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 the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of T(V). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bi-algebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure. Note: In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct.
Construction Let V be a vector space over a field K. For any nonnegative integer k, we define the kth tensor power of V to be the tensor product of V with itself k times:
T k V = V ⊗ k = V ⊗ V ⊗ ⋯ ⊗ V . {\displaystyle T^{k}V=V^{\otimes k}=V\otimes V\otimes \cdots \otimes V.}
That is, TkV consists of all tensors on V of order k. By convention T0V is the ground field K (as a one-dimensional vector space over itself). We then construct T(V) as the direct sum of TkV for k = 0,1,2,…
T ( V ) = ⨁ k = 0 ∞ T k V = K ⊕ V ⊕ ( V ⊗ V ) ⊕ ( V ⊗ V ⊗ V ) ⊕ ⋯ . {\displaystyle T(V)=\bigoplus _{k=0}^{\infty }T^{k}V=K\oplus V\oplus (V\otimes V)\oplus (V\otimes V\otimes V)\oplus \cdots .}
The multiplication in T(V) is determined by the canonical isomorphism
T k V ⊗ T ℓ V → T k + ℓ V {\displaystyle T^{k}V\otimes T^{\ell }V\to T^{k+\ell }V}
given by the tensor product, which is then extended by linearity to all of T(V). This multiplication rule implies that the tensor algebra T(V) is naturally a graded algebra with TkV serving as the grade-k subspace. This grading can be extended to a Z-grading by appending subspaces T k V = { 0 } {\displaystyle T^{k}V=\{0\}} for negative integers k. The construction generalizes in a straightforward manner to the tensor algebra of any module M over a commutative ring. If R is a non-commutative ring, one can still perform the construction for any R-R bimodule M. (It does not work for ordinary R-modules because the iterated tensor products cannot be formed.)
Adjunction and universal property The tensor algebra T(V) is also called the free algebra on the vector space V, and is functorial; this means that the map V ↦ T ( V ) {\displaystyle V\mapsto T(V)} extends to linear maps for forming a functor from the category of K-vector spaces to the category of associative algebras. Similarly with other free constructions, the functor T is left adjoint to the forgetful functor that sends each associative K-algebra to its underlying vector space. Explicitly, the tensor algebra satisfies the following universal property, which formally expresses the statement that it is the most general algebra containing V:
Any linear map f : V → A {\displaystyle f:V\to A} from V to an associative algebra A over K can be uniquely extended to an algebra homomorphism from T(V) to A as indicated by the following commutative diagram:
Here i is the canonical inclusion of V into T(V). As for other universal properties, the tensor algebra T(V) can be defined as the unique algebra satisfying this property (specifically, it is unique up to a unique isomorphism), but this definition requires to prove that an object satisfying this property exists. The above universal property implies that T is a functor from the category of vector spaces over K, to the category of K-algebras. This means that any linear map between K-vector spaces U and W extends uniquely to a K-algebra homomorphism from T(U) to T(W).
Non-commutative polynomials If V has finite dimension n, another way of looking at the tensor algebra is as the "algebra of polynomials over K in n non-commuting variables". If we take basis vectors for V, those become non-commuting variables (or indeterminates) in T(V), subject to no constraints beyond associativity, the distributive law and K-linearity. Note that the algebra of polynomials on V is not T ( V ) {\displaystyle T(V)} , but rather T ( V ∗ ) {\displaystyle T(V^{*})} : a (homogeneous) linear function on V is an element of V ∗ , {\displaystyle V^{*},} for example coordinates x 1 , … , x n {\displaystyle x^{1},\dots ,x^{n}} on a vector space are covectors, as they take in a vector and give out a scalar (the given coordinate of the vector).
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