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Integrability for multidimensional lattice models
Abstract
The generating principle for the algebraic construction of the hierarchy of the d-simplex equations generalizing the Yang-Baxter equation in any dimension d is given. Following this principle, we construct the generalization of the Lax equations for multidimensional integrable systems. We show that this contribution leads, as in d=2, to the existence of an infinite series of conserved charges for such models. In the d=3 case a reinterpretation of these results is given in terms of functionals of loops, leading to the notion of extended gauge symmetry and gauge fields associated to quantum groups.
... [2, 3, 11-14, 24, 28] and references therein). The Yang-Baxter and tetrahedron equations are members of the family of n-simplex equations [4,5,8,22,24,25], and they correspond to the cases of 2-simplex and 3-simplex, respectively. Presently, the relations of tetrahedron maps with integrable systems and with algebraic structures (including groups and rings) are a very active area of research (see, e.g. ...
... can be called the Maillet-Nijhoff equation [24,25] in Korepanov's form, but it is usually called the local Yang-Baxter equation in the literature. A well-known method [6,12,26] for constructing tetrahedron maps by means of (4) is as follows. ...
... As discussed in remark 2.5, these maps can be linearised by a change of variables, and the corresponding linear tetrahedron maps can be found in the work of Hietarinta [8] in a very different context, but invariants and Lax representations were not known for them. Also, we constructed new tetrahedron maps (20), (21), (25) and (26) of the form G 3 → G 3 with Lax representations, where G is an arbitrary group. Furthermore, we obtained new tetrahedron maps (28). ...
We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and their matrix Lax representations defined by the local Yang--Baxter equation.
Sergeev [S.M. Sergeev 1998 Lett. Math. Phys. 45, 113--119] presented classification results on three-dimensional tetrahedron maps obtained from the local Yang--Baxter equation for a certain class of matrix-functions in the situation when the equation possesses a unique solution which determines a tetrahedron map. In this paper, using correspondences arising from the local Yang--Baxter equation for some simple $2\times 2$ matrix-functions, we show that there are (non-unique) solutions to the local Yang--Baxter equation which define tetrahedron maps that do not belong to the Sergeev list; this paves the way for a new, wider classification of tetrahedron maps. We present invariants for the derived tetrahedron maps and prove Liouville integrability for some of them.
Furthermore, using the approach of solving correspondences arising from the local Yang--Baxter equation, we obtain several new birational tetrahedron maps with Lax representations and invariants, including maps on arbitrary groups, a nine-dimensional map associated with a Darboux transformation for the derivative nonlinear Schr"odinger (NLS) equation, and a nine-dimensional generalisation of the three-dimensional Hirota map.
... It has appeared in several guises and studied from various point of view. See for example [3,12,18,19,21,23] and the references therein. A survey from a quantum group theoretical perspective is available in [14]. ...
... where L Z 124 , L Z 135 , L Z 236 are given by (19) ...
... where L Z 135 and L Z 236 are given by (19) with (r, s, t, w) = (r 2 , s 2 , t 2 , w 2 ) and (r 3 , s 3 , t 3 , w 3 ), respectively. In this case, R ∈ End(F + ⊗ F ⊗ F) and the sum (25) extends over a ∈ Z ≥0 and b, c ∈ Z. ...
We present a family of new solutions to the tetrahedron equation of the form RLLL=LLLR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$RLLL=LLLR$$\end{document}, where L operator may be regarded as a quantized six-vertex model whose Boltzmann weights are specific representations of the q-oscillator or q-Weyl algebras. When the three L’s are associated with the q-oscillator algebra, R coincides with the known intertwiner of the quantized coordinate ring Aq(sl3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_q(sl_3)$$\end{document}. On the other hand, L’s based on the q-Weyl algebra lead to new R’s whose elements are either factorized or expressed as a terminating q-hypergeometric type series.
... The tetrahedron equation is a higher-dimensional analogue of the famous quantum Yang-Baxter equation and has applications in many diverse branches of physics and mathematics, including statistical mechanics, quantum field theories, combinatorics, low-dimensional topology, and the theory of integrable systems (see, e.g., [2,3,11,12,13,19,23] and references therein). The Yang-Baxter and tetrahedron equations are members of the family of n-simplex equations [4,5,8,17,19,20], and they correspond to the cases of 2-simplex and 3-simplex, respectively. Presently, the relations of tetrahedron maps with integrable systems and with algebraic structures (including groups and rings) are a very active area of research (see, e.g., [1,6,9,10,13,17,23,25]). ...
... can be called the Maillet-Nijhoff equation [19,20] in Korepanov's form, but it is usually called the local Yang-Baxter equation in the literature. A well-known method [6,12,21] for constructing tetrahedron maps by means of (4) is as follows. ...
... If the group G is noncommutative, the maps (20), (21) are noninvolutive. ...
We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and their matrix Lax representations defined by the local Yang--Baxter equation. Sergeev \cite{Kashaev-Sergeev, Sergeev} presented classification results on three-dimensional tetrahedron maps obtained from the local Yang--Baxter equation for a certain class of matrix-functions in the situation when the equation possesses a unique solution which determines a tetrahedron map. In this paper, using correspondences arising from the local Yang--Baxter equation for some simple $2\times 2$ matrix-functions, we show that there are (non-unique) solutions to the local Yang--Baxter equation which define tetrahedron maps that do not belong to the Sergeev list; this paves the way for a new, wider classification of tetrahedron maps. We present invariants for the derived tetrahedron maps and prove Liouville integrability for some of them. Furthermore, using the approach of solving correspondences arising from the local Yang--Baxter equation, we obtain new birational tetrahedron maps with matrix Lax representations on arbitrary groups.
