Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A
Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392] and [Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages], we undertake a detailed study of twisted traces on quantizations of Kleinian singularities of type ₙ₋₁. In particular, we give explicit integral formulas...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
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2021
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| Zitieren: | Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A. Pavel Etingof, Daniil Klyuev, Eric Rains and Douglas Stryker. SIGMA 17 (2021), 029, 31 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862632769496023040 |
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| author | Etingof, Pavel Klyuev, Daniil Rains, Eric Stryker, Douglas |
| author_facet | Etingof, Pavel Klyuev, Daniil Rains, Eric Stryker, Douglas |
| citation_txt | Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A. Pavel Etingof, Daniil Klyuev, Eric Rains and Douglas Stryker. SIGMA 17 (2021), 029, 31 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392] and [Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages], we undertake a detailed study of twisted traces on quantizations of Kleinian singularities of type ₙ₋₁. In particular, we give explicit integral formulas for these traces and use them to determine when a trace defines a positive Hermitian form on the corresponding algebra. This leads to a classification of unitary short star-products for such quantizations, a problem posed by Beem, Peelaers, and Rastelli in connection with 3-dimensional superconformal field theory. In particular, we confirm their conjecture that for ≤ 4 a unitary short star-product is unique and compute its parameter as a function of the quantization parameters, giving exact formulas for the numerical functions by Beem, Peelaers, and Rastelli. If = 2, this, in particular, recovers the theory of unitary spherical Harish-Chandra bimodules for ₂. Thus, the results of this paper may be viewed as a starting point for a generalization of the theory of unitary Harish-Chandra bimodules over enveloping algebras of reductive Lie algebras [Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we derive recurrences to compute the coefficients of short star-products corresponding to twisted traces, which are generalizations of discrete Painlevé systems.
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| first_indexed | 2026-03-14T20:34:07Z |
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| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T20:34:07Z |
| publishDate | 2021 |
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| spelling | Etingof, Pavel Klyuev, Daniil Rains, Eric Stryker, Douglas 2025-12-29T11:10:43Z 2021 Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A. Pavel Etingof, Daniil Klyuev, Eric Rains and Douglas Stryker. SIGMA 17 (2021), 029, 31 pages 1815-0659 2020 Mathematics Subject Classification: 16W70; 33C47 arXiv:2009.09437 https://nasplib.isofts.kiev.ua/handle/123456789/211320 https://doi.org/10.3842/SIGMA.2021.029 Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392] and [Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages], we undertake a detailed study of twisted traces on quantizations of Kleinian singularities of type ₙ₋₁. In particular, we give explicit integral formulas for these traces and use them to determine when a trace defines a positive Hermitian form on the corresponding algebra. This leads to a classification of unitary short star-products for such quantizations, a problem posed by Beem, Peelaers, and Rastelli in connection with 3-dimensional superconformal field theory. In particular, we confirm their conjecture that for ≤ 4 a unitary short star-product is unique and compute its parameter as a function of the quantization parameters, giving exact formulas for the numerical functions by Beem, Peelaers, and Rastelli. If = 2, this, in particular, recovers the theory of unitary spherical Harish-Chandra bimodules for ₂. Thus, the results of this paper may be viewed as a starting point for a generalization of the theory of unitary Harish-Chandra bimodules over enveloping algebras of reductive Lie algebras [Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we derive recurrences to compute the coefficients of short star-products corresponding to twisted traces, which are generalizations of discrete Painlevé systems. The work of P.E. was partially supported by the NSF grant DMS-1502244. P.E. is grateful to Anton Kapustin for introducing him to the topic of this paper, and to Chris Beem, Mykola Dedushenko, and Leonardo Rastelli for useful discussions. E.R. would like to thank Nicholas Witte for pointing out the reference [12]. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A Article published earlier |
| spellingShingle | Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A Etingof, Pavel Klyuev, Daniil Rains, Eric Stryker, Douglas |
| title | Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A |
| title_full | Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A |
| title_fullStr | Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A |
| title_full_unstemmed | Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A |
| title_short | Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A |
| title_sort | twisted traces and positive forms on quantized kleinian singularities of type a |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211320 |
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