Vertex Operator Algebra - A Trivial Example

A Trivial Example

To make things more concrete a trivial example of a commutative vertex algebra is presented. We take the vector space V to consist of 2-vectors, v, with components v=(p,q)T. If we think of this vector as a complex number, p+qi, the Y function can be seen as taking it from its vector form to a 2x2 matrix form of (p-qz)+qi. The p and q are real numbers.

So that multiplication between two vectors u=(p,q)T and v=(r,s)T is defined by:

The identity element Id_V = (1,0)T so that:

and

and the operator T in this example is:

it is easy to check:

Since the Y matrices commute, locality is satisfied trivially. Non-trivial vertex algebras and vertex operator algebras would require infinite sized matrices to represent them. All the terms with powers of zn with n < 0 must be zero when applied to the identity vector Id_V. i.e. of the form:

but adding a single term like this won't commute with the other terms. Hence our example would no longer be trivial since we may need an infinite number of terms to restore locality.