... They belong to the most fundamental equations in mathematical physics and have applications in many diverse branches of physics and mathematics, including statistical mechanics, quantum field theories, algebraic topology, and the theory of integrable systems. The Yang-Baxter and tetrahedron equations are members of the family of n-simplex equations [4,11,16,30,32,33], where they correspond to the cases of two-simplex and three-simplex, respectively. Some applications of the tetrahedron equation can be found in [5,6,11,12,15,21,22,25,32,33,38,40] and references therein. ...
... The Yang-Baxter and tetrahedron equations are members of the family of n-simplex equations [4,11,16,30,32,33], where they correspond to the cases of two-simplex and three-simplex, respectively. Some applications of the tetrahedron equation can be found in [5,6,11,12,15,21,22,25,32,33,38,40] and references therein. ...
... It is known that the local Yang-Baxter equation [32] can be viewed as a 'Lax equation' or 'Lax system' for the tetrahedron equation (see, e.g. [11], and references therein). ...
We present several algebraic and differential-geometric constructions of tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. In particular, we obtain a family of new (nonlinear) polynomial tetrahedron maps on the space of square matrices of arbitrary size, using a matrix refactorisation equation, which does not coincide with the standard local Yang–Baxter equation. Liouville integrability is established for some of these maps. Also, we show how to derive linear tetrahedron maps as linear approximations of nonlinear ones, using Lax representations and the differentials of nonlinear tetrahedron maps on manifolds. We apply this construction to two nonlinear maps: a tetrahedron map obtained in [10] in a study of soliton solutions of vector KP equations and a tetrahedron map obtained in [27] in a study of a matrix trifactorisation problem related to a Darboux matrix associated with a Lax operator for the NLS equation. We derive parametric families of new linear tetrahedron maps (with nonlinear dependence on parameters), which are linear approximations for these nonlinear ones. Furthermore, we present (nonlinear) matrix generalisations of a tetrahedron map from Sergeev’s classification [37]. These matrix generalisations can be regarded as tetrahedron maps in noncommutative variables. Besides, several tetrahedron maps on arbitrary groups are constructed.
... In particular, this observation makes it possible to use the Bethe ansatz to solve models of statistical physics, including answers to purely physical questions such as the description of the spontaneous magnetization effect. In § 4 we generalize the relationship between statistical models and lattice spin systems to the case of dimension 3. The presentation there mainly follows [15], which is in a certain sense a development of results due to Maillet and Nijhoff [16]. ...
... A partial analogue of this statement for dimension 3 is given by the result in [16]. ...
... The transfer matrix is expressed as the trace of the monodromy matrix (16), ...
... They belong to the most fundamental equations in mathematical physics and have applications in many diverse branches of physics and mathematics, including statistical mechanics, quantum field theories, algebraic topology, and the theory of integrable systems. The Yang-Baxter and tetrahedron equations are members of the family of n-simplex equations [3,10,15,26,28,29], where they correspond to the cases of 2-simplex and 3-simplex, respectively. Some applications of the tetrahedron equation can be found in [4,5,10,11,14,18,20,21,28,29,35,38] and references therein. ...
... The Yang-Baxter and tetrahedron equations are members of the family of n-simplex equations [3,10,15,26,28,29], where they correspond to the cases of 2-simplex and 3-simplex, respectively. Some applications of the tetrahedron equation can be found in [4,5,10,11,14,18,20,21,28,29,35,38] and references therein. This paper is devoted to tetrahedron maps and their relations with Yang-Baxter maps, which are set-theoretical solutions to the tetrahedron equation and the Yang-Baxter equation, respectively. ...
... It is known that the local Yang-Baxter equation [28] can be viewed as a "Lax equation" or "Lax system" for the tetrahedron equation (see, e.g., [10] and references therein). This allows one to introduce the notion of Lax representations for tetrahedron maps, see Remark 5.10 for details. ...
We present several algebraic and differential-geometric constructions of tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. In particular, we obtain a family of new (nonlinear) polynomial tetrahedron maps on the space of square matrices of arbitrary size, using a matrix refactorisation equation, which does not coincide with the standard local Yang--Baxter equation. Liouville integrability is established for some of these maps. Also, we show how to derive linear tetrahedron maps as linear approximations of nonlinear ones, using Lax representations and the differentials of nonlinear tetrahedron maps on manifolds. We apply this construction to two nonlinear maps: a tetrahedron map obtained in [arXiv:1708.05694] in a study of soliton solutions of vector KP equations and a tetrahedron map obtained in [arXiv:2005.13574] in a study of a matrix trifactorisation problem related to a Darboux matrix associated with a Lax operator for the NLS equation. We derive parametric families of new linear tetrahedron maps, which are linear approximations for these nonlinear ones. Another result is a (nonlinear) matrix generalisation of a tetrahedron map from Sergeev's classification [arXiv:solv-int/9709006]. This matrix generalisation can be regarded as a tetrahedron map in noncommutative variables. Furthermore, we present several tetrahedron maps on arbitrary groups.
... All classified lattice equations, even in non-commutative version, have appeared earlier in the literature, see [21]. The second was obtained by Sergeev in [29], see also [9,8], where a classification of tetrahedron maps was presented, based on the local Yang-Baxter equations [19,20], which serve as a zero-curvature condition of the corresponding maps. We will show that the invariants of the symmetry groups of transformations of the integrable lattice equations on Z 3 , become the variables to express the tetrahedron maps from the latter list. ...
... where , denotes the usual inner product between a vector and a dual vector. Combining (19) and (20) we conclude that the differential dI of a solution of (19) annihilates the vector v. Moreover, the latter is true for any 1-form ω which is a non zero scalar multiple of the differential dI, i.e. ...
... where , denotes the usual inner product between a vector and a dual vector. Combining (19) and (20) we conclude that the differential dI of a solution of (19) annihilates the vector v. Moreover, the latter is true for any 1-form ω which is a non zero scalar multiple of the differential dI, i.e. ...
