Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 009, 28 pages      arXiv:2105.12652      https://doi.org/10.3842/SIGMA.2022.009

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

Daniil Klyuev
Department of Mathematics, Massachusetts Institute of Technology, USA

Received May 27, 2021, in final form January 17, 2022; Published online January 30, 2022

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.

Key words: quantization; trace; inner product; star-product.

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