Hecke algebraic properties of dynamical R -matrices. Application to related quantum matrix algebras

Abstract

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.

Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral su(2) WZNW Model

“…Thus the quantum matrix a intertwines between the U q covariant monodromy M and the diagonal one M p ( [22,16]):…”

“…Introduce, following Lusztig (see [27] and references therein), the "divided powers" 22) where the q-binomial coefficients 3 a b defined, for integer a and non-negative integer b , as…”

Quantum matrix algebra for theSU(n) WZNW model

“…Its accurate formulation in the continuum limit has only been attempted in the case of G = SU(2) (see [35,36,26]). The identification (in [45]) of the zero mode (U q )vertex operators a i α (the "U qoscillators" of the SU(2) case [35]) with the generators of a quantum matrix algebra defined by a pair of (dynamical) R-matrices allows to extend this approach to the case of G = SU(n) .…”

“…. , n (in the notation of [45] to be recapitulated in Sec-tion 1 below). The operator valued coefficients a i α of the resulting expansion intertwine finite dimensional irreducible representations (IR) of U q ≡ U q (sl n ) that are labeled by the same weights.…”

“…A new point here is the accurate treatment of the path dependence of the exchange relations in both the x and the z = e ix pictures (Proposition 1.3). In Section 2 we carry out the factorization of the chiral field u(x) into CVO and U q vertex operators and review relevant results of [45] computing, in particular, the determinant of the quantum matrix a as a function of the U q (sl n ) weights. The discussion of the interrelation between the braiding properties of four-point blocks and the exchange relations among zero modes presented in Section 2.2 is new; so are some technical results like Proposition 2.3 used in the sequel.…”