A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on $\mathbb{Z}^3$ 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.
... All classified lattice equations, even in non-commutative version, have appeared earlier in the literature, see [21]. The second was obtained by Sergeev in [29], see also [9,8], where a classification of tetrahedron maps was presented, based on the local Yang-Baxter equations [19,20], which serve as a zero-curvature condition of the corresponding maps. We will show that the invariants of the symmetry groups of transformations of the integrable lattice equations on Z 3 , become the variables to express the tetrahedron maps from the latter list. ...
... where , denotes the usual inner product between a vector and a dual vector. Combining (19) and (20) we conclude that the differential dI of a solution of (19) annihilates the vector v. Moreover, the latter is true for any 1-form ω which is a non zero scalar multiple of the differential dI, i.e. ...
... where , denotes the usual inner product between a vector and a dual vector. Combining (19) and (20) we conclude that the differential dI of a solution of (19) annihilates the vector v. Moreover, the latter is true for any 1-form ω which is a non zero scalar multiple of the differential dI, i.e. ...
We present two lists of multi-component systems of integrable difference equations defined on the edges of a $\mathbb{Z}^2$ graph. The integrability of these systems is manifested by their Lax formulation which is a consequence of the multi-dimensional compatibility of these systems. Imposing constraints consistent with the systems of difference equations, we recover known integrable quad-equations including the discrete version of the Krichever-Novikov equation. The systems of difference equations allow us for a straightforward reformulation as Yang-Baxter maps. Certain two-component systems of equation defined on the vertices of a $\mathbb{Z}^2$ lattice, their non-potential form and integrable equations defined on 5-point stencils, are also obtained.
... The Yang-Baxter equation is a member of a family, called simplex equations [18] (also see, e.g., [65,67,69,39]). The N -simplex equation is an equation imposed on a mapR : V ⊗N → V ⊗N , respectivelyR : U N → U N for the set-theoretical version. The next-to-Yang-Baxter equation, the 3-simplex equation,R 123R145R246R356 =R 356R246R145R123 , is also called tetrahedron equation or Zamolodchikov equation. ...
... The last system is also known as the tetrahedral Zamolodchikov algebra (also see [58,16]). Analogously, there is a Lax system for the Yang-Baxter equation [88], consisting of 1-simplex equations, which is the Zamolodchikov-Faddeev algebra [61], and this structure extends to all simplex equations [67,65,66,69]. The underlying idea of relaxing a system of N -simplex equations in the above way, by introducing an objectR, such that consistency imposes the (N + 1)-simplex equation on it, is the "obstruction method" in [67,65,66,69,74,44,25]. ...
... Analogously, there is a Lax system for the Yang-Baxter equation [88], consisting of 1-simplex equations, which is the Zamolodchikov-Faddeev algebra [61], and this structure extends to all simplex equations [67,65,66,69]. The underlying idea of relaxing a system of N -simplex equations in the above way, by introducing an objectR, such that consistency imposes the (N + 1)-simplex equation on it, is the "obstruction method" in [67,65,66,69,74,44,25]. Also see [50,51,57,85,9] for a formulation in the setting of 2-categories. ...
It is shown that higher Bruhat orders admit a decomposition into a higher
Tamari order, the corresponding dual Tamari order, and a "mixed order". We
describe simplex equations (including the Yang-Baxter equation) as realizations
of higher Bruhat orders. Correspondingly, a family of "polygon equations"
realizes higher Tamari orders. They generalize the well-known pentagon
equation. The structure of simplex and polygon equations is visualized in terms
of deformations of maximal chains in posets forming 1-skeletons of polyhedra.
The decomposition of higher Bruhat orders induces a reduction of the N-simplex
equation to the (N+1)-gon equation, its dual, and a compatibility equation.
... One important relation between the tetrahedron equation and the Yang-Baxter equation is that the solutions to the local Yang-Baxter equation are possible solutions to the tetrahedron equation [27,17]. In fact, a map which is derived by substitution of a square matrix to the local Yang-Baxter equation may satisfy the tetrahedron equation [27]. ...
... One important relation between the tetrahedron equation and the Yang-Baxter equation is that the solutions to the local Yang-Baxter equation are possible solutions to the tetrahedron equation [27,17]. In fact, a map which is derived by substitution of a square matrix to the local Yang-Baxter equation may satisfy the tetrahedron equation [27]. In the commutative case, the proof that a map is a tetrahedron map is a matter of straightforward substitution to the tetrahedron equation. ...
It is known that the local Yang--Baxter equation is a generator of potential solutions to Zamolodchikov's tetrahedron equation. In this paper, we show under which additional conditions the solutions to the local Yang--Baxter equation are tetrahedron maps, namely solutions to the set-theoretical tetrahedron equation. This is exceptionally useful when one wants to prove that noncommutative maps satisfy the Zamolodchikov's tetrahedron equation. We construct new noncommutative maps and we prove that they possess the tetrahedron property. Moreover, by employing Darboux transformations with noncommutative variables, we derive noncommutative tetrahedron maps. In particular, we derive a noncommutative nonlinear Schr\"odinger type of tetrahedron map which can be restricted to a noncommutative version of Sergeev's map on invariant leaves. We prove that these maps are tetrahedron maps.
... They belong to the most fundamental equations in mathematical physics and have applications in many diverse branches of physics and mathematics, including statistical mechanics, quantum field theories, algebraic topology, and the theory of integrable systems. Some applications of the tetrahedron equation can be found in [4,6,13,15,19,25,26,30,40,43,44,49,50] and references therein. ...
... • The Yang-Baxter and tetrahedron equations are members of the family of n-simplex equations [5,13,21,40,43,44], where they correspond to the cases of two-simplex and three-simplex, respectively. We suggest to try to extend the results of this paper and of [8] to the case of n-simplex equations for n 4. Some results on linear set-theoretical solutions to n-simplex equations (which can be called linear n-simplex maps) are presented in [11,21]. ...
