Poisson kernel on the quantum Lobachevsky spaces (Q1876554)

The Poisson transform on the quantum Lobachevsky spaces is defined and the properties of the transform are discussed. The idea of the note is to define an analog of the Poisson transform on the quantum Lobachevsky space based on the classical Lobachevsky space \(L^3\) equipped with the hyperbolic metric. Thus, the Poisson kernel has the integral representation \(P_\nu(x^*,H,y)=\sigma F(Q_\nu)\) with \(FQ_\nu (y^*,H,y)\) as Fourier transform. The notations are more complex, so the readers can refer to the original paper. At the end of the paper, the authors give a construction idea by a commutative diagram \[ \begin{tikzcd} (\psi\in K, \Xi_q) \ar[r,"F^{-1}"]\ar[d,"Q_\mu\times" '] & (\varphi\in Z,\Xi_q)\ar[d,"P_\nu *"]\\ (\widetilde W_\nu, \widetilde L_{\delta,q})\ar[r,"\sigma F" '] & (F_\nu\in W_\nu,L_{\delta,q})\rlap{\,.}\end{tikzcd} \]