Abstract. The quantum discrete Liouville model in the strongly coupled regime, 1 < c < 25, is formulated as a well defined quantum mechanical problem with unitary evolution operator. The theory is selfdual: there are two exponential fields related by Hermitean conjugation, satisfying two discrete quantum Liouville equations, and living in mutually commuting subalgebras of the quantum algebra of observables.

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Abstract.The hamiltonian formalism is developed for the Sine-Gordon model on the spacetime light-like lattice, first introduced by Hirota. The evolution operator is explicitely constructed in the quantum variant of the model, the integrability of corresponding classical finite-dimensional system is established.

Abstract. A certain special function of the generalized hypergeometric variety is shown to fulfill a host of useful noncommutative identities.

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Just like decent classical difference-difference systems define symplectic maps on suitable phase spaces, their counterparts with properly ordered noncommutative entries come as Heisenberg equations of motion for corresponding quantum discrete-discrete models. We observe how this idea applies to a difference-difference counterpart of the Liouville equation. We produce explicit forms of of its evolution operator for the two natural space-time coordinate systems. We discover that discrete-discrete models inherit crucial features of their continuoustime parents like locality and integrability while the new-found algebraic transparency promises a useful progress in some branches of Quantum Inverse Scattering Method.

In the present paper, we prove the Yang-Baxter relations for a specific R-matrix dependent on the spectral parameter. For the simpliest Lax operator connected with the trigonometric case of the main commutation relation, this R-matrix plays the role of the fundamental R-matrix in the sense of [1]. Now we describe this Lax operator.Let P and Q be the Heisenberg pair of self-adjoint operators with the commutation relationWe regard the real number 7 as a dimensionless coupling constant and put the Planck constant h equal to 1. We construct the Weyl pair of unitary operatorssatisfying the commutation relationThe quantum space 7/, where P and Q are represented irreducibly, can be realized as L2(IR) with r such that Or = ~r pc(x) _ ~ d i ~ r (4)(coordinate representation). In the sequel, a specific realization of P and Q is not significant. The Lax operator acts in the-tensor product of the quantum space 7/and the auxiliary space 12 = C 2 and can be given by a 2 • 2-matrix whose entries are operators on 7/, U xV)Here, z is a complex number called the spectral parameter. The indices f and a stand for the quantum and auxiliary spaces where Lf, a(x ) acts. In this notation, the main commutation relation (cf.[2]) can conveniently be written in the following form:which is regarded as a relation in 7"l @ 1)1 | ~22. Here, Ral,a2(x) is an operator in the tensor product 1) | 1) = C 4 which can be given by the 4

Modified universal R-matrices, associated with the central extension (through the Drinfeld's double construction) of the quantum groups Uq(sl N ), are realized through an infinite dimensional spectral parameter dependent solution for the tetrahedron equation, provided a certain identity on q-exponentials holds true.Date: December 1998.