In this post I am going to look at the centralizer of non-central elements in the first Weyl algebra over some field of characteristic zero. Recall that the first Weyl algebra is defined to be the -algebra generated by and with the relation It then follows easily that for every where It is easily seen that the center of is Also, every non-zero element of can be uniquely written in the form where and We call the degree of It is easy to see that is a domain. For every we will denote by the centralizer of in The goal is to show that if then is a commutative algebra and also a free -module of finite rank. This result is due to Amitsur.

**Remark 1**. If then where means the -th derivative of with respect to This follows easily by induction and the fact that

**Remark 2**. If then This is easy to see: clearly Conversely, if commutes with and then comparing the coefficients in both sides of will give us which is a contradiction. Thus and so

So, by the above remark, we only need to find the centralizer of an element of in the form

**Lemma 1**. Let be a field of characteristic zero and let Let and be two elements of Then for some

*Proof*. Since for any induction on shows that for any integer Therefore the coefficient of in and are and respectively. Thus, since we must have

Hence, since is commutative, we will have

A similar arguemnt shows that implies that

Now, multiplying both sides of (1) by and both sides of (2) by and then subtracting the resulting identities will give us Thus

because is a domain, and the characteristic of is zero. Now look at as a subalgebra of the field of rational functions Then, since by (3) we have and hence i.e. for some

In part (2) we will prove that is a free -module of finite rank.