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The zero modes' Fock space for the extended chiral $su(2)$ WZNW model givesroom to a realization of the Grothendieck fusion ring of representations of therestricted $U_q sl(2)$ quantum universal enveloping algebra (QUEA) at an even($2h$-th) root of unity, and of its extension by the Lusztig operators. It isshown that expressing the Drinfeld images of canonical characters in terms ofChebyshev polynomials of the Casimir invariant $C$ allows a streamlinedderivation of the characteristic equation of $C$ from the defining relations ofthe restricted QUEA. The properties of the fusion ring of the Lusztig'sextension of the QUEA in the zero modes' Fock space are related to the braidingproperties of correlation functions of primary fields of the extended$su(2)_{h-2}$ current algebra model.

Zero modes' fusion ring and braid group representations for the extended chiral su(2) WZNW model