### 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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References

1. Bavula V.V., Generalized Weyl algebras and their representations, St. Petersburg Math. J. 4 (1993), 71-92.
2. Beem C., Peelaers W., Rastelli L., Deformation quantization and superconformal symmetry in three dimensions, Comm. Math. Phys. 354 (2017), 345-392, arXiv:1601.05378.
3. Dedushenko M., Fan Y., Pufu S.S., Yacoby R., Coulomb branch operators and mirror symmetry in three dimensions, J. High Energy Phys. 2018 (2018), no. 4, 037, 111 pages, arXiv:1712.09384.
4. Dedushenko M., Gaiotto D., Algebras, traces, and boundary correlators in ${\mathcal N}=4$ SYM, J. High Energy Phys. 2021 (2021), no. 12, 050, 62 pages, arXiv:2009.11197.
5. Dedushenko M., Pufu S.S., Yacoby R., A one-dimensional theory for Higgs branch operators, J. High Energy Phys. 2018 (2018), no. 3, 138, 83 pages, arXiv:1610.00740.
6. Etingof P., Klyuev D., Rains E., Stryker D., Twisted traces and positive forms on quantized Kleinian singularities of type A, SIGMA 17 (2021), 029, 31 pages, arXiv:2009.09437.
7. Etingof P., Stryker D., Short star-products for filtered quantizations, I, SIGMA 16 (2020), 014, 28 pages, arXiv:1909.13588.
8. Klyuev D., On unitarizable Harish-Chandra bimodules for deformations of Kleinian singularities, arXiv:2003.11508.
9. Klyuev D., Generalized star-products and unitarizability of bimodules over deformations and $q$-deformations of Kleinian singularities of type A, in preparation.
10. Mumford D., Tata lectures on theta. I, Progress in Mathematics, Vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983.
11. Pusz W., Irreducible unitary representations of quantum Lorentz group, Comm. Math. Phys. 152 (1993), 591-626.