Tetrahedron Equation Research Articles

Tetrahedron and 3D Reflection Equation from PBW Bases of the Nilpotent Subalgebra of Quantum Superalgebras

In this paper, we study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with three-dimensional (3D) integrability. We obtain new solutions to the Zamolodchikov tetrahedron equation via type A and the 3D reflection equation via type B, where the latter equation was proposed by Isaev and Kulish as a 3D analog of the reflection equation of Cherednik. As a by-product of our approach, the Bazhanov–Sergeev solution to the Zamolodchikov tetrahedron equation is characterized as the transition matrix for a particular case of type A, which clarifies an algebraic origin of it. Our work is inspired by the recent developments connecting transition matrices for quantum non-super algebras with intertwiners of irreducible representations of quantum coordinate rings. We also discuss the crystal limit of transition matrices, which gives a super analog of transition maps of Lusztig’s parametrizations of the canonical basis.

Solution of tetrahedron equation and cluster algebras

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.

Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence

In this note, we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.

Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation

We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle ($Y-\Delta$) transformation at the critical point $n=2$. We deduce the theorem of Matiyasevich and its inverse from the Biggs formula, and we apply this relation to construct the recursion on the parameter $n$. We provide two different proofs of the Zamolodchikov tetrahedron equation satisfied by the star-triangle transformation in the case of $n=2$ multivariate Tutte polynomial, we extend the latter to the case of valency 2 points and show that the Biggs formula and the star-triangle transformation commute.

Higher Level q-Oscillator Representations for $$U_q(C_n^{(1)}),U_q(C^{(2)}(n+1))$$ and $$U_q(B^{(1)}(0,n))$$

We introduce higher level $q$-oscillator representations for the quantum affine (super)algebras of type $C_n^{(1)},C^{(2)}(n+1)$ and $B^{(1)}(0,n)$. These representations are constructed by applying the fusion procedure to the level one $q$-oscillator representations which were obtained through the studies of the tetrahedron equation. We prove that these higher level $q$-oscillator representations are irreducible. For type $C_n^{(1)}$ and $C^{(2)}(n+1)$, we compute their characters explicitly in terms of Schur polynomials.

Tetrahedron equation: algebra, topology, and integrability

Уравнение тетраэдров Замолодчикова наследует почти все богатство структур и сюжетов, в которых фигурирует уравнение Янга-Бакстера. Вместе с тем этот переход символизирует рост порядка задачи, шаг от уравнения Янга-Бакстера к локальному уравнению Янга-Бакстера, от алгебры Ли к $2$-Ли алгебре, от обычных узлов в $\mathbb{R}^3$ к $2$-узлам в $\mathbb{R}^4$. Мы проследим за этими переходами в нескольких примерах, а также поговорим о проявлении уравнения тетраэдров в давно стоящем вопросе интегрируемости трехмерной модели Изинга и связанной с ней модели теории нейронных сетей - модели Хопфилда на двумерной решетке. Библиография: 82 названия.

Nonlinear Schrödinger type tetrahedron maps

This paper is concerned with the construction of new solutions in terms of birational maps to the functional tetrahedron equation and parametric tetrahedron equation. We present a method for constructing solutions to the parametric tetrahedron equation via Darboux transformations. In particular, we study matrix refactorisation problems for Darboux transformations associated with the nonlinear Schrödinger (NLS) and the derivative nonlinear Schrödinger (DNLS) equation, and we construct novel nine-dimensional tetrahedron maps. We show that the latter can be restricted to six-dimensional parametric tetrahedron maps on invariant leaves. Finally, we construct parametric tetrahedron maps employing degenerated Darboux transformations of NLS and DNLS type.

Non-commutative birational maps satisfying Zamolodchikov equation, and Desargues lattices

We present new solutions of the functional Zamolodchikov tetrahedron equation in terms of birational maps in totally non-commutative variables. All the maps originate from Desargues lattices, which provide geometric realization of solutions to the non-Abelian Hirota–Miwa system. The first map is derived using the original Hirota’s gauge for the corresponding linear problem, and the second one is derived from its affine (non-homogeneous) description. We also provide an interpretation of the maps within the local Yang–Baxter equation approach. We exploit the decomposition of the second map into two simpler maps, which, as we show, satisfy the pentagonal condition. We also provide geometric meaning of the matching ten-term condition between the pentagonal maps. The generic description of Desargues lattices in homogeneous coordinates allows us to define another solution of the Zamolodchikov equation, but with a functional parameter that should be adjusted in a particular way. Its ultra-local reduction produces a birational quantum map (with two central parameters) with the Zamolodchikov property, which preserves Weyl commutation relations. In the classical limit, our construction gives the corresponding Poisson map, satisfying the Zamolodchikov condition.

Tetrahedron maps and symmetries of three dimensional integrable discrete equations

A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on Z3 is investigated. Our approach is a generalization of a method developed in the context of Yang–Baxter maps, based on the invariants of symmetry groups of the lattice equations. The method is demonstrated by a case–by–case analysis of the octahedron type lattice equations classified recently, leading to some new examples of tetrahedron maps and integrable coupled lattice equations.

