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Symmetry, Integrability and Geometry: Methods and Applications, 2021, Volume 17, 029, 31 pp.
DOI: https://doi.org/10.3842/SIGMA.2021.029
(Mi sigma1712)
 

This article is cited in 2 scientific papers (total in 2 papers)

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

Pavel Etingofa, Daniil Klyueva, Eric Rainsb, Douglas Strykera

a Department of Mathematics, Massachusetts Institute of Technology, USA
b Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA
Full-text PDF (524 kB) Citations (2)
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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 $n\le 4$ 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 ${\mathfrak{sl}}_2$. 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.
Keywords: star-product, orthogonal polynomial, quantization, trace.
Funding agency Grant number
National Science Foundation DMS-1502244
The work of P.E. was partially supported by the NSF grant DMS-1502244.
Received: September 22, 2020; in final form March 8, 2021; Published online March 25, 2021
Bibliographic databases:
Document Type: Article
MSC: 16W70, 33C47
Language: English
Citation: Pavel Etingof, Daniil Klyuev, Eric Rains, Douglas Stryker, “Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A”, SIGMA, 17 (2021), 029, 31 pp.
Citation in format AMSBIB
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\by Pavel~Etingof, Daniil~Klyuev, Eric~Rains, Douglas~Stryker
\paper Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type~A
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\yr 2021
\vol 17
\papernumber 029
\totalpages 31
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Symmetry, Integrability and Geometry: Methods and Applications
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