Tensor algebra

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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. The tensor algebra is, in a sense, the "most general" algebra containing V. This notion of generality is formally expressed by a certain universal property (see below).

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

1 Construction
2 Universal property
3 Quotients
4 See also

[edit] Construction

Let V be a vector space over a field K. For any nonnegative integer k, we define the k^th 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.

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.)

[edit] Universal property

The fact that the tensor algebra is the most general algebra containing V is expressed by the following universal property: 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). 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).

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).

The tensor algebra T(V) is also called the free algebra on the vector space V. As with other free constructions, the functor T is left adjoint to some forgetful functor, specifically the functor which sends each K-algebra to its underlying vector space.

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).

[edit] Quotients

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