Let be a field. We proved here that every derivation of a finite dimensional central simple -algebra is inner. In this post I will give an example of an infinite dimensional central simple -algebra all of whose derivations are inner. As usual, we will denote by the -th Weyl algebra over Recall that is the -algebra generated by with the relations for all When we just write instead of If then is an infinite dimensional central simple -algebra and we can formally differentiate and integrate an element of with respect to or exactly the way we do in calculus. Let me clarify “integration” in For every we denote by and the derivations of with respect to and respectively. Let be such that Then and so lies in the centralizer of which is So For example, if then if and only if for some We will write
Theorem. If then every derivation of is inner.
Proof. I will prove the theorem for the idea of the proof for the general case is similar. Suppose that is a derivation of Since is -linear and the -vector space is generated by the set an easy induction over shows that is inner if and only if there exists some such that and But and Thus is inner if and only if there exists some which satisfies the following conditions
Also, taking of both sides of the relation will give us
From we have for some It is now easy to see that
will satisfy both conditions in