Global Pseudo-differential Operators on the Quantum Group $SU_q(2)$

TL;DR

This paper develops a theory of global pseudo-differential operators on the quantum group $SU_q(2)$, including spectral analysis and a $*$-representation, extending classical analysis to a quantum setting.

Contribution

It introduces a new framework for pseudo-differential operators on $SU_q(2)$, including a Fourier-based graduation and a bounded operator representation.

Findings

Spectral results for pseudo-differential operators on $SU_q(2)$

A Fourier decomposition-based graduation of the algebra

A $*$-representation on $L^2(S^1)$

Abstract

In this paper, following [1], we develop the theory of global pseudo-differential operators defined on the quantum group $SU_q(2)$, and provide some spectral results concerning these operators. We define a graduation for this algebra of pseudo-differential operators in terms of its natural Fourier decomposition and, using the infinite-dimensional representations introduced by Woronowicz [15], we also provide a $*$-representation of $SU_q(2)$ as bounded pseudo-differential operators acting on $L^2(\mathbb{S}^1)$.