A Note on Regularities of The Standard Quantized Matrix Algebra $M_q(n)$

TL;DR

This paper investigates the algebraic properties of the standard quantized matrix algebra $M_q(n)$, establishing its regularity, Cohen-Macaulay, and maximal order properties, thereby deepening understanding of its structure.

Contribution

The paper proves that $M_q(n)$ is Auslander regular, Cohen-Macaulay, Artin-Schelter regular, and a maximal order, providing new insights into its algebraic regularities and structural properties.

Findings

$M_q(n)$ is Auslander regular.

$M_q(n)$ is Cohen-Macaulay.

$M_q(n)$ is a maximal order in its quotient division algebra.

Abstract

Let $M_q(n)$ be the standard quantized matrix algebra (introduced by Faddeev, Reshetikhin, and Takhtajan). It is shown that $M_q(n)$ is Auslander regular, Cohen-Macaulay, Artin-Schelter regular, and a maximal order in its quotient division algebra.