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Module algebra structures of nonstandard quantum group $X_{q}(A_{1})$ on $\C_{q}[x,y,z]$

Dong Su

TL;DR

This work classifies all $X_q(A_1)$-module algebra structures on the quantum polynomial algebra $\mathbb{C}_q[x,y,z]$ for the nonstandard quantum group. Using a weight-decomposition approach, the authors analyze actions via full action matrices and constrain them with $K_i$-weights and the relation $EF-FE=\frac{K_2K_1^{-1}-K_2^{-1}K_1}{q-q^{-1}}$, distinguishing the $t=0$ and $t\neq 0$ cases. For $t=0$, they obtain a complete classification up to isomorphism: all nontrivial structures have $E=F=0$ and diagonal $K_i$ actions, yielding a three-parameter family of pairwise nonisomorphic module algebras. For $t\neq 0$, the analysis allows $K_i$ actions with $xz$-terms and yields two families of nonisomorphic structures, both with $E=F=0$, parameterized by explicit weight data and auxiliary parameters. The results advance understanding of Hopf actions of nonstandard quantum groups on quantum coordinate algebras and provide explicit, disjoint families of module algebra structures for further study.

Abstract

In this paper, the module algebra structures of $X_{q}(A_{1})$ on quantum polynomial algebra $\C_{q}[x,y,z]$ are investigated, and a complete classification of $X_{q}(A_{1})$-module algebra structures on $\C_{q}[x,y,z]$ is given

Module algebra structures of nonstandard quantum group $X_{q}(A_{1})$ on $\C_{q}[x,y,z]$

TL;DR

This work classifies all -module algebra structures on the quantum polynomial algebra for the nonstandard quantum group. Using a weight-decomposition approach, the authors analyze actions via full action matrices and constrain them with -weights and the relation , distinguishing the and cases. For , they obtain a complete classification up to isomorphism: all nontrivial structures have and diagonal actions, yielding a three-parameter family of pairwise nonisomorphic module algebras. For , the analysis allows actions with -terms and yields two families of nonisomorphic structures, both with , parameterized by explicit weight data and auxiliary parameters. The results advance understanding of Hopf actions of nonstandard quantum groups on quantum coordinate algebras and provide explicit, disjoint families of module algebra structures for further study.

Abstract

In this paper, the module algebra structures of on quantum polynomial algebra are investigated, and a complete classification of -module algebra structures on is given
Paper Structure (7 sections, 12 theorems, 107 equations)

This paper contains 7 sections, 12 theorems, 107 equations.

Table of Contents

  1. Preliminaries
  2. When $t=0$, classification of $X_{q}(A_{1})$-module algebra structures on $\mathbb{C}_{q}[x,y,z]$
  3. Properties of $X_{q}(A_{1})$-module algebras on $\mathbb{C}_{q}[x,y,z]$
  4. The structures of $X_{q}(A_{1})$-module algebra on $\mathbb{C}_{q}[x,y,z]$
  5. When $t\neq0$, classification of $X_{q}(A_{1})$-module algebra structures on $\mathbb{C}_{q}[x,y,z]$
  6. Properties of $X_{q}(A_{1})$-module algebras on $\mathbb{C}_{q}[x,y,z]$
  7. The structures of $X_{q}(A_{1})$-module algebra on $\mathbb{C}_{q}[x,y,z]$

Key Result

Lemma 2.1

There are $7$ cases for the $0$-st homogeneous component $\left(M_{EF}\right)_{0}$ of $M_{EF}$, as follows:

Theorems & Definitions (21)

  • Definition 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Lemma 3.1
  • ...and 11 more