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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
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.
Received: September 22, 2020; in final form March 8, 2021; Published online March 25, 2021
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.
Linking options:
https://www.mathnet.ru/eng/sigma1712 https://www.mathnet.ru/eng/sigma/v17/p29
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