### Abstract

A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator R_{1,2,3} in the space of a triple Weyl algebra. R_{1,2,3} is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for R_{1,2,3} follows without further calculation. If the Weyl parameter is taken to be a root of unity, R_{1,2,3} decomposes into a matrix conjugation operator R _{1,2,3} and a c-number functional mapping R_{1,2,3} ^{(f)}. The operator R_{1,2,3} satisfies a modified tetrahedron equation (MTE) in which the "rapidities" are solutions of a classical integrable Hirota-type equations. R_{1,2,3} 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 | 19 |

Issue number | SUPPL. 2 |

Publication status | Published - May 2004 |

Externally published | Yes |

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### Cite this

*International Journal of Modern Physics A*,

*19*(SUPPL. 2), 179-204.

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*International Journal of Modern Physics A*, vol. 19, no. SUPPL. 2, pp. 179-204.

**The modified tetrahedron equation and its solutions.** / Von Gehlen, Gunter; Pakuliak, Stanislav; Sergeev, S.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The modified tetrahedron equation and its solutions

AU - Von Gehlen, Gunter

AU - Pakuliak, Stanislav

AU - Sergeev, S.

PY - 2004/5

Y1 - 2004/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=4944256781&partnerID=8YFLogxK

M3 - Article

VL - 19

SP - 179

EP - 204

JO - International Journal of Modern Physics A

JF - International Journal of Modern Physics A

SN - 0217-751X

IS - SUPPL. 2

ER -