scholarly journals Multispecies TASEP and the tetrahedron equation

2016 ◽  
Vol 49 (11) ◽  
pp. 114001 ◽  
Author(s):  
Atsuo Kuniba ◽  
Shouya Maruyama ◽  
Masato Okado
Keyword(s):  
Tetrahedron Equation

scholarly journals Solution of tetrahedron equation and cluster algebras

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
P. Gavrylenko ◽  
M. Semenyakin ◽  
Y. Zenkevich
Keyword(s):  
Integrable System ◽  
Cluster Algebra ◽  
Newton Polygon ◽  
Cluster Algebras ◽  
Weyl Groups ◽  
Newton Polygons ◽  
Tetrahedron Equation ◽  
Affine Weyl Groups ◽  
New Formalism ◽  
Bruhat Cell

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

Solutions to the Vector Tetrahedron Equation

1965 ◽  
Vol 87 (2) ◽  
pp. 228-234 ◽  
Author(s):  
Milton A. Chace
Keyword(s):  
Closed Form ◽  
Digital Computer ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Spherical Coordinates ◽  
Tetrahedron Equation ◽  
Position Determination ◽  
Closed Form Solutions

A set of nine closed-form solutions are presented to the single, three-dimensional vector tetrahedron equation, sum of vectors equals zero. The set represents all possible combinations of unknown spherical coordinates among the vectors, assuming the coordinates are functionally independent. Optimum use is made of symmetry. The solutions are interpretable and can be evaluated reliably by digital computer in milliseconds. They have been successfully applied to position determination of many three-dimensional mechanisms.

scholarly journals Cohomologies of n-simplex relations

2016 ◽  
Vol 161 (2) ◽  
pp. 203-222 ◽  
Author(s):  
IGOR G. KOREPANOV ◽  
GEORGY I. SHARYGIN ◽  
DMITRY V. TALALAEV
Keyword(s):  
Tetrahedron Equation

AbstractA theory of (co)homologies related to set-theoretic n-simplex relations is constructed in analogy with the known quandle and Yang–Baxter (co)homologies, with emphasis made on the tetrahedron case. In particular, this permits us to generalise Hietarinta's idea of “permutation-type” solutions to the quantum (or “tensor”) n-simplex equations. Explicit examples of solutions to the tetrahedron equation involving nontrivial cocycles are presented.

scholarly journals Tetrahedron Equation, Weyl Group, and Quantum Dilogarithm

2014 ◽  
Vol 105 (1) ◽  
pp. 45-61
Author(s):  
Andrei Bytsko ◽  
Alexander Volkov
Keyword(s):  
Weyl Group ◽  
Tetrahedron Equation ◽  
Quantum Dilogarithm

scholarly journals Tetrahedron Reflection Equations

1997 ◽  
Vol 12 (06) ◽  
pp. 427-437 ◽  
Author(s):  
A. P. Isaev ◽  
P. P. Kulish
Keyword(s):  
Half Plane ◽  
Consistency Condition ◽  
Boundary Plane ◽  
Tetrahedron Equation ◽  
Reflection Equation ◽  
Reflection Equations ◽  
Geometric Picture

Reflection equation for the scattering of lines moving in half-plane is obtained. The corresponding geometric picture is related with configurations of half-planes touching the boundary plane in (2+1) dimensions. This equation can be obtained as an addition to the tetrahedron equation consistency condition for a modified Zamolodchikov algebra.

Two-Dimensional R-Matrices — Descendants of Three-Dimensional R-Matrices

1997 ◽  
Vol 12 (19) ◽  
pp. 1393-1410 ◽  
Author(s):  
S. M. Sergeev
Keyword(s):  
Three Dimensional ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Affine Algebra ◽  
Tetrahedron Equation ◽  
Drinfeld Double ◽  
R Matrix ◽  
Operator Solution ◽  
Dimensional Models ◽  
Three Dimensional Models

Finite layers of three-dimensional models can be regarded as two-dimensional with complicated multi-stated weights. The tetrahedron equation in 3D provides the Yang–Baxter equation for this composite weights in 2D. Such solutions of the Yang–Baxter equation are constructed for the simplest operator solution of the tetrahedron equation. These R-matrices can be regarded as a special projection of universal R-matrix for some Drinfeld double [Formula: see text], associated with the affine algebra [Formula: see text]. Usual R-matrix for [Formula: see text] is another projection of [Formula: see text].

scholarly journals Explicit free parametrization of the modified tetrahedron equation

2003 ◽  
Vol 36 (4) ◽  
pp. 975-998 ◽  
Author(s):  
G von Gehlen ◽  
S Pakuliak ◽  
S Sergeev
Keyword(s):  
Tetrahedron Equation

Some solutions of the Zamolodchikov tetrahedron equation

1993 ◽  
Vol 26 (17) ◽  
pp. L887-L891 ◽  
Author(s):  
A Liguori ◽  
M Mintchev
Keyword(s):  
Tetrahedron Equation

scholarly journals MODIFIED TETRAHEDRON EQUATION AND RELATED 3D INTEGRABLE MODELS, II

1996 ◽  
Vol 11 (02) ◽  
pp. 313-327 ◽  
Author(s):  
H.E. BOOS
Keyword(s):  
General Model ◽  
Spin Model ◽  
Parametric Family ◽  
Layer Transfer ◽  
Transfer Matrices ◽  
Tetrahedron Equation ◽  
Cubic Lattice ◽  
Analytical Formulae ◽  
Integrable Spin ◽  
Free Parameters

This work is a continuation of Ref. 13, where the Boltzmann weights for the N-state integrable spin model on the cubic lattice has been obtained only numerically. In this paper we present the analytical formulae for this model in a particular case. Here the Boltzmann weights depend on six free parameters including the elliptic modulus. The obtained solution allows us to construct a two-parametric family of the commuting two-layer transfer matrices. The presented model is expected to be simpler for a further investigation in comparison with a more general model mentioned above.

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