AN EXPLICIT CONSTRUCTION OF THE QUANTUM GROUP IN CHIRAL WZW-MODELS

It is shown how a chiral Wess-Zumino-Witten theory with globally defin ed vertex operators and a one-to-one correspondence between fields and states can be constructed. The Hilbert space of this theory is the di rect sum of tensor products of representations of the chiral algebra a nd finite dimensional internal parameter spaces. On this enlarged spac e there exists a natural action of Drinfeld's quasi-quantum group A(g, t), which commutes with the action of the chiral algebra and plays the role of an internal symmetry algebra. The R matrix describes the brai ding of the chiral vertex operators and the coassociator Phi gives ris e to a modification of the duality property. For generic q the quasi-q uantum group is isomorphic to the coassociative quantum group U-q(g) a nd thus the duality property of the chiral theory can be restored. Thi s construction has to be modified for the physically relevant case of integer level. The quantum group has to be replaced by the correspondi ng truncated quasiquantum group, which is not coassociative because of the truncation. This exhibits the truncated quantum group as the inte rnal symmetry algebra of the chiral WZW model, which therefore has onl y a modified duality property. The case of g = su(2) is worked out in detail.