SubjectPredicateObjectConfidenceSrc
Tensor algebrais afree algebra0.90text
Tensor algebrarelated to Adjunction and universal propertyThe0.60section
Tensor algebrarelated to Adjunction and universal propertyK-vector0.60section
Tensor algebrarelated to Adjunction and universal propertySimilarly0.60section
Tensor algebrarelated to Adjunction and universal propertyK-algebra0.60section
Tensor algebrarelated to Adjunction and universal propertyExplicitly0.60section
Tensor algebrarelated to CoalgebraThe0.60section
Tensor algebrarelated to CoalgebraOne0.60section
Tensor algebrarelated to CoalgebraHopf0.60section
Tensor algebrarelated to CoalgebraThis0.60section
Tensor algebrarelated to CoalgebraThat0.60section
Tensor algebrarelated to Cofree cocomplete coalgebraOne0.60section

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