### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 031, 14 pages      arXiv:1505.06579      https://doi.org/10.3842/SIGMA.2017.031
Contribution to the Special Issue on Recent Advances in Quantum Integrable Systems

### Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of $2d$ Quantum Integrable Systems

Dmitry V. Talalaev
Geometry and Topology Department, Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia

Received January 17, 2017, in final form May 13, 2017; Published online May 22, 2017

Abstract
The main aim of this work is to develop a method of constructing higher Hamiltonians of quantum integrable systems associated with the solution of the Zamolodchikov tetrahedral equation. As opposed to the result of V.V. Bazhanov and S.M. Sergeev the approach presented here is effective for generic solutions of the tetrahedral equation without spectral parameter. In a sense, this result is a two-dimensional generalization of the method by J.-M. Maillet. The work is a part of the project relating the tetrahedral equation with the quasi-invariants of 2-knots.

Key words: Zamolodchikov tetrahedral equation; quantum integrable systems; star-triangle transformation.

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