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

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 $Π$ is introdused. This means that the representation $Π$ is invariant under coproduct by arbitrary representation. Algebra of functions on the quantum group $SU_{q}(2)$ in the representation $Π$ is a factor of a type II$_{\infty}$. Formula for the trace in the representation $Π$ is given . The invariant integral of Woronovich on $SU_{q}(2)$ will take the form $\int f dμ= tr(f cc^{*})$.