Towards integrable structure in 3d Ising model

In this paper, we proceed to the study of tetrahedral symmetry in the 3D Ising model, which is considered as the first forerunner of integrability. The weight matrix of the model on a regular cubic lattice satisfying the twisted tetrahedron equation (TTE) is constructed. The latter is a modification of the Zamolodchikov tetrahedron equation, which appeared in integrable 3D statistical models. The method is based on the theory of the n-simplicial complex and the original recursion procedure on the space of n-simplex solutions. This recursion deserves its own investigation. Surprisingly, the weight matrix has some properties inherent for the hypercube combinatorics and coding theory.

Stochasticization of Solutions to the Yang–Baxter Equation

In this paper we introduce a procedure that, given a solution to the Yang-Baxter equation as input, produces a stochastic (or Markovian) solution to (a possibly dynamical version of) the Yang-Baxter equation. We then apply this "stochasticization procedure" to obtain three new, stochastic solutions to several different forms of the Yang-Baxter equation. The first is a stochastic, elliptic solution to the dynamical Yang-Baxter equation; the second is a stochastic, higher rank solution to the dynamical Yang-Baxter equation; and the third is a stochastic solution to a dynamical variant of the tetrahedron equation.

Matrix KP: tropical limit and Yang\u2013Baxter maps

We study soliton solutions of matrix Kadomtsev–Petviashvili (KP) equations in a tropical limit, in which their support at fixed time is a planar graph and polarizations are attached to its constituting lines. There is a subclass of “pure line soliton solutions” for which we find that, in this limit, the distribution of polarizations is fully determined by a Yang–Baxter map. For a vector KP equation, this map is given by an R-matrix, whereas it is a nonlinear map in the case of a more general matrix KP equation. We also consider the corresponding Korteweg–deVries reduction. Furthermore, exploiting the fine structure of soliton interactions in the tropical limit, we obtain an apparently new solution of the tetrahedron (or Zamolodchikov) equation. Moreover, a solution of the functional tetrahedron equation arises from the parameter dependence of the vector KP R-matrix.

Tetrahedron Equation and Quantum R Matrices for q-Oscillator Representations Mixing Particles and Holes

We construct $2^n+1$ solutions to the Yang-Baxter equation associated with the quantum affine algebras $U_q\big(A^{(1)}_{n-1}\big)$, $U_q\big(A^{(2)}_{2n}\big)$, $U_q\big(C^{(1)}_n\big)$ and $U_q\big(D^{(2)}_{n+1}\big)$. They act on the Fock spaces of arbitrary mixture of particles and holes in general. Our method is based on new reductions of the tetrahedron equation and an embedding of the quantum affine algebras into $n$ copies of the $q$-oscillator algebra which admits an automorphism interchanging particles and holes.

Totally positive matrices and dilogarithm identities

We show that two involutions on the variety Nn+ of upper triangular totally positive matrices are related, on the one hand, to the tetrahedron equation and, on the other hand, to the action of the symmetric group S3 on some subvariety of Nn+ and on the set of certain functions on Nn+. Using these involutions, we obtain a family of dilogarithm identities involving minors of totally positive matrices. These identities admit a form manifestly invariant under the action of the symmetric group S3.

Integrable Structure of Multispecies Zero Range Process

We present a brief review on integrability of multispecies zero range process in one dimension introduced recently. The topics range over stochastic $R$ matrices of quantum affine algebra $U_q (A^{(1)}_n)$, matrix product construction of stationary states for periodic systems, $q$-boson representation of Zamolodchikov-Faddeev algebra, etc. We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of $R$ matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter.

Cohomology of the rational “electric” tetrahedron relation

A tetrahedral cochain complex is generalized to the case of the functional “electric” solution of the tetrahedron equation, which is expressed in terms of rational functions. The nontrivial part of the 3-cohomology group for this solution is calculated. It turns out to be the free Abelian group with one generator; the generator is explicitly specified.

Combinatorial Yang–Baxter maps arising from the tetrahedron equation

We survey the matrix product solutions of the Yang-Baxter equation obtained recently from the tetrahedron equation. They form a family of quantum $R$ matrices of generalized quantum groups interpolating the symmetric tensor representations of $U_q(A^{(1)}_{n-1})$ and the anti-symmetric tensor representations of $U_{-q^{-1}}(A^{(1)}_{n-1})$. We show that at $q=0$ they all reduce to the Yang-Baxter maps called combinatorial $R$, and describe the latter by explicit algorithm.

Cohomologies ofn-simplex relations

AbstractA theory of (co)homologies related to set-theoreticn-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.

Multispecies TASEP and the tetrahedron equation

We introduce a family of layer to layer transfer matrices in a three-dimensional (3D) lattice model which can be viewed as partition functions of the q-oscillator valued six-vertex model on an m × n square lattice. By invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions. At q = 0 and m = n, they ultimately yield a new proof of the steady state formula for the n-species totally asymmetric simple exclusion process obtained recently by the authors, revealing the 3D integrability in the matrix product construction.

Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation

We consider a three-dimensional lattice model associated with the intertwiner of the quantized coordinate ring $A\_q(sl\_3)$, and introduce a family of layer to layer transfer matrices on $m\times n$ square lattice. By using the tetrahedron equation we derive their commutativity and bilinear relations mixing various boundary conditions. At $q=0$ and $m=n$, they lead to a new proof of the steady state probability of the $n$-species totally asymmetric zero range process obtained recently by the authors, revealing the three-dimensional integrability in the matrix product construction.