Abstract
The tetrahedron equation arises as a generalization of the famous Yang-Baxter equation to the2+1-dimensional quantum field theory and three-dimensional statistical mechanics. Not much is known about its solutions. In the present paper, a systematic method of constructing nontrivial solutions to the tetrahedron equation with spin-like variables on the links is described. The essence of this method is the use of the so-called tetrahedral Zamolodchikov algebras. Bibliography:12 titles.
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Literature Cited
A. B. Zamolodchikov, “Tetrahedron equations and the relativisticS-matrix of straight-strings in 2+1dimensions,”Commun. Math. Phys.,79, 489–505 (1981).
R. J. Baxter, “On Zamolodchikov's solution of the tetrahedron equations,”Commun. Math. Phys.,88, 158–205 (1983).
I. G. Korepanov, “Tetrahedral Zamolodchikov algebra and the two-layer flat model in statistical mechanics,”Modern Phys. Lett. B,3, No. 3, 201–206 (1989).
I. M. Krichever, “Baxter equations and algebraic geometry,”Funkst. Anal. Prilozhen.,15, No. 2, 435–454 (1981).
B. U. Felderhof, “Diagonalization of the transfer matrix of the fermion model,”Physica,66, No. 2, 279–298 (1973).
V. V. Bazhanov and A. B. Zamolodchikov, “Tetrahedron equations and the relativisticS-matrix of straight-strings in 2+1-dimensions,”Commun. Math. Phys.,79, 489–505 (1981).
A. N. Tyurin, “Classification of the vector bundles over algebraic curves,”Izv. AN SSSR, Ser. Mat.,29, 658–680 (1965).
I. G. Korepanov, “New solutions to the tetrahedron equation,”Dep. in the VINITI Physica, Chelyabinsk,89, No. 1751 (1989).
I. G. Korepanov, “Applications of the algebro-geometrical constructions to the triangle and tetrahedron equations,”Zap. Nauchn. Semin. LOMI (1989).
V. V. Bazhanov and R. J. Baxter, “New solvable lattice models in three dimensions,” to appearJ. Stat. Phys. (1992).
V. V. Bazhanov, and R. J. Baxter, “Star-triangle relation for a three dimensional model,” to appearJ. Stat. Phys. (1992).
R. M. Kashaev, V. V. Mangazeev, Yu. G. Stroganov, V. V. Bazhanov, and R. J. Baxter, “Spatial symmetry, local integrability and tetrahedron equations in the Baxter-Bazhanov model,” IHEP Preprint 92-63, Protvino.
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Published inZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 137–149.
Translated by I. G. Korepanov.
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Korepanov, I.G. The tetrahedron equation and algebraic geometry. J Math Sci 83, 85–92 (1997). https://doi.org/10.1007/BF02398463
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DOI: https://doi.org/10.1007/BF02398463