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Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A

Etingof, Pavel and Klyuev, Daniil and Rains, Eric and Stryker, Douglas (2021) Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A. Symmetry, Integrability and Geometry, Methods and Applications (SIGMA), 17 . Art. No. 29. ISSN 1815-0659. doi:10.3842/sigma.2021.029. https://resolver.caltech.edu/CaltechAUTHORS:20210602-134422197

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Abstract

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 A_(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 n≤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 n=2, this, in particular, recovers the theory of unitary spherical Harish-Chandra bimodules for sl₂. 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.3842/sigma.2021.029DOIArticle
https://arxiv.org/abs/2009.09437arXivDiscussion Paper
Additional Information:© 2021 The authors retain the copyright for their papers published in SIGMA under the terms of the Creative Commons Attribution-ShareAlike License. Received September 22, 2020, in final form March 08, 2021; Published online March 25, 2021. This paper is a contribution to the Special Issue on Representation Theory and Integrable Systems in honor of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html. 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].
Funders:
Funding AgencyGrant Number
NSFDMS-1502244
Subject Keywords:star-product; orthogonal polynomial; quantization; trace
Classification Code:2020 Mathematics Subject Classification: 16W70; 33C47
DOI:10.3842/sigma.2021.029
Record Number:CaltechAUTHORS:20210602-134422197
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210602-134422197
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:109347
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:02 Jun 2021 21:15
Last Modified:02 Jun 2021 21:15

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