We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and Yang–Baxter maps, which are set-theoretical solutions to the quantum Yang–Baxter equation. In particular, we clarify the structure of the nonlinear algebraic relations which define linear (parametric) tetrahedron maps (with nonlinear dependence on parameters), and we present several transformations which allow one to obtain new such maps from known ones. Furthermore, we prove that the differential of a (nonlinear) tetrahedron map on a manifold is a tetrahedron map as well. Similar results on the differentials of Yang–Baxter and entwining Yang–Baxter maps are also presented. Using the obtained general results, we construct new examples of (parametric) Yang–Baxter and tetrahedron maps. The considered examples include maps associated with integrable systems and matrix groups. In particular, we obtain a parametric family of new linear tetrahedron maps, which are linear approximations for the nonlinear tetrahedron map constructed by Dimakis and Müller-Hoissen [9] in a study of soliton solutions of vector Kadomtsev–Petviashvili (KP) equations. Also, we present invariants for this nonlinear tetrahedron map.
... They belong to the most fundamental equations in mathematical physics and have applications in many diverse branches of physics and mathematics, including statistical mechanics, quantum field theories, algebraic topology, and the theory of integrable systems. Some applications of the tetrahedron equation can be found in [4,6,13,15,19,25,26,30,40,43,44,49,50] and references therein. ...
... • The Yang-Baxter and tetrahedron equations are members of the family of n-simplex equations [5,13,21,40,43,44], where they correspond to the cases of two-simplex and three-simplex, respectively. We suggest to try to extend the results of this paper and of [8] to the case of n-simplex equations for n 4. Some results on linear set-theoretical solutions to n-simplex equations (which can be called linear n-simplex maps) are presented in [11,21]. ...
Yang--Baxter maps (YB maps) are set-theoretical solutions to the quantum Yang--Baxter equation. For a set $X=\Omega\times V$, where $V$ is a vector space and $\Omega$ is regarded as a space of parameters, a linear parametric YB map is a YB map $Y\colon X\times X\to X\times X$ such that $Y$ is linear with respect to $V$ and one has $\pi Y=\pi$ for the projection $\pi\colon X\times X\to\Omega\times\Omega$. These conditions are equivalent to certain nonlinear algebraic relations for the components of $Y$. Such a map $Y$ may be nonlinear with respect to parameters from $\Omega$. We present general results on such maps, including clarification of the structure of the algebraic relations that define them and several transformations which allow one to obtain new such maps from known ones. Also, methods for constructing such maps are described. In particular, developing an idea from [Konstantinou-Rizos S and Mikhailov A V 2013 J. Phys. A: Math. Theor. 46 425201], we demonstrate how to obtain linear parametric YB maps from nonlinear Darboux transformations of some Lax operators using linear approximations of matrix refactorisation problems corresponding to Darboux matrices. New linear parametric YB maps with nonlinear dependence on parameters are presented.
... Namely, 3D models appear in the local Yang -Baxter equation (LYBE) approach. LYBE (i.e. a Yang -Baxter equation with different "spectral" parameters in the left and right hand sides) can be adapted to a discrete space -time evolution of the triangulated two dimensional oriented surface as a kind of zero curvature condition [15,16,17]. In few words, if a matrix L i,j (x), acting as usual in a tensor product of two finite dimensional spaces labelled by numbers i and j, with some fixed functional structure and depending on a set of parameters x, obeys the equation ...
... The algebraic equivalence usually called zero curvature, and LYBE as the zero curvature condition as well as functional evolution models was considered in [15,16,17]. Another formulation of the algebraic equivalence, different to the LYBE approach, is Korepanov's matrix model (see [8,31] and references therein). ...
A quantum evolution model in 2 + 1 discrete spacetime, connected with a 3D fundamental map , is investigated. Map is derived as a map providing a zero curvature of a 2D linear lattice system called `the current system'. In a special case of the local Weyl algebra for dynamical variables the map appears to be canonical and it corresponds to the known operator-valued -matrix. The current system is a type of the linear problem for the 2 + 1 evolution model. A generating function for the integrals of motion for the evolution is derived with the help of the current system. Thus, the complete integrability in 3D is proved directly.
... The specific q-dependent factor in (21), involving the Pochhammer symbol ...
... This equation is sometimes called the local Yang-Baxter equation[21]. Note, that it is not equivalent to the Yang-Baxter equation. Even thought it has the same matrix structure, the L-matrices in the LHS and RHS of (6) are different. ...
In this paper we construct a three-dimensional (3D) solvable lattice model
with non-negative Boltzmann weights. The spin variables in the model are
assigned to edges of the 3D cubic lattice and run over an infinite number of
discrete states. The Boltzmann weights satisfy the tetrahedron equation, which
is a 3D generalisation of the Yang-Baxter equation. The weights depend on a
free parameter 0<q<1 and three continuous field variables. The layer-to-layer
transfer matrices of the model form a two-parameter commutative family. This is
the first example of a solvable 3D lattice model with non-negative Boltzmann
weights.
... Many physical integrable systems, such as the classical Toda, Korteweig-de Vries and the Kadomtsev-Petviashvili hierarchies [1], as well as the XXX/XXZ/XYZ family of quantum spin chains [9], can be transformed in this way to the canonical integrable system on g˚for certain Lie bialgebras g. However, these are all one dimensional, and it is generally a difficult task to identify the notion of integrability for higher dimensional systems; see for example for some proposals [10,11]. ...
