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Twisted Traces and Positive Forms on Generalized $q$-Weyl Algebras
Daniil Klyuev Department of Mathematics, Massachusetts Institute of Technology, USA
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.
Keywords:
quantization, trace, inner product, star-product.
Received: May 27, 2021; in final form January 17, 2022; Published online January 30, 2022
Citation:
Daniil Klyuev, “Twisted Traces and Positive Forms on Generalized $q$-Weyl Algebras”, SIGMA, 18 (2022), 009, 28 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1804 https://www.mathnet.ru/eng/sigma/v18/p9
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Abstract page: | 70 | Full-text PDF : | 22 | References: | 6 |
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