Tensor algebra

In mathematics, the tensor algebra of a vector space V, written as T(V) or T^•(V), is a way to work with tensors. Tensors are special math objects that can have different ranks. In tensor algebra, these tensors are multiplied using something called the tensor product.

Tensor algebra is like the most basic and general algebra that includes the vector space V. It is very important in many areas of advanced math.

Many other types of algebras can be created from tensor algebra. These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra, and universal enveloping algebras. These algebras are useful in physics and geometry.

Tensor algebra also has special parts called coalgebras. One type is simple, and the other is more complicated. The more complicated one can be expanded to form a Hopf algebra. These special structures help mathematicians see how different math systems are connected. In this article, all algebras are 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

Universal property of the tensor algebra

The tensor algebra T(V) is called the free algebra on the vector space V. It works like a special kind of map. This means that the way we go from V to T(V) can be used with linear maps, which are straight 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, the tensor algebra is like a set of polynomials over a field K in n special variables. When we choose basis vectors for V, they act like variables in T(V) where their order matters — they don’t commute — but they still follow normal algebra rules like associativity and the distributive law.

The usual polynomial algebra on V is different from T(V). Instead, it 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. We can create many other important algebras by starting with the tensor algebra and then adding 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 turned into something called a bialgebra. We can then make this 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 work with the exterior algebra and the symmetric algebra, giving them Hopf algebra structures too. These structures help show how different algebraic systems are connected.

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