Algebras

If we have a vector space \(V\) over a field \(\mathbb{F}\), we already have two operations available: vector addition and scalar multiplication. The natural way to extend this concept, is to define a map that combines two vectors into another vector. That is exactly how we end up with an algebra.

If add a bilinear operator \(\star\) to a vector space \(V\) over \(\mathbb{F}\), then we will call \(V\) an algebra (with \(\star\)) over \(V\). Again, there is an unfortunate notation in which both \(V\) represents the algebra, as well as the set over which it is defined.

In some sense we could say that algebras are a generalization of fields in the way that field multiplication is now generalized to the bilinear operation of the algebra. In a sense, we can call the field multiplication a bilinear operation (in which the vector space associated to the bilinear operation is the field over itself).

Let \(\require{physics} \set{\vb{e}_i ; i=1,\dots,n}\) be a basis for the underlying \(n\)-dimensional vector space \(V\) of the algebra. It is then possible, in much the same way as operators can be represented as matrices in a certain basis, to characterize the the multiplication \(\star\) of the algebra as \[\begin{equation} \vb{e}_i \star \vb{e}_j = \sum_k^n f_{ijk} \vb{e}_k \thinspace , \end{equation}\] in which \(f_{ijk}\) are called the structure constants of the algebra.

If \(\star\) is associative, i.e. \[\begin{equation} \forall \vb{u}, \vb{v}, \vb{w}: \vb{u} \star (\vb{v} \star \vb{w}) = (\vb{u} \star \vb{v}) \star \vb{w} \thinspace , \end{equation}\] then the algebra is called associative.

We all know examples of algebras, with the easiest example being the \((n \times n)\)-matrices with matrix multiplication.

There also exist important classes of algebras, named Lie algebras.