The theory of Poisson-Lie groups and Lie bialgebras plays a major role in the study of one dimensional integrable systems. Many families of integrable systems can be recovered from a Lax pair which is constructed from a Lie bialgebra associated to a Poisson-Lie group. Some categorified notions of Poisson Lie groups and Lie bialgebras has been proposed using $L_2$-algebras (arXiv:1109.1344 and arxiv:1202.0079), which gave rise to the notions of (strict) Lie 2-bialgebras and Poisson-Lie 2-groups . In this paper, we use these structures to generalize the construction of a Lax pair and introduce an appropriate notion of {categorified integrability}. Within this framework, we explicitly construct and analyze the 2-dimensional version of the XXX model, whose dynamics is governed by an underlying Lie 2-bialgebra. We emphasize that the 2-graded form of our categorified notion of integrability directly implies the 2-dimensional nature of the degrees-of-freedom in our theory.
... An n-simplex equation may be understood as describing integrable systems. 4-simplex equation appeared in the work [170], n-simplex equations have been studied for example in [166,171,172]. ...
This review is a collection of various methods and observations relevant to structures in three-dimensional systems similar to those responsible for integrability of two-dimensional systems. Particular focus is given to Nambu structures and loop variables naturally appearing in membrane dynamics. While reviewing each topic in more details we emphasize connections between them and speculate on possible relations to membrane integrability.
... Moreover, n-simplex maps can be generated by matrix refactorisation problems. In particular, a quite interesting fact is that the local (n − 1)-simplex equation is a generator of solutions to the n-simplex equation [23,25]. However, in the literature there are only examples of 2-and 3-simplex maps generated by the local 1-and 2-simplex equations, respectively. ...
The 4-simplex equation is a higher-dimensional analogue of Zamolodchikov's tetrahedron equation and the Yang--Baxter equation which are two of the most fundamental equations of mathematical physics. In this paper, we introduce a method for constructing 4-simplex maps, namely solutions to the set-theoretical 4-simplex equation, using Lax matrix refactorisation problems. Employing this method, we construct 4-simplex maps which at a certain limit give tetrahedron maps classified by Kashaev, Korepanov and Sergeev. Moreover, we construct a Kadomtsev--Petviashvili type of 4-simplex map. Finally, we introduce a method for constructing 4-simplex maps which can be restricted on level sets to parametric 4-simplex maps using Darboux transformations of integrable PDEs. We construct a nonlinear Schr\"odinger type parametric 4-simplex map which is the first parametric 4-simplex map in the literature.
... It has appeared in several guises and studied from various point of view. See for example [18,11,20,5,17,22] and the references therein. A survey from a quantum group theoretical perspective is available in [13]. ...
We present a family of new solutions to the tetrahedron equation of the form $RLLL=LLLR$, where $L$ operator may be regarded as a quantized six-vertex model whose Boltzmann weights are specific representations of the $q$-oscillator or $q$-Weyl algebras. When the three $L$'s are associated with the $q$-oscillator algebra, $R$ coincides with the known intertwiner of the quantized coordinate ring $A_q(sl_3)$. On the other hand, $L$'s based on the $q$-Weyl algebra lead to new $R$'s whose elements are either factorized or expressed as a terminating $q$-hypergeometric type series.
... is the well known star-triangle transformation for resistances in electric networks. The identity (5.3.20) is related to the local Yang-Baxter equation [231] and is also rewritten in the operator form [230] W (q | α 1 ) W (p | α −1 2 ) W (q | α 3 ) = W (p |ᾱ −1 3 ) W (q |ᾱ 2 ) W (p |ᾱ −1 1 ) . It is tempting to apply identities (5.3.20) ...
The principles of the theory of quantum groups are reviewed from the point of view of the possibility of their use for deformations of symmetries in physical models. The R-matrix approach to the theory of quantum groups is discussed in detail and is taken as the basis of the quantization of classical Lie groups, as well as some Lie supergroups. We start by laying out the foundations of non-commutative and non-cocommutative Hopf algebras. Much attention has been paid to Hecke and Birman-Murakami-Wenzl (BMW) R-matrices and related quantum matrix algebras. Trigonometric solutions of the Yang-Baxter equation associated with the quantum groups GL_q(N), SO_q(N), Sp_q(2n) and supergroups GL_q(N|M), Osp_q(N|2m), as well as their rational (Yangian) limits, are presented. Rational R-matrices for exceptional Lie algebras and elliptic solutions of the Yang-Baxter equation are also considered. The basic concepts of the group algebra of the braid group and its finite dimensional quotients (such as Hecke and BMW algebras) are outlined. A sketch of the representation theories of the Hecke and BMW algebras is given (including methods for finding idempotents and their quantum dimensions). Applications of the theory of quantum groups and Yang-Baxter equations in various areas of theoretical physics are briefly discussed.
... On the other hand, the 3D L was obtained by a heuristic quantization of the solution to the local Yang-Baxter equation [33] by . They made an ansatz that the 3D L gives an operator-valued solution to the local Yang-Baxter equation, which is equivalent to the tetrahedron equation (1.2), and solved (1.2) for R. It also gives an alternative derivation of the 3D R. As a remarkable result related to the 3D L, it is known that the layer-to-layer transfer matrix of size m × n associated with the 3D L gives the spectral duality between different row-to-row transfer matrices: sl(m) spin chain of system size n and sl(n) spin chain of system size m [4]. ...
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.
... On the other hand, the 3D L was obtained by a heuristic quantization of the solution to the local Yang-Baxter equation [33] by . They made an ansatz that the 3D L gives an operatorvalued solution to the local Yang-Baxter equation, which is equivalent to the tetrahedron equation (1.2), and solved (1.2) for R. It also gives an alternative derivation of the 3D R. As a remarkable result related to the 3D L, it is known that the layer-to-layer transfer matrix of size m × n associated with the 3D L gives the spectral duality between different row-to-row transfer matrices: sl(m) spin chain of system size n and sl(n) spin chain of system size m [4]. ...
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.
