The discrete quantum group $su_q(2)$ and its dual

TL;DR

This paper explores the duality between the discrete quantum group $su_q(2)$, constructed via Hopf algebra deformation, and the compact quantum group $SU_q(2)$, within the framework of algebraic quantum groups.

Contribution

It provides a detailed analysis of the duality between $su_q(2)$ and $SU_q(2)$, emphasizing the construction from Hopf algebra deformation rather than duality from the compact quantum group.

Findings

The dual of the discrete quantum group $su_q(2)$ is the compact quantum group $SU_q(2)$.

The paper clarifies the construction of $su_q(2)$ from the Hopf algebra deformation of the Lie algebra.

The duality between discrete and compact quantum groups is exemplified in the case of $su_q(2)$ and $SU_q(2)$.

Abstract

Discrete quantum groups were introduced as duals of compact quantum groups by Podle\'s and Woronowicz in 1990. They have been studied intrinsically by Effros and Ruan (1994) and by the author (1996). In a more recent note (2025), we have given a slightly updated treatment, viewing the duality between discrete and compact quantum groups as a special case of the more general duality of algebraic quantum groups. Along these lines, we start in this paper with the discrete quantum group $su_q(2)$, not constructed as the dual of the compact quantum group $SU_q(2)$ but rather from the Hopf algebra deformation of the enveloping algebra of the Lie algebra of $SU(2)$, as given by Jimbo (1985). The passage to the discrete quantum group as studied in earlier papers is not completely trivial as we will see. This is a known phenomenon. We consider the dual of this discrete quantum group in the sense of duality of algebraic quantum groups and see that this indeed is the compact quantum group $SU_q(2)$.