The Prime ideal Stratification and The Automorphism Group of $U_{r,s}^{+}(B_2)$

Let $\mathfrak{g}$ be a finite dimensional complex simple Lie algebra, and let $r,s\in {\mathbb{C}}^{\ast }$ be transcendental over $\mathbb{Q}$ such that $r^m{s}^n=1$ implies $m=n=0$. We will obtain some basic properties of the two-parameter quantized enveloping algebra $U_{r,s}^{+}(\mathfrak{g})$. In particular, we will verify that the algebra $U_{r,s}^{+}(\mathfrak{g})$ satisfies many nice properties such as having normal separation, catenarity and Dixmier-Moeglin equivalence. We shall study a concrete example, the algebra $U_{r,s}^{+}(B_2)$ in detail. We will first determine the normal elements, prime ideals and primitive ideals for the algebra $U_{r,s}^{+}(B_2)$, and study their stratifications. Then we will prove that the algebra automorphism group of the algebra $U_{r,s}^{+}(B_2)$ is isomorphic to $({\mathbb{C}}^{\ast }{)}^2$.