... Another refactorization problem, called the local Yang-Baxter equation [32,38], allows to find solutions to the Zamolodchikov tetrahedron equation [58]. In particular, such solutions connected to the non-Abelian Hirota-Miwa system (1.2), but without imposing the periodicity constraint have been found recently in [15]. ...
We study double-sided continued fractions whose coefficients are non-commuting symbols. We work within the formal approach of the Mal’cev–Neumann series and free division rings. We start with presenting the analogs of the standard results from the theory of continued fractions, including their (right and left) simple fractions decomposition, the Euler–Minding summation formulas, and the relations between nominators and denominators of the simple fraction decompositions. We also transfer to the non-commutative double-sided setting the standard description of the continued fractions in terms of 2 × 2 matrices presenting also a weak version of the Serret theorem. The equivalence transformations between the double continued fractions are described, including also the transformation from generic such fractions to their simplest form. Then we give the description of the double-sided continued fractions within the theory of quasideterminants and we present the corresponding version of the LR and qd-algorithms. We study also (strictly and ultimately) periodic double-sided non-commutative continued fractions and we give the corresponding version of the Euler theorem. Finally we present a weak version of the Galois theorem and we give its relation to the non-commutative KP map, recently studied in the theory of discrete integrable systems.
... Let us describe the geometric interpretation of the Hirota map and of its linear problem (in all the gauges considered in the paper) within the local Yang-Baxter equation approach [38,23,45,30]. It is known that such a description of the mapping provides a simple proof of its Zamolodchikov property. ...
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 from its affine (non-homogeneous) description. We provide also an interpretation of the maps within the local Yang-Baxter equation approach. We exploit decomposition of the second map into two simpler maps which, as we show, satisfy the pentagonal condition. We provide also geometric meaning of the matching ten-term condition between the pentagonal maps. The generic description of Desargues lattices in homogeneous coordinates allows to define another solution of the Zamolodchikov equation, but with functional parameter which should be adjusted in a particular way. Its ultra-local reduction produces a birational quantum map (with two central parameters) with Zamolodchikov property, which preserves Weyl commutation relations. In the classical limit our construction gives the corresponding Poisson map satisfying the Zamolodchikov condition.
... The term Yang-Baxter maps was proposed by Veselov [72] as an alternative name to Drinfeld's one. Early results on Yang-Baxter maps were provided in [1,57,38,52]. ...
... If separability of variables on the invariants is imposed, then higher rank analogues of the Yang-Baxter maps of Propositions 3.3, 3.9 and 3.11 are expected. Moreover, solutions of the functional tetrahedron equation [41,42,49,64], or even of higher simplex equations [17,53,54] are anticipated. For example if we consider the following, different than (6.1), choice of invariants: ...
... The term Yang-Baxter maps was proposed by Veselov [36] as an alternative name to Drinfeld's one. Early results on Yang-Baxter maps were provided in [37,38,39,40]. ...
A QRT map is the composition of two involutions on a biquadratic curve: one switching the $x$-coordinates of two intersection points with a given horizontal line, and the other switching the $y$-coordinates of two intersections with a vertical line. Given a QRT map, a natural question is to ask whether it allows a decomposition into further involutions. Here we provide new answers to this question and show how they lead to a new class of maps, as well as known HKY maps and quadrirational Yang-Baxter maps.
... Remark. Another refactorization problem, called the local Yang-Baxter equation [38,32], allows to find solutions to the Zamolodchikov tetrahedron equation [58]. In particular, such solutions connected to the non-Abelian Hirota-Miwa system (1.2), but without imposing the periodicity constraint have been found recently in [15]. ...
We study double-sided continued fractions whose coefficients are non-commuting symbols. We work within the formal approach of the Mal'cev-Neumann series and free division rings. We start with presenting the analogs of the standard results from the theory of continued fractions, including their (right and left) simple fractions decomposition, the Euler-Minding summation formulas, and the relations between nominators and denominators of the simple fraction decompositions. We also transfer to the non-commutative double-sided setting the standard description of the continued fractions in terms of $2\times 2$ matrices presenting also a weak version of the Serret theorem. The equivalence transformations between the double continued fractions are described, including also the transformation from generic such fractions to their simplest form. Then we give the description of the double-sided continued fractions within the theory of quasideterminants and we present the corresponding version of the $LR$ and $qd$-algorithms. We study also (strictly and ultimately) periodic double-sided non-commutative continued fractions and we give the corresponding version of the Euler theorem. Finally we present a weak version of the Galois theorem and we give its relation to the non-commutative KP map, recently studied in the theory of discrete integrable systems.
... If separability of variables on the invariants is imposed, then higher rank analogues of the Yang-Baxter maps of propositions 3.3, 3.9 and 3.11 are expected. Moreover, solutions of the functional tetrahedron equation [50,42,43,64], or even of higher simplex equations [54,55,17] are anticipated. For example if we consider the following, different than the (32), choice of invariants: They are exactly the Hirota's map [42,43,64], i.e. the map R : (u, v, w) → (U, V, W ), where U = uv u + w , V = u + w, W = vw u + w , acting on (123), (145), (246) and (356) coordinates respectively. ...
We present the explicit form of a family of Liouville integrable maps in 3 variables, the so-called {\it triad family of maps} and we propose a multi-field generalisation of the later. We show that by imposing separability of variables to the invariants of this family of maps, the $H_I, H_{II}$ and $H_{III}^A$ Yang-Baxter maps in general position of singularities emerge. Two different methods to obtain entwining Yang-Baxter maps are also presented. The outcomes of the first method are entwining maps associated with the $H_I, H_{II}$ and $H_{III}^A$ Yang-Baxter maps, whereas by the second method we obtain non-periodic entwining maps associated with the whole $F$ and $H-$list of quadrirational Yang-Baxter maps. Finally, we show how the transfer maps associated with the $H-$list of Yang-Baxter maps can be considered as the $(k-1)$-iteration of some maps of simpler form. We refer to these maps as {\it extended transfer maps} and in turn they lead to $k-$point alternating recurrences which can be considered as alternating versions of some hierarchies of discrete Painlev\'e equations.
