The isomorphism problem for quantum affine spaces, homogenized quantized Weyl algebras, and quantum matrix algebras

TL;DR

This paper solves the isomorphism problem for certain quantum algebras by exploiting graded algebra properties and an iterative process to analyze parameters, advancing understanding of their structural relationships.

Contribution

It extends the isomorphism classification to quantum affine spaces, quantum matrix algebras, and homogenized quantized Weyl algebras using a novel iterative method.

Findings

Established criteria for algebra isomorphisms based on degree one normal elements.

Developed an iterative process to determine relationships between parameters.

Provided a classification framework for quantum algebra isomorphisms.

Abstract

Bell and Zhang have shown that if $A$ and $B$ are two connected graded algebras finitely generated in degree one that are isomorphic as ungraded algebras, then they are isomorphic as graded algebras. We exploit this result to solve the isomorphism problem in the cases of quantum affine spaces, quantum matrix algebras, and homogenized multiparameter quantized Weyl algebras. Our result involves determining the degree one normal elements, factoring out, and then repeating. This creates an iterative process that allows one to determine relationships between relative parameters.