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Conditions of commutativity of transfer matrices on a multidimensional lattice

V. V. Bazhanov, Yu. G. Stroganov


Abstract: It is shown that the transfer matrices for vertex models on a three-dimensional lattice commute if the statistical weights of the vertices satisfy a certain algebraic system of equations, which is called the “tetrahedron equation”. A multidimensional generalization of this proposition is also obtained, and it leads to "equations of $d$-simpliees".

Received: 29.04.1981

 English version: Theoretical and Mathematical Physics, 1982, 52:1, 685–691

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