... In Sect. 10, we reveal the structure of the vector KP R-matrix, which leads us to a more general two-parameter R-matrix. Its parameter dependence determines, via a "local" Yang-Baxter equation [18] (also see [9]), a solution of the functional tetrahedron equation (see, e.g., [9,15,20]), i.e., the set-theoretical version of the tetrahedron equation. Finally, Sect. ...
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 non-linear map in case of a more general matrix KP equation. We also consider the corresponding Korteweg-deVries (KdV) reduction. Furthermore, exploiting the fine structure of soliton interactions in the tropical limit, we obtain a 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.
... Our next goal is to isolate solutions of functional tetrahedron equations within the family of maps (25). The question arises whether within the families of 2 n -rational maps presented here (14), one can find solutions of higher simplex equations [37][38][39]. We leave this question open for future investigations. ...
We present a natural extension of the notion of nondegenerate rational maps (quadrirational maps) to arbitrary dimensions. We refer to these maps as 2 ⁿ -rational maps. In this note we construct a rich family of 2ⁿ-rational maps. These maps by construction are involutions and highly symmetric in the sense that the maps and their companion maps have the same functional form.
... The lattice BKP equation was derived by Miwa [8] (therefore also referred to as the Miwa equation) as a four-term bilinear equation and its nonlinear form in terms of multi-ratios was later given by Nimmo & Schief [13] (see also [14]). The lattice CKP equation was obtained from the star-triangle transform in the Ising model by Kashaev [15] based on the idea of non-local Yang-Baxter maps [16]. It was named CKP by Schief [17], who revealed that Kashaev's lattice model is the superposition property of the continuous CKP equation. ...
A unified framework is presented for the solution structure of 3D discrete integrable systems, including the lattice AKP, BKP and CKP equations. This is done through the so-called direct linearising transform which establishes a general class of integral transforms between solutions. As a particular application, novel soliton-type solutions for the lattice CKP equation are obtained.
... Our next goal is to isolate solutions of functional tetrahedron equations within the family of maps (25). We also anticipate the connection of 2 n −rational maps presented here (14) with solutions of higher simplex equations [32,33,34]. ...
We present a natural extension of the notion of nondegenerate rational maps
(quadrirational maps) to arbitrary dimensions. We refer to these maps as
$2^n-$rational maps. In this note we construct a rich family of $2^n-$rational
maps. These maps by construction are involutions and highly symmetric in the
sense that the maps and their companion maps have the same functional form.
... The relation (3.5) is equivalent to quantum Korepanov equation, it can be also seen as the tetrahedral Zamolodchikov algebra/local Yang-Baxter equation for the adjoint action of R. See the long story of [12,13,14,15,4,11] for details. We fix the normalization of R by ...
It is known that a solution of the tetrahedron equation generates infinitely many solutions of the Yang-Baxter equation via suitable reductions. In this paper this scheme is applied to an oscillator solution of the tetrahedron equation involving bosons and fermions by using special 3d boundary conditions. The resulting solutions of the Yang-Baxter equation are identified with the quantum R matrices for the spin representations of B (1) n , D (1) n and D (2) n+1 .
... The first candidate for auxiliary problem was the local Yang-Baxter equation proposed by J.-M. Maillet and F. Nijhoff [69]. Basically, the [46,47,81]. ...
... The relation (3.5) is equivalent to quantum Korepanov equation, it can be also seen as the tetrahedral Zamolodchikov algebra/local Yang-Baxter equation for the adjoint action of R. See the long story of [12,13,14,15,4,11] for details. We fix the normalization of R by ...
It is known that a solution of the tetrahedron equation generates infinitely
many solutions of the Yang-Baxter equation via suitable reductions. In this
paper this scheme is applied to an oscillator solution of the tetrahedron
equation involving bosons and fermions by using special 3d boundary conditions.
The resulting solutions of the Yang-Baxter equation are identified with the
quantum R matrices for the spin representations of B^{(1)}_n, D^{(1)}_n and
D^{(2)}_{n+1}.
... At the end of this Section we note that the local Yang-Baxter equations were introduced in [12] and applied to the investigations of 3d integrable systems in many papers (see e.g. [13,14]). ...
R-matrix acting in the tensor product of two spinor representation
spaces of Lie algebra so(d) is considered thoroughly. Corresponding
Yang-Baxter equation is proved. The relation to the local Yang-Baxter
relation is established.
... The role of the tetrahedron equation for the effective Yang-Baxter equations and commutativity of transfer matrices was established in [1] while the first solution of TE was suggested in [2] in the framework of scattering theory. The method of the local Yang-Baxter equation was proposed in [3]. A lot of exercises with the functional tetrahedron equation may be found in [4,5,6,7], however the q-oscillator quantization was obtained in [8]. ...
Lectures at the Workshop \Geometry and Integrability", Melbourne, 6 15 February 2008. Style = Draft, even a sketch, useless notes. Lecture 1. Elementary ideas of 3D integrability What is the quantum tetrahedron equation and what are its beneflts. Classical limit. The tetrahedron equation is the three-dimensional generalization of the Yang-Baxter (triangle) equation. To link together two- and three-dimensional methods, we commence with a short reminder of 2D quantum inverse scattering method. Sketch of OUISM and YBE. The most useful object in 2D is the L-operator (1) Lfi;i ¡ an operator acting in the tensor product of two spaces Vfi › Vi All spaces may be equipped by independent C-valued (spectral) parameters. In what follows, we imply that an \auxiliary" space Vfi is simple (e.g. dimVfi = 2) while \quantum" space Vi may be more complicated.
