the tensor algebra of a vector space

May 29th, 2015
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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. It is the free algebra on V, in the sense of being left adjoint to the forgetful

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This articledoes notciteanyreferences or sources.Please helpimprove this articleby adding citations toreliable sources. Unsourced material may bechallengedandremoved.(March 2009)Inmathematics, thetensor algebraof avector spaceV, denotedT(V) orT(V), is thealgebraoftensorsonV(of any rank) with multiplication being thetensor product. It is thefree algebraonV, in the sense of beingleft adjointto the forgetful functor from algebras to vector spaces: it is the "most general" algebra containingV, in the sense of the correspondinguniversal property(see below).The tensor algebra also has twocoalgebra structures; one simple one, which does not make it a bialgebra, and a more complicated one, which yields abialgebra, and can be extended with an antipode to aHopf algebrastructure.Note: In this article, all algebras are assumed to beunitalandassociative.Contents[hide]1Construction2Adjunction and universal property3Non-commutative polynomials4Quotients5Coalgebra structures5.1Simple coalgebra structure5.2Bialgebra and Hopf algebra structure6See also[edit]ConstructionLetVbe avector spaceover afieldK. For any nonnegativeintegerk, we define thekthtensor powerofVto be thetensor productofVwith itselfktimes:That is,TkVconsists of all tensors onVofrankk. By conventionT0Vis the ground fieldK(as a one-dimensional vector sp

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