An Extension of the Character Ring of sl(3) and Its Quantisation

We construct a commutative ring with identity which extends the ring of characters of finite dimensional representations of sl(3). It is generated by characters with values in the group ring $Z[\tilde{W}]$ of the extended affine Weyl group of $\widehat{sl}(3{)}_k$ at $k\in ̸Q$. The `quantised’ version at rational level $k+3=3\mathrm{/}p$ realises the fusion rules of a WZW conformal field theory based on admissible representations of $\widehat{sl}(3{)}_k$.