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On Simple Modules over the Quantum Matrix algebra at roots of unity

Sanu Bera, Snehashis Mukherjee

Abstract

This article investigates the two-parameter quantum matrix algebra at roots of unity. In the roots of unity setting, this algebra becomes a Polynomial Identity (PI) algebra and it is known that simple modules over such algebra are finite-dimensional with dimension at most the PI degree. We determine the center, compute the PI degree, and classify simple modules for two-parameter quantum matrix algebra, up to isomorphism, over an algebraically closed field of arbitrary characteristics.

On Simple Modules over the Quantum Matrix algebra at roots of unity

Abstract

This article investigates the two-parameter quantum matrix algebra at roots of unity. In the roots of unity setting, this algebra becomes a Polynomial Identity (PI) algebra and it is known that simple modules over such algebra are finite-dimensional with dimension at most the PI degree. We determine the center, compute the PI degree, and classify simple modules for two-parameter quantum matrix algebra, up to isomorphism, over an algebraically closed field of arbitrary characteristics.

Paper Structure

This paper contains 16 sections, 22 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Torsion and Torsionfree Modules
  4. Skew Polynomial Presentation
  5. Polynomial Identity Algebras
  6. The Center of $M_2(\alpha,\beta)$
  7. PI Degree for $M_2(\alpha,\beta)$
  8. PI degree of $M_2(\alpha,\beta)$ when $\alpha\beta=1$
  9. PI degree of $M_2(\alpha,\beta)$ when $\alpha\beta\neq 1$
  10. Construction of Simple Modules over $M_2(\alpha,\beta)$
  11. Classification of simple $M_2(\alpha,\beta)$-modules
  12. Simple $X_{12}$-torsion $M_2(\alpha,\beta)$-modules
  13. Simple $X_{21}$-torsion $M_2(\alpha,\beta)$-modules
  14. Simple $D$-torsion $M_2(\alpha,\beta)$-modules
  15. Simple $X_{12},X_{21}$ and $D$-torsionfree $M_2(\alpha,\beta)$-modules
  16. ...and 1 more sections

Key Result

Lemma 2.1

Suppose that $A$ is an algebra, $x\in A$ is a normal element and $M$ is a simple $A$-module. Then either $Mx=0$ (if $M$ is $x$-torsion) or the map $x_{M}:M\rightarrow M$ given by $m\mapsto mx$ is an isomorphism (if $M$ is $x$-torsionfree).

Theorems & Definitions (34)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 24 more