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Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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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, trace.
Mots-clés : orthogonal polynomial, quantization 
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     author = {Pavel Etingof and Daniil Klyuev and Eric Rains and Douglas Stryker},
     title = {Twisted {Traces} and {Positive} {Forms} on {Quantized} {Kleinian} {Singularities} of {Type~A}},
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Pavel Etingof; Daniil Klyuev; Eric Rains; Douglas Stryker. Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a28/

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