On a Morita equivalence between the duals of quantum SU(2) and quantum E(2)

TL;DR

This paper establishes a Morita equivalence between the duals of quantum SU(2) and quantum E(2), showing their von Neumann algebraic quantum groups are related via unitary cocycle deformations, revealing deep structural similarities.

Contribution

It proves that the duals of SU_q(2) and E_q(2) are Morita equivalent as locally compact quantum groups through unitary cocycle deformations.

Findings

The von Neumann algebraic quantum groups of SU_q(2) and E_q(2) are unitary cocycle deformations of each other.

Their duals are Morita equivalent as locally compact quantum groups.

The result links the quantum deformations of classical Lie groups through quantum group theory.

Abstract

Let SU_q(2) and E_q(2) be Woronowicz' q-deformations of respectively the compact Lie group SU(2) and the non-trivial double cover of the Lie group E(2) of Euclidian transformations of the plane. We prove that, in some sense, their duals are `Morita equivalent locally compact quantum groups'. In more concrete terms, we prove that the von Neumann algebraic quantum groups of `bounded measurable functions' on SU_q(2) and E_q(2) are unitary cocycle deformations of each other.