PRIMITIVE IDEALS AND AUTOMORPHISM GROUP OF $U_{q}^{+}(B_{2})$

Abstract

Let 𝔤 be a complex simple Lie algebra of type B2 and q be a nonzero complex number which is not a root of unity. In the classical case, a theorem of Dixmier asserts that the simple factor algebras of Gelfand–Kirillov dimension 2 of the positive part U+(𝔤) of the enveloping algebra of 𝔤 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 [Formula: see text] of the quantized enveloping algebra of 𝔤 and then we study the simple factor algebras of Gelfand–Kirillov dimension 2 of [Formula: see text]. 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 allow us to prove that the automorphism group of [Formula: see text] is isomorphic to the torus (ℂ*)2, as conjectured by Andruskiewitsch and Dumas.