Simple Modules and PI Structure of the Two-Parameter Quantized Algebra $U_{r,s}^{+}(B_2)$

We study the two-parameter quantized enveloping algebra $U_{r,s}^{+}(B_2)$ at roots of unity and investigate its structure and representations. We first show that when $r$ and $s$ are roots of unity, the algebra becomes a PI algebra, and we compute its PI degree explicitly using De Concini-Procesi method. We construct and classify finite-dimensional simple modules for $U_{r,s}^{+}(B_2)$ by analyzing a subalgebra $B\subset U_{r,s}^{+}(B_2)$. Simple modules are categorized into torsion-free and torsion types with respect to a distinguished normal element. We classify all torsion-free simple $B$-modules and lift them to $U_{r,s}^{+}(B_2)$. The remaining simple modules are constructed in the nilpotent case. This work provides a complete classification of simple $U_{r,s}^{+}(B_2)$-modules at roots of unity and contributes to the understanding of two-parameter quantum groups in type $B_2$.