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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 A be a generalized q-Weyl algebra, it is generated by u, v, Z, Z−1 with relations ZuZ−1=q2u, ZvZ−1=q−2v, uv=P(q−1Z), vu=P(qZ), where P is a Laurent polynomial. A Hermitian form (⋅,⋅) on A is called invariant if (Za,b)=(a,bZ−1), (ua,b)=(a,sbv), (va,b)=(a,s−1bu) for some s∈C with |s|=1 and all a,b∈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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