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3.1 Algebras over Vector Spaces

In this section, we will introduce the concept of algebras over vector spaces.

Table of Contents​

Introduction​

In many physical applications, there are often natural ways to "multiply" vectors together. We can multiply matrices together, we can take the cross product of vectors in , and we can multiply complex numbers together. Pauli matrices, used in quantum mechanics, act as a set of basis elements for a vector space. Quaternions, which extend complex numbers, also form a vector space with a natural multiplication operation. So we can see that in many cases, vector spaces come equipped with a natural multiplication operation, where the product of two vectors is another vector in the same space. We call any vector space endowed with such a multiplication operation an algebra over a field.

Definition 3.1.1 (Algebra over a Field)

An algebra over a field is a vector space over equipped with a bilinear map (called multiplication) . The image of under this map is denoted by for all . This multiplication must satisfy the following properties for all and all :

  1. Linearity in the first argument: ,
  2. Linearity in the second argument: .

is

  • associative if for all ,
  • commutative if for all , and
  • unital if there exists an element such that for all . The identity element is sometimes also denoted by .

As multiplication is not necessarily commutative or associative, the notion of inverses becomes more complicated. Leting , we say that is a left inverse of if , and a right inverse of if .

Properties of Algebras​

First, if is the zero vector in (which always exists as is a vector space), then for any , we have . This follows from the bilinearity of the multiplication operation:

and similarly for .

Second, in an associative algebra, left and right inverses coincide. If is a left inverse of and is a right inverse of , then

Third, in an associative algebra, this (both-sided) inverse is unique.

Fourth, the identity element in a unital algebra is unique. If and are both identity elements, then obviously

Let's put this together in a theorem.

Theorem 3.1.2 (Properties of Associative Algebras)

Let be an associative algebra over a field . Let . Then,

  1. If has a left inverse and a right inverse , then .
  2. If has an inverse, then it is unique.
  3. If and are invertible, then so is their product , and we have

Proof. We have already shown parts 1 and 2 from above. For part 3, we can verify that is indeed the inverse of by showing that they multiply to identity:

and similarly,

Thus part 3 is proven.

Next, as vector spaces have subspaces, we can also define subsets of algebras that are closed under the multiplication operation.

Definition 3.1.3 (Subalgebra)

A subalgebra of an algebra over a field is a subset that is itself an algebra over with the same multiplication operation as . Trivially, is closed under addition, scalar multiplication, and the multiplication operation of .

A subalgebra generated by a subset is the smallest subalgebra of that contains . It is formed by taking all finite linear combinations and products of elements in . If contains a single element , then the subalgebra generated by is just the set of all polynomials in with coefficients from the field .