1.2.2 Tensor Algebra Definition

Tensor Algebra Definition is the precise characterization of the tensor algebra of a vector space as the direct sum of all tensor powers of that space, equipped with a multiplication operation given by the tensor product, forming the most general associative algebra that can be built from the vector space through this construction. It specifies exactly which algebraic object is meant by "the tensor algebra," distinguishing this single canonical construction from the broader family of tensors and tensor operations studied under the wider heading of tensor algebra as a field.

Constructing the Tensor Algebra

Given a vector space over a field, the tensor algebra of that space is built by first forming every tensor power of the space: the field itself, regarded as the zeroth tensor power, the vector space itself as the first tensor power, the tensor product of the space with itself as the second tensor power, and so on for every non-negative integer. The tensor algebra is then defined as the direct sum of all of these tensor powers, taken together as a single vector space.

The expression above defines the tensor algebra of a vector space as the direct sum of every tensor power of that space, from the zeroth power upward.

Multiplication in the Tensor Algebra

The vector space structure obtained from this direct sum is turned into an algebra by defining multiplication through the tensor product: an element belonging to the k-th tensor power, multiplied by an element belonging to the m-th tensor power, produces an element of the (k+m)-th tensor power, obtained simply by taking their tensor product. This multiplication is associative, since the tensor product itself is associative, but it is generally not commutative, since reordering the factors in a tensor product produces, in general, a different element.

This grading of the tensor algebra by the tensor power to which each element belongs is an essential feature of its structure: the tensor algebra is not merely an algebra, but a graded algebra, in which every element can be decomposed into homogeneous pieces, each lying entirely within a single tensor power.

The Universal Property

The tensor algebra is defined, up to the specific model chosen for its construction, by a universal property: for any associative algebra and any linear map from the original vector space into that algebra, there exists a unique algebra homomorphism from the tensor algebra into the given algebra that extends the original linear map. This property expresses precisely in what sense the tensor algebra is the "most general" or "freest" associative algebra containing the vector space, since any other associative algebra built from that same vector space can be obtained from the tensor algebra by imposing additional relations.

This universal characterization is the reason the tensor algebra serves as the common ancestor from which more specialized structures, such as the symmetric algebra and the exterior algebra, are derived: each of these is constructed as a quotient of the tensor algebra by an ideal generated by specific relations, such as commutativity in the symmetric case or antisymmetry in the exterior case.

Relationship to the Broader Notion of a Tensor

It is important to distinguish the tensor algebra, in this precise sense, from the general study of tensors and their properties, sometimes referred to loosely by the same name. A tensor of type (p, q), as introduced in the general definition of a tensor, is an element of a tensor product involving both a vector space and its dual space. The tensor algebra, by contrast, is constructed using only the tensor powers of a single vector space, without incorporating the dual space, though an analogous construction using both a vector space and its dual can be formed to accommodate mixed tensors of every type simultaneously.

Significance of the Definition

Defining the tensor algebra precisely as a direct sum of tensor powers with tensor-product multiplication accomplishes two things: it provides a single, well-defined algebraic object in which every tensor power of a vector space is embedded simultaneously, allowing tensors of different ranks to be added, multiplied, and otherwise combined within one common structure, and it establishes, through the universal property, the sense in which this construction is canonical rather than one arbitrary choice among several possible ways of building an algebra from a vector space.