On the representation of derivative algebras in characteristic $p \gt 0$

Abstract

In this paper we show that neither the Weyl algebra $A_n(K)$ nor the derivative algebra $DA_n(K)$ has infinite irreducible representations in the case when the ground field $K$ has characteristic $p \gt 0$. We also give a complete classification of irreducible representations of the first derivative algebra $DA_1$ when $K$ is algebraically closed. Finally, we present an algorithm that determines, in finitely many steps, whether $DA_1/L$ is a simple $DA_1$-module, where $L$ is any left ideal of $DA_1$.