Module algebras on truncated quantum planes over small quantum groups

Wanxia Wang; Shilin Yang; Wanxia Wang; Shilin Yang

doi:10.3934/era.2026220

Electronic Research Archive

2026, Volume 34, Issue 7: 4968-4998. doi: 10.3934/era.2026220

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Research article

Module algebras on truncated quantum planes over small quantum groups

Wanxia Wang ¹,
Shilin Yang ^{2,3
,
,}

1.
College of Big Data and Software Engineering, Zhejiang Wanli University, Ningbo 315100, China
2.
School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China
3.
Department of General Education, Shandong Xiehe University, Jinan 250109, China

Received: 31 March 2026 Revised: 27 May 2026 Accepted: 02 June 2026 Published: 09 June 2026

Let $ q $ be a $ 2n $-th primitive root of unity and $ {\mathfrak{u}_q({\mathfrak{sl}}_2)} $ a small quantum group in the usual way. Under reasonable simplifying assumptions, we provided a complete classification of all $ {\mathfrak{u}_q({\mathfrak{sl}}_2)} $-module algebras on the truncated quantum plane $ {A_{n, q}[x, y]} $, which was generated by $ x $, $ y $ subject to the relations: $ yx = qxy, \ x^n = y^n = 0. $
- truncated quantum plane,
- small quantum group,
- module algebra
Citation: Wanxia Wang, Shilin Yang. Module algebras on truncated quantum planes over small quantum groups[J]. Electronic Research Archive, 2026, 34(7): 4968-4998. doi: 10.3934/era.2026220

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Abstract

Let $ q $ be a $ 2n $-th primitive root of unity and $ {\mathfrak{u}_q({\mathfrak{sl}}_2)} $ a small quantum group in the usual way. Under reasonable simplifying assumptions, we provided a complete classification of all $ {\mathfrak{u}_q({\mathfrak{sl}}_2)} $-module algebras on the truncated quantum plane $ {A_{n, q}[x, y]} $, which was generated by $ x $, $ y $ subject to the relations: $ yx = qxy, \ x^n = y^n = 0. $

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