A Dichotomy for the Gelfand–Kirillov Dimensions of Simple Modules over Simple Differential Rings

Abstract

The Gelfand-Kirillov dimension has gained importance since its introduction as an tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand-Kirillov dimension of simple modules over certain simple rings of differential operators. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. J. Algebra 76 (1982), no. 2, 489-493 concerning this question for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand-Kirillov dimension of an induced module. * Corresponding author † Graduate student.