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

Abstract

Let $\mathcal{A}$ be a generalized $q$-Weyl algebra, it is generated by $u$, $v$, $Z$, $Z^{-1}$ with relations $Zu{Z}^{-1}=q^{2}u$, $Zv{Z}^{-1}=q^{-2}v$, $uv=P\left(q^{-1}Z\right)$, $vu=P(qZ)$, where $P$ is a Laurent polynomial. A Hermitian form $(\cdot ,\cdot )$ on $\mathcal{A}$ is called invariant if $(Za,b)=(a,b{Z}^{-1})$, $(ua,b)=(a,sbv)$, $(va,b)=(a,s^{-1}bu)$ 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.

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Cited by in corpus (1)

• Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A