An extension of the character ring of sl(3) and its quantisation (Q1297673)

The authors study the characters (one-dimensional representations) of the fusion algebra of \(\mathfrak{g}=\widehat{sl}(n)_k\) WZW conformal models at rational (shifted) level \(\kappa=k+n=p'/p\) for \(n=3=p'\), \((p,3)=1\) and their ``classical'' counterparts. They 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 \(\mathbb{Z}[\widetilde{W}]\) of the extended affine Weyl group of \(\widehat{sl}(3)_k\) at \(k\not\in\mathbb{Q}\). This construction of ``classical characters'' takes a considerable part of the paper and is its main novel result. Then the ``quantized'' version at rational levels \(k+3=3/p\) realizes the fusion rules of a WZW conformal field theory based on admissible representations of \(\widehat{sl}(3)_k\).