### Abstract

An invariant approach to quantum integrable models in wholly discrete 2 + 1-dimensional space–time is considered. An auxiliary linear problem on two dimensional lattices generalizing quantum chains is formulated. A method of constructing a complete set of integrals of motion is given. For the two dimensional lattices, we formulate and solve a zero-curvature representation allowing us to construct integrable evolutionary mappings. We place special emphasis on finite-dimensional representations of the algebra of observables,

which exist if the Weyl algebra parameter is at a rational point of a unit circle (so-called “root of unity”). For this case, we derive a universal functional eigenvalue equation for integrals of motion. A groupoid of isospectral deformations is constructed for the finite-dimensional representations of the algebra of observables. Because the systems under consideration are finite-dimensional at the root of unity, the integrable systems can be treated as models of statistical mechanics on three-dimensional lattices. We formulate a method of constructing eigenstates of the models under consideration; the method is based on isospectral deformations (the method of quantum

separation of variables for 2 + 1-dimensional models)

which exist if the Weyl algebra parameter is at a rational point of a unit circle (so-called “root of unity”). For this case, we derive a universal functional eigenvalue equation for integrals of motion. A groupoid of isospectral deformations is constructed for the finite-dimensional representations of the algebra of observables. Because the systems under consideration are finite-dimensional at the root of unity, the integrable systems can be treated as models of statistical mechanics on three-dimensional lattices. We formulate a method of constructing eigenstates of the models under consideration; the method is based on isospectral deformations (the method of quantum

separation of variables for 2 + 1-dimensional models)

Original language | English |
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Pages (from-to) | 1-31 |

Number of pages | 61 |

Journal | Physics of Particles and Nuclei |

Volume | 35 |

Issue number | 5 |

Publication status | Published - 2004 |

Externally published | Yes |