This is a review of recent work on the chiral extensions of the WZNW phase
space describing both the extensions based on fields with generic monodromy as
well as those using Bloch waves with diagonal monodromy. The symplectic form on
the extended phase space is inverted in both cases and the chiral WZNW fields
are found to satisfy quadratic Poisson bracket relations characterized by
monodromy dependent exchange r-matrices. Explicit expressions for the exchange
r-matrices in terms of the arbitrary monodromy dependent 2-form appearing in
the chiral WZNW symplectic form are given. The exchange r-matrices in the
general case 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 exhibited that are in one-to-one correspondence with the
possible PL structures on G and admit them as PL symmetries.Comment: Based on a lecture by L.F. at the Seminaire de Mathematiques
Superieures, Montreal, 1999; LaTeX, 21 page
