  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.
