Primitive ideals and automorphism group of $U_q^+(B_2)$

Abstract

Let g be a complex simple Lie algebra of type B_2 and q be a non-zero complex number which is not a root of unity. In the classical case, a theorem of Dixmier asserts that the simple factor algebras of the positive part U^+(g) of the enveloping algebra of g, whose Gelfand-Kirillov dimension is equal to 2, are isomorphic to the first Weyl algebra. In order to obtain some new quantized analogues of the first Weyl algebra, we explicitly describe the prime and primitive spectra of the positive part U_q^+(g) of the quantized enveloping algebra of g and then we study the simple factor algebras of U_q^+(g) whose Gelfand-Kirillov dimension is equal to 2. In particular, we show that the centers of such simple factor algebras are reduced to the ground field C and we compute their group of invertible elements. These computations suggest that we distinguish between two families of such simple factor algebras of U_q^+(g). The first family consists of those simple factor algebras whose group of units is non-trivial; algebras in this first family are the so-called Weyl-Hayashi algebras and turn out to be examples of generalized Weyl algebras over a Laurent polynomial ring in one indeterminate. In contrast, the second family consists of those simple factor algebras whose group of units is trivial; algebras in this second class can not be presented as generalized Weyl algebras over a (Laurent) polynomial ring in one indeterminate. Finally we use these results to describe the structure of the automorphism group of U_q^+(g). More precisely, we prove that this group is isomorphic to the torus (C^*)^2, as conjectured by Andruskiewitsch and Dumas. As a corollary, we obtain that the action of this group on the set of all (left) primitive ideals of U_q^+(g) has exactly 8 orbits that we describe explicitly.