Abstract
A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator [Formula: see text] in the space of a triple Weyl algebra. [Formula: see text] is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for [Formula: see text] follows without further calculation. If the Weyl parameter is taken to be a root of unity, [Formula: see text] decomposes into a matrix conjugation operator R1,2,3 and a c-number functional mapping [Formula: see text]. The operator R1,2,3 satisfies a modified tetrahedron equation (MTE) in which the "rapidities" are solutions of a classical integrable Hirota-type equations. R1,2,3 can be represented in terms of the Bazhanov-Baxter Fermat curve cyclic functions, or alternatively in terms of Gauss functions. The paper summarizes several recent publications on the subject.
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