de Mathématiques de Luminy
On classification and realization of Poisson-Lie anomalies.
We introduce the concept of (nonlinear) Poisson-Lie cohomology of Lie algebras and show that it classifies possible anomalies of Poisson-Lie symmetries. Given a Poisson-Lie 2-cocycle, we also explicitely construct a symplectic manifold which realizes the corresponding anomaly. We call this symplectic manifold an anomalous Heisenberg double since it can be viewed as a generalisation of the Heisenberg double of Semenov-Tian-Shansky. We show that particular classes of Poisson-Lie anomalies naturally arise by taking a limit q → ∞ of known Poisson-Lie symmetric dynamical systems. In particular, we establish that the q → ∞ limit of the quasitriangular q-WZW model is underlied by a particular anomalous Heisenberg double of the loop group and we evaluate the corresponding anomalous q → ∞ current algebra brackets. We also show that the q → ∞ WZW model admits a chiral decomposition compatible with the Poisson-Lie symmetry and the chiral theory can be dualized in a remarkable way. Finally, we evaluate the exchange (braiding) relations for the q → ∞ chiral WZW fields in both the original and the dual pictures.