Quantum integrable systems, non-skew-symmetric r -matrices and algebraic Bethe ansatzThe quantum dynamical Yang-Baxter ͑or Gervais-Neveu-Felder͒ equation defines an R-matrix R (p), where p stands for a set of mutually commuting variables. A family of SL(n)-type solutions of this equation provides a new realization of the Hecke algebra. We define quantum antisymmetrizers, introduce the notion of quantum determinant and compute the inverse quantum matrix for matrix algebras of the type R (p)a 1 a 2 ϭa 1 a 2 R . It is pointed out that such a quantum matrix algebra arises in the operator realization of the chiral zero modes of the WZNW model.

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Decoupling the chiral dynamics in the canonical approach to the WZNW model requires an extended phase space that includes left and right monodromy variables. Earlier work on the subject, which traced back the quantum qroup symmetry of the model to the Lie-Poisson symmetry of the chiral symplectic form, left some open questions: - How to reconcile the monodromy invariance of the local 2D group valued field (i.e., equality of the left and right monodromies) with the fact that the latter obey different exchange relations? - What is the status of the quantum group symmetry in the 2D theory in which the chiral fields commute? - Is there a consistent operator formalism in the chiral and in the extended 2D theory in the continuum limit? We propose a constructive affirmative answer to these questions for G=SU(2) by presenting the chiral quantum fields as sums of chiral vertex operators and q-Bose creation and annihilation operators.Comment: 18 pages, LATE

MSC codes:17B37 Quantum groups (quantum enveloping algebras) and related deformations 53D05 Symplectic manifolds 81S10 Geometry and quantization, symplectic methods Keywords AbstractWe define the chiral zero modes' phase space of the G = SU (n) Wess-Zumino-Novikov-Witten (WZNW) model as an (n − 1)(n + 2)-dimensional manifold M q equipped with a symplectic form Ω q involving a Wess-Zumino (WZ) term ρ which depends on the monodromy M and is implicitly defined (on an open dense neighbourhood of the group unit) byThis classical system exhibits a Poisson-Lie symmetry that evolves upon quantization into an U q (sℓ n ) symmetry for q a primitive even root of 1 . For each (non-degenerate, constant) solution of the classical Yang-Baxter equation we write down explicitly a ρ(M ) satisfying Eq.( * ) and invert the form Ω q , thus computing the Poisson bivector of the system. The resulting Poisson brackets (PB) appear as the classical counterpart of the exchange relations of the quantum matrix algebra studied previously in [32]. We argue that it is advantageous to equate the determinant D of the zero modes' matrix (a j α ) to a pseudoinvariant under permutations q-polynomial in the SU (n) weights, rather than to adopt the familiar convention D = 1 . A finite dimensional "Fock space" operator realization of the factor algebra M q /I h , where I h is an appropriate ideal in M q for q h = −1 , is briefly discussed.