Isomorphisms of some quantum spaces

TL;DR

This paper investigates conditions under which various quantum algebraic structures, such as quantum planes, matrix algebras, and Weyl algebras, are isomorphic, revealing specific parameter relationships that determine isomorphism.

Contribution

It extends the understanding of isomorphisms among quantum spaces, including quantum matrix algebras and Weyl algebras, with new criteria for their equivalence.

Findings

Quantum Weyl algebras are isomorphic iff their parameters are equal or inverses.

Conditions for isomorphism of quantum planes and matrix algebras are established.

Modified a key result by Alev and Dumas regarding quantum Weyl algebras.

Abstract

We consider a series of questions that grew out of determining when two quantum planes are isomorphic. In particular, we consider a similar question for quantum matrix algebras and certain ambiskew polynomial rings. Additionally, we modify a result by Alev and Dumas to show that two quantum Weyl algebras are isomorphic if and only if their parameters are equal or inverses of each other.