Primitive ideals of U-q(Sl(n)(+))

Abstract

Let U-q(g(+)) be the quantized enveloping algebra of the nilpotent Lie algebra g(+) = sl(n+1)(+) which occurs as the positive part in the triangular decomposition of the simple Lie algebra sl(n+1) of type A(n). Assuming the base field K is algebraically closed and of characteristic 0, and that the parameter q is an element of K* is not a root of unity, we define and study certain quotients of U-q(g(+)) which coincide with the Weyl-Hayashi algebra when n = 2 (see Alev and Dumas, 1996, Hayashi, 1990; Kirkman and Small, 1993). 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 U-q(sl(4)(+)) in detail, somewhat in the spirit of Launois (to appear). We determine all primitive ideals of U-q(sl(4)(+)), find a set of generators for each one, compute their heights and find a simple U-q(sl(4)(+))-module corresponding to each primitive ideal of U-q(sl(4)(+)).