### Abstract

A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator R1,2,3 in the space of a triple Weyl algebra. R1,2,3 is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for R1,2,3 follows without further calculation. If the Weyl parameter is taken to be a root of unity, R1,2,3 decomposes into a matrix conjugation operator R 1,2,3 and a c-number functional mapping R1,2,3(f). 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

Original language | English |
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Pages (from-to) | 179-204 |

Number of pages | 26 |

Journal | International Journal of Modern Physics A |

Volume | A19S2 |

Publication status | Published - 2004 |

Externally published | Yes |

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## Cite this

Von Gehlen, G., Pakuliak, S., & Sergeev, S. (2004). The modified tetrahedron equation amd its solutions.

*International Journal of Modern Physics A*,*A19S2*, 179-204.