Cyclic Representations of the Quantum Matrix Algebras

In this paper we give a complete classiication of the minimal cyclic M q (n)-modules and construct them explicitly. Also, we give a complete classiica-tion of the minimal cyclic modules of the so-called Dipper-Donkin quantum matrix algebra as well as of two other natural quantized matrix algebras. In the last part of the paper we relate the results to the De Concini { Procesi conjecture. 1. introduction At present, much progress has been made in the representation theory of quantum groups both in the generic case where the quantum parameter q is not a root of unity and in the special cases where q is a root of unity. In the latter case, a completely new class of representations, cyclic representations, have been constructed and studied by diierent authors (e.g. 1], 6]). The name cyclic actually is a little unfortunate since it does not mean \having a generating vector" but we shall keep the notation since the terminology seems to have become widely accepted. See Deenition 2.1 below for the precise deenition. The cyclic representations of the quantum enveloping algebra U q (g) at a root of unity have been studied by De Concini and Kac 2] for an arbitrary nite dimensional Lie algebra g. Some cyclic representations of U q (sl(n + 1)) were constructed in 6]. In 1], the minimal cyclic representations of U q (g) were constructed explicitly in case g is of type A n , B n , or C n. The quantum matrix algebra M q (n) is an associative algebra over C generated by