The Standard Quantized Matrix Algebra $M_q(n)$ is A Solvable Polynomial Algebra

TL;DR

This paper proves that the standard quantized matrix algebra $M_q(n)$ is a solvable polynomial algebra by constructing a specific monomial ordering, enabling further structural analysis and computational methods.

Contribution

It introduces a monomial ordering on $M_q(n)$'s PBW basis, demonstrating its status as a solvable polynomial algebra for the first time.

Findings

$M_q(n)$ is a solvable polynomial algebra

Structural properties of $M_q(n)$ can be systematically studied

Enables computational approaches to modules over $M_q(n)$

Abstract

Let $M_q(n)$ be the standard quantized matrix algebra, introduced by Faddeev, Reshetikhin, and Takhtajan. It is shown, by constructing an appropriate monomial ordering $\prec$ on its PBW $K$-basis ${\cal B}$ , that $M_q(n)$ is a solvable polynomial algebra. Consequently, further structural properties of $M_q(n)$ and their modules may be established and realized in a constructive-computational way.