Twisted Traces and Positive Forms on Generalized $q$-Weyl Algebras

TL;DR

This paper classifies positive definite invariant Hermitian forms on generalized q-Weyl algebras, which are algebraic structures generated by elements with specific relations involving a Laurent polynomial.

Contribution

It provides a complete classification of positive definite invariant Hermitian forms on generalized q-Weyl algebras, extending understanding of their algebraic and Hermitian structure.

Findings

Classification of positive definite invariant Hermitian forms achieved

Conditions for invariance under algebra automorphisms established

New insights into the structure of generalized q-Weyl algebras obtained

Abstract

Let ${\mathcal A}$ be a generalized $q$-Weyl algebra, it is generated by $u$, $v$, $Z$, $Z^{-1}$ with relations $ZuZ^{-1}=q^2u$, $ZvZ^{-1}=q^{-2}v$, $uv=P\big(q^{-1}Z\big)$, $vu=P(qZ)$, where $P$ is a Laurent polynomial. A Hermitian form $(\cdot,\cdot)$ on ${\mathcal A}$ is called invariant if $(Za,b)=\big(a,bZ^{-1}\big)$, $(ua,b)=(a,sbv)$, $(va,b)=\big(a,s^{-1}bu\big)$ for some $s\in {\mathbb C}$ with $|s|=1$ and all $a,b\in {\mathcal A}$. In this paper we classify positive definite invariant Hermitian forms on generalized $q$-Weyl algebras.