Tensor algebra

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, meaning it is the most general algebra that contains V. This makes it a fundamental structure in many areas of advanced mathematics.

The tensor algebra is important because many other algebras can be built from it. These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra, and universal enveloping algebras. These algebras have many applications in physics and geometry.

Tensor algebra also has special structures called coalgebras. One is simple, while the other is more complex and can be extended to create a Hopf algebra. These structures help mathematicians understand how different algebraic systems relate to each other. In this article, all algebras are assumed to be unital and associative.

Construction

Let’s explore how we build something called the tensor algebra from a vector space. Imagine we have a special kind of space filled with vectors, and we want to combine these vectors in many ways.

We start by taking the tensor product of our vectors over and over again. For example, we can combine a vector with itself once, twice, three times, and so on. Each time we combine them, we create something new called a tensor of a certain order. When we combine them zero times, we simply get our original field of numbers. By putting all these different combinations together, we create the tensor algebra. This algebra has a special way of multiplying these tensors, making it a graded algebra where each level corresponds to how many times we combined the vectors.

Adjunction and universal property

The tensor algebra T(V) is called the free algebra on the vector space V, and it works like a special kind of map. This means that the way we go from V to T(V) can be extended to work with linear maps, which are straight-line paths between vector spaces.

Because of this, the tensor algebra has a special rule called the universal property. This rule says that any linear path from V to another algebra A can be stretched out in a unique way to a full algebra path from T(V) to A. This makes the tensor algebra the most general algebra that includes V.

Non-commutative polynomials

If a vector space V has a finite number of dimensions, we can think of the tensor algebra as the algebra of polynomials over a field K in n non-commuting variables. When we choose basis vectors for V, they act like variables in T(V) that don’t commute — meaning their order matters — but they still follow normal algebra rules like associativity and the distributive law.

Note that the usual polynomial algebra on V is not the same as T(V), but instead relates to T(V∗). A simple example is how coordinates like x₁ to xₙ work; these are covectors that take a vector and return a number, such as a specific coordinate of that vector.

Quotients

The tensor algebra is very general, which means we can create many other important algebras by starting with the tensor algebra and then adding some rules for how the pieces fit together. This process is called creating "quotient algebras" of the tensor algebra.

Some famous examples of algebras built this way include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra, and universal enveloping algebras. These algebras have many uses in advanced mathematics and physics.

Coalgebra

The tensor algebra has two different coalgebra structures. One structure works well with the tensor product and can be extended into something called a bialgebra, which can then be made into a Hopf algebra by adding an antipode. The other structure is simpler but cannot be made into a bialgebra.

The first structure can also be used with the exterior algebra and the symmetric algebra, giving them Hopf algebra structures as well. These structures help show how different algebraic systems are related.

There are also different ways to define a coproduct in the tensor algebra. One simpler method creates a coalgebra that is dual to the algebra structure on the dual vector space. This approach can be adjusted to form a bialgebra with a specific product rule.