The Prime ideal Stratification and The Automorphism Group of $U^{+}_{r,s}(B_{2})$

TL;DR

This paper investigates the structure and automorphisms of a two-parameter quantum algebra associated with a simple Lie algebra, specifically analyzing prime and primitive ideals, stratification, and automorphism groups.

Contribution

It provides a detailed study of the algebra $U_{r,s}^{+}(B_{2})$, including its prime and primitive ideals, stratification, and automorphism group, expanding understanding of two-parameter quantum groups.

Findings

Determined the normal elements, prime ideals, and primitive ideals of $U_{r,s}^{+}(B_{2})$.

Proved the automorphism group of $U_{r,s}^{+}(B_{2})$ is isomorphic to $( ext{C}^*)^2$.

Verified properties like normal separation, catenarity, and Dixmier-Moeglin equivalence for $U_{r,s}^{+}(rak g)$.

Abstract

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 $(\mathbb{C}^{\ast})^{2}$.