The Stable Representation of the algebra of functions on the quantum group $SU_{q}(2)$

TL;DR

This paper investigates the coproduct operation on representations of the quantum group $SU_q(2)$, introduces stable representations invariant under coproduct, and characterizes the algebra of functions as a type II$_{ ext{infinity}}$ factor with a specific trace formula.

Contribution

It introduces the notion of stable representations for the quantum group $SU_q(2)$ and analyzes their properties, including the algebra's classification and trace formula.

Findings

The algebra of functions on $SU_q(2)$ in a stable representation is a type II$_{ ext{infinity}}$ factor.

A formula for the trace in the stable representation is provided.

The invariant integral on $SU_q(2)$ is expressed as a trace of a specific operator.

Abstract

An operation of a coproduct of representations of a bialgebra is defined. The coproduct operation for representations of the Hopf algebra of functions on the quantum group $SU_{q}(2)$ is investigated. A notion of a stable representation $\Pi$ is introdused. This means that the representation $\Pi$ is invariant under coproduct by arbitrary representation. Algebra of functions on the quantum group $SU_{q}(2)$ in the representation $\Pi$ is a factor of a type II$_{\infty}$. Formula for the trace in the representation $\Pi$ is given . The invariant integral of Woronovich on $SU_{q}(2)$ will take the form $\int f d\mu = tr(f cc^{*})$.