PDF《HU-EP-19/02, ZMP-HH/19-2》

2 C quantised, in particular, the algebra of quantum T -generators is T1T2 = T2T1R12 and its consistency is guaranteed by the fact that the matrix R, being a quantisation of r, is triangular, R12R21 = ?, and obeys the quantum Yang-Baxter equation. The quantum L-operator is then introduced as L = T ?1 gT and it is an invariant under the action of F. In [19] the same formula for Ik as given above2 was derived by eliminating from the commuting operators Trgk = TrT Lk T ?1 an element T . To build up the hyperbolic RS model, one can start from the Heisenberg double associated to a Lie group G. As a manifold, the Heisenberg double is G*G and it has a well-de?ned Poisson structure being a deformation of the one on T ? G [24]. However, an attempt to repeat the same steps of the reduction procedure meets an obstacle: since the action of G on the Heisenberg double is Poisson, rather than hamiltonian, the Poisson bracket of two Frobenius invariants, {L1, L2}, is not closed, i.e. it is not expressed via L'

s alone. Moreover, for the same reason, the Poisson bracket {pi, pj} does not vanish on the Heisenberg double. On the other hand, a part of the non-abelian moment map generates second class constraints and to ?nd the Poisson structure on the reduced manifold one has to resort to the Dirac bracket construction.3 In this paper we work out the Dirac brackets for Frobenius invariants and show in detail how the cancellation of the non-invariant terms happens on the constraint surface. This leads to the canonical set of brackets for the degrees of freedom (pi, qi) on the reduced manifold, the physical phase space of the RS model. However, continuing along the same path as in the rational case [19] does not seem to yield {T, L} and {T, T } brackets. The variable T is not invariant with respect to the stability subgroup of the moment map and computation of such brackets requires ?xing a gauge, which makes the whole approach rather obscure. Moreover, the very simple and elegant bracket {L1, L2} emerging on the reduced phase space looks the same as in the rational case, with one exception: now the r-matrix r12 entering this bracket is not skew-symmetric, i.e. r12 = r21. We then ?nd a quantisation of r12: a simple quantum R-matrix R+ satisfying R+12R?21 = ?, where R?12 is another solution of the quantum Yang-Baxter equation. In the absence of the triangular property for R+12, assuming, for instance, the same algebra for T '

s as in the rational case - that is T1T2 = T2T1R+12 - would be inconsistent. Thus, at this point we simply conjecture that the integrals of the hyperbolic model have absolutely the same form as in the rational case, with the exception that the rational R-matrices are replaced by their hyperbolic analogues, which we explicitly ?nd. That this conjecture yields integrals of motion can then be veri?ed by tedious but direct computation and indeed holds true. Working out explicit expressions for these integrals for small numbers N of particles we ?nd the determinant formulae relating these integrals to the standard basis of Macdonald operators. The rest of the paper is devoted to the model whose formulation includes the spectral parameter. Neither for the rational nor for the hyperbolic case the spectral parameter is actually needed to demonstrate their Liouville integrability, but its introduction leads to interesting algebraic structures and clari?es the origin of the shifted Yang-Baxter equation [22] and its scale-violating solutions. The paper is organised as follows. In the next section we show how to obtain the hyperbolic RS model by the Poisson reduction of the Heisenberg double. This includes the derivation of the Poisson algebra of the Lax matrix via the Dirac bracket construction. We also introduce the spectral parameter and build up the theory based on spectral parameter-dependent (baxterised) r-matrices. We also describe a freedom in the de?nition of r-matrices that does not change the Poisson algebra of L'