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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 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 also has two coalgebra structures; one simple one, which does not make it a bialgebra, and a more complicated one, which yields a bialgebra, and can be extended with an antipode to a Hopf algebra structure.

Note: In this article, all algebras are assumed to be unital and associative.

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:

\( T^kV = V^{\otimes k} = V\otimes V \otimes \cdots \otimes V. \)

That is, T^kV consists of all tensors on V of rank 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 T^kV for k = 0,1,2,…

\( T(V)= \bigoplus_{k=0}^\infty T^kV = K\oplus V \oplus (V\otimes V) \oplus (V\otimes V\otimes V) \oplus \cdots.\)

The multiplication in T(V) is determined by the canonical isomorphism

\( T^kV \otimes T^\ell V \to T^{k + \ell}V\)

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 T^kV serving as the grade-k subspace. This grading can be extended to a Z grading by appending subspaces \(T^{k}V=\{0\}\) for negative integers k.

The construction generalizes in 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. As with other free constructions, the functor T is left adjoint to some forgetful functor. In this case, it's the functor which sends each 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 transformation f : V → A from V to an algebra A over K can be uniquely extended to an algebra homomorphism from T(V) to A as indicated by the following commutative diagram:

Universal property of the tensor algebra

Here i is the canonical inclusion of V into T(V) (the unit of the adjunction). One can, in fact, define the tensor algebra T(V) as the unique algebra satisfying this property (specifically, it is unique up to a unique isomorphism), but one must still prove that an object satisfying this property exists.

The above universal property shows that the construction of the tensor algebra is functorial in nature. That is, T is a functor from the K-Vect, category of vector spaces over K, to K-Alg, the category of K-algebras. The functoriality of T means that any linear map from V to W extends uniquely to an algebra homomorphism from T(V) 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 indeterminants) 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), but rather \( T(V^*)\) : a (homogeneous) linear function on V is an element of \( V^* \) , for example coordinates \( x^1,\dots,x^n \) on a vector space are covectors, as they take in a vector and give out a scalar (the given coordinate of the vector).

Quotients

Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e. by constructing certain quotient algebras of T(V). Examples of this are the exterior algebra, the symmetric algebra, Clifford algebras and universal enveloping algebras.

Coalgebra structures

The tensor algebra has two coalgebra structures; one simple one, which does not make it a bialgebra, and a more complicated one, which yields a bialgebra, and can be extended with an antipode to a Hopf algebra structure.

Simple coalgebra structure

The simple coalgebra structure on the tensor algebra is given as follows. The coproduct Δ is defined by

\( \Delta(v_1 \otimes \dots \otimes v_m ) := \sum_{i=0}^{m} (v_1 \otimes \dots \otimes v_i) \otimes (v_{i+1} \otimes \dots \otimes v_m)\)

extended by linearity to all of TV. The counit is given by

\( \varepsilon\left(v\right)=v for every v\in T^0\left(V\right) \) and
\( \varepsilon\left(v\right)=0 for every v\in T^k\left(V\right)\) for every k > 0.

Note that Δ : TV → TV ⊗ TV respects the grading

\( T^mV \to \bigoplus_{i+j=m} T^iV \otimes T^jV\)

and ε is also compatible with the grading.

The tensor algebra is not a bialgebra with this coproduct.

Bialgebra and Hopf algebra structure

However, the following more complicated coproduct does yield a bialgebra:

\( \Delta(x_1\otimes\dots\otimes x_m) = \sum_{p=0}^m \sum_{\sigma\in\mathrm{Sh}_{p,m-p}} \left(x_{\sigma(1)}\otimes\dots\otimes x_{\sigma(p)}\right)\otimes\left(x_{\sigma(p+1)}\otimes\dots\otimes x_{\sigma(m)}\right)\)

where the summation is taken over all (p,m-p)-shuffles.

Finally, the tensor algebra becomes a Hopf algebra with antipode given by

\( S(x_1\otimes\dots\otimes x_m) = (-1)^mx_m\otimes\dots\otimes x_1\)

extended linearly to all of TV.

This is just the standard Hopf algebra structure on a free algebra, where one defines the comultiplication on \(T^1(V)=V \) by

\( \Delta(x)=x\otimes1+1\otimes x\)

and then extends to T^m(V) via

\( \Delta(x_1\otimes\dots\otimes x_m) = \Delta(x_1)\Delta(x_2)\cdots\Delta(x_m).\)

Similarly one defines the antipode on \( T^1(V)=V \) by

S(x)=-x

and then extends the antipode as the unique antiautomorphism of T(V) with this property, i.e. we define the antipode on \(T^m(V) \) via

\( S(x_1\otimes\dots\otimes x_m) = S(x_m)S(x_{m-1})\cdots S(x_2)S(x_1).\)

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Tensor algebra