Primitive Ideals of

Abstract

Abstract Let U q (+) be the quantized enveloping algebra of the nilpotent Lie algebra + = n+1 + which occurs as the positive part in the triangular decomposition of the simple Lie algebra  n+1 of type A n . Assuming the base field  is algebraically closed and of characteristic 0, and that the parameter q ∊ ∗ is not a root of unity, we define and study certain quotients of U q (+) which coincide with the Weyl–Hayashi algebra when n = 2 (see Alev and Dumas, Citation1996; Hayashi, Citation1990; Kirkman and Small, Citation1993). We show that these are simple Noetherian domains, with a trivial center and even Gelfand-Kirillov dimension. Hence, they play a role analogous to that played by the Weyl algebras in the classical case. In the remainder of the article, we study the primitive spectrum of Uq (4 + in detail, somewhat in the spirit of Launois (Citationto appear). We determine all primitive ideals of Uq (4 +, find a set of generators for each one, compute their heights and find a simple Uq (4 +-module corresponding to each primitive ideal of Uq (4 +. Key Words: Nilpotent Lie algebraPrimitive idealsQuantized enveloping algebraWeyl algebraMathematics Subject Classification: 16W3516D3017B3716380 ACKNOWLEDGMENTS The author would like to express his gratitude to the referee for a thorough reading of the manuscript, improvements on some arguments, and in particular for suggesting a proof of Proposition 4.6. The above work is partially supported by Centro de Matemática da Universidade do Porto (CMUP), financed by FCT (Portugal) through the programs POCTI (Programa Operacional Ciência, Tecnologia, Inovaç cão) and POSI (Programa Operacional Sociedade da Informaç cão), with national and European Community structural funds, and NSF Grant #DMS0245082. Notes Communicated by J. Aleu.