Chiral extensions of the WZNW phase space, Poisson–Lie symmetries and groupoids Original Research Article

The chiral WZNW symplectic form Ωρchir is inverted in the general case. Thereby a precise relationship between the arbitrary monodromy dependent 2-form appearing in Ωρchir and the exchange r-matrix that governs the Poisson brackets of the group valued chiral fields is established. The exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang–Baxter (YB) equation and Poisson–Lie (PL) groupoids are constructed that encode this equation analogously as PL groups encode the classical YB equation. For an arbitrary simple Lie group G, exchange r-matrices are found that are in one-to-one correspondence with the possible PL structures on G and admit them as PL symmetries.