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# THE GRADED ALGEBRA GENERATED BY TWO EULERIAN DERIVATIVES

Abstract. We study the algebra R p;q generated by the Eulerian derivatives for two parameters p and q. Subject to certain conditions on the parameters, we show that R p;q is a finitely presented N-graded algebra of Gelfand-Kirillov dimension 3. We establis

THE GRADED ALGEBRA GENERATED BY TWO EULERIAN DERIVATIVES DAVID A. JORDAN Abstract. We study the algebra R

generated by the Eulerian derivatives for two parameters p and q. Subject to certain conditions on the parameters, we show that R is a nitely presented N -graded algebra of Gelfand-Kirillov dimension 3. We establish a criterion for the cyclic module R=R f to be Noetherian, where f is homogeneous of degree 1. For some choices of the parameters, this criterion always holds and we know of no situation where it fails. It is not known whether R is Noetherian. We classify the point modules for R and determine the normal elements and graded automorphisms for R . p;q p;q p;q p;q p;q p;q p;q

1. Introduction Let F be a eld and let V be the Laurent polynomial ring F y]. For 0 6= h 2 F and 1 6= q 2 F, let dh; xq 2 EndF V be as follows: ) dh: f (y) 7! f (y+ hh? f (y); xq: f (y) 7! f (qy)? f (y): qy? y Let x= d= d=dy, the usual derivative, so that, when F is a sub eld of C, limq! xq= x P limh! dh. For q 2 F and a non-negative= integer n, let n]q= in? qi and let?n]q=?q?n n]q . Then, for n 2 Z, xq: yn 7! n]q yn?: The operator xq is variously known as the q-di erence operator 8], q-derivative operator 6] or Eulerian derivative 7] and was studied, in the context of real-valued functions, by F. H. Jackson in a sequence of papers including 10]. It has signi cance in the theory of hypergeometric series 6] and has also been of interest in operator calculus, for example 7], and noncommutative ring theory, for example 8], where, due to the identity xq y? qyxq= 1, it features in a representation of the quantized Weyl algebra Aq as an algebra of operators on V, with y acting by multiplication. For h; k 2 F the operators dh and dk commute with each other and if char F= 0 and either the additive subgroup of F generated by h 1 1 0 1 1 1 1=0 0 1

Key words and phrases. Graded ring, Eulerian derivative, point module. 1

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