Chiral extensions of the WZNW phase space, Poisson-Lie symmetries and groupoids

The chiral WZNW symplectic form Ohm(chir)(rho), is inverted in the general case, Thereby a precise relationship between the arbitrary monodromy depend ent 2-form appearing in Omega(chir)(rho) and the exchange r-matrix that gov erns the Poisson brackets of the group valued chiral fields is established, The exchange r-matrices are shown to satisfy a new dynamical generalizatio n of the classical modified Yang-Baxter (YB) equation and Poisson-Lie (PL) groupoids are constructed that encode this equation analogously as PL group s encode the classical YB equation. For an arbitrary simple Lie group G, ex change r-matrices are found that are in one-to-one correspondence with the possible PL structures on G and admit them as PL symmetries, (C) 2000 Elsev ier Science B.V. All rights reserved.