$U^{+}_{r,s}(B_{2})$的素理想层与自同构群
The Prime ideal Stratification and The Automorphism Group of $U^{+}_{r,s}(B_{2})$
摘要：
设${\mathfrak g}$是有限维复单李代数，$r,s\in \mathbb{C}^{\ast}$是$\mathbb{Q}$上的超越代数，使得$r^{m}s^{n}=1$蕴涵$m=n=0$ .我们将得到两参数量子化包络代数$U_{r,s}^{+}(\mathfrak g)$的一些基本性质。特别地，我们将证明代数$U_{r,s}^{+}(\mathfrak g)$满足许多很好的性质，如具有正常分离，链和Dixmier-Moeglin等价。我们将详细研究一个具体的例子，代数$U_{r,s}^{+}(B_{2})$。首先确定了代数$U_{r,s}^{+}(B_{2})$的正规元、素理想和本原理想，并研究了它们的分层。然后证明了代数$U_{r,s}^{+}(B_{2})$的代数自同构群与$(\mathbb{C}^{\ast})^{2}$同构。
英文摘要：
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 $(\mathbb{C}^{\ast})^{2}$.
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