Simple Modules and PI Structure of the Two-Parameter Quantized Algebra $U^+_{r,s}(B_2)$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909708032409600 |
|---|---|
| author | Mukherjee, Snehashis Pandey, Ritesh Kumar |
| author_facet | Mukherjee, Snehashis Pandey, Ritesh Kumar |
| contents | 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 $B_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21856 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simple Modules and PI Structure of the Two-Parameter Quantized Algebra $U^+_{r,s}(B_2)$ Mukherjee, Snehashis Pandey, Ritesh Kumar Representation Theory Quantum Algebra 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 $B_2$. |
| title | Simple Modules and PI Structure of the Two-Parameter Quantized Algebra $U^+_{r,s}(B_2)$ |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2506.21856 |