Non-formal star-exponential on contracted one-sheeted hyperboloids
Pierre Bieliavsky, Axel de Goursac, Yoshiaki Maeda, Florian Spinnler
TL;DR
This paper constructs a non-formal star-exponential for SL(2,R) on hyperboloid orbits, using Bessel functions, and establishes a new identity on Bessel functions as an application.
Contribution
It provides the first explicit geometric realization of the star-exponential on hyperboloid orbits with a natural star-product, including a new Bessel function identity.
Findings
Explicit expression of star-exponential with Bessel functions
Continuous group homomorphism into von Neumann algebra
New identity on Bessel functions
Abstract
In this paper, we exhibit the non-formal star-exponential of the Lie group SL(2,R) realized geometrically on the curvature contraction of its one-sheeted hyperboloid orbits endowed with its natural non-formal star-product. It is done by a direct resolution of the defining equation of the star-exponential and produces an expression with Bessel functions. This yields a continuous group homomorphism from SL(2,R) into the von Neumann algebra of multipliers of the Hilbert algebra underlied by this natural star-product. As an application, we prove a new identity on Bessel functions.
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