... is the well known star-triangle transformation for resistances in electric networks. The identity (5.3.12) is related to the local Yang-Baxter equation [161] and is also rewritten in the operator form [160] W (q | α 1 ) W (p | α −1 2 ) W (q | α 3 ) = W (p |ᾱ −1 3 ) W (q |ᾱ 2 ) W (p |ᾱ −1 1 ) . (5.3.14) ...
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.
It is known that the local Yang–Baxter equation is a generator of potential solutions to Zamolodchikov’s tetrahedron equation. In this paper, we show under which additional conditions the solutions to the local Yang–Baxter equation are tetrahedron maps, namely solutions to the set-theoretical tetrahedron equation. This is exceptionally useful when one wants to prove that noncommutative maps satisfy the Zamolodchikov’s tetrahedron equation. We construct new noncommutative maps and we prove that they possess the tetrahedron property. Moreover, by employing Darboux transformations with noncommutative variables, we derive noncommutative tetrahedron maps. In particular, we derive a noncommutative nonlinear Schrödinger type of tetrahedron map which can be restricted to a noncommutative version of Sergeev’s map on invariant leaves. We prove that these maps are tetrahedron maps.
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.
We study tetrahedron maps, which are set-theoretical solutions to Zamolodchikov's functional tetrahedron equation, and their relations with Yang-Baxter maps, which are set-theoretical solutions to the quantum Yang-Baxter equation. In particular, we clarify the structure of the nonlinear algebraic relations which define linear (parametric) tetrahedron maps (with nonlinear dependence on parameters), and we present several transformations which allow one to obtain new such maps from known ones. Also, we prove that the differential of a (nonlinear) tetrahedron map on a manifold is a tetrahedron map as well. Using the obtained general results, we construct new examples of (parametric) Yang-Baxter and tetrahedron maps. Considered examples include maps associated with integrable systems and matrix groups. In particular, we obtain a parametric family of new linear tetrahedron maps, which are linear approximations for the nonlinear tetrahedron map constructed by Dimakis and M\"uller-Hoissen [arXiv:1708.05694] in a study of soliton solutions of matrix Kadomtsev-Petviashvili (KP) equations. Also, we present invariants for this nonlinear tetrahedron map.
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.
Given is an overview of integrable models of quantum, classical as well as statistical mechanics, defined as evolution models in a wholly discrete 2+1-dimensional space-time, and based on a special type of auxiliary linear problem.
A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator $\rscr{R}_{1,2,3}$ in the space of a triple Weyl algebra. $\rscr{R}_{1,2,3}$ is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for $\rscr{R}_{1,2,3}$ follows without further calculation. If the Weyl parameter is taken to be a root of unity, $\rscr{R}_{1,2,3}$ decomposes into a matrix conjugation operator R1,2,3 and a c-number functional mapping $\rscr{R}^{(f)}_{1,2,3}$. The operator R1,2,3 satisfies a modified tetrahedron equation (MTE) in which the "rapidities" are solutions of a classical integrable Hirota-type equations. R1,2,3 can be represented in terms of the Bazhanov-Baxter Fermat curve cyclic functions, or alternatively in terms of Gauss functions. The paper summarizes several recent publications on the subject.
The tetrahedron equation and the four-simplex equation are multidimensional generalizations of the Yang-Baxter or triangle equations. We discuss common features of these members of the family of “simplex equations”. Zamolodchikov's solution of the tetrahedron equation is rewritten in an algebraic form and a generalization of it to the four-simplex case is proposed. Relevance of the simplex equation for the understanding of multidimensional integrability is briefly discussed.
The paper is the expanded text of a report to the International Mathematical Congress in Berkeley (1986). In it a new algebraic formalism connected with the quantum method of the inverse problem is developed.
Examples are constructed of noncommutative Hopf algebras and their connection with solutions of the Yang-Baxter quantum identity are discussed.
The repulsive $$\delta${}$ interaction problem in one dimension for $N$ particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations $R$ of the permutation group ${S}_{N}$. For some $R'\mathrm{s}$ this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-\textonehalf{} fermions is reduced to a generalized Fredholm equation.
A method to construct a commuting transfer matrix has been proposed for three-dimensional fermion field theory. The method is based on the use of ``tetrahedron equations''. For the case of free fermions, the commuting transfer matrix structure has been studied completely and some solution has been obtained for the tetrahedron equations.
A linearizing integral transform is proposed which relates solutions of a spectral problem associated with a class of integrable partial difference equations to any given solution of the spectral problem. Examples of this class are lattice versions of the isotropic Heisenberg spin chain, the nonlinear Schrödinger equation and the (complex) sine-Gordon equation.
Aq-difference analogue of the universal enveloping algebra U(g) of a simple Lie algebra g is introduced. Its structure and representations are studied in the simplest case g=sl(2). It is then applied to determine the eigenvalues of the trigonometric solution of the Yang-Baxter equation related to sl(2) in an arbitrary finite-dimensional irreducible representation.
The quantumS-matrix theory of straight-strings (infinite one-dimensional objects like straight domain walls) in 2+1-dimensions is considered. TheS-matrix is supposed to be purely elastic and factorized. The tetrahedron equations (which are the factorization conditions) are investigated for the special two-colour model. The relativistic three-stringS-matrix, which apparently satisfies this tetrahedron equation, is proposed.
We study for g=g[(N+1) the structure and representations of the algebra (g), a q-analogue of the universal enveloping algebra U(g). Applying the result, we construct trigonometric solutions of the Yang-Baxter equation associated with higher representations of g.
We report the explicit form of the quantumR matrix in the fundamental representation for the generalized Toda system associated with non-exceptional affine Lie algebras.
The partition function of the zero-field “Eight-Vertex” model on a square M by N lattice is calculated exactly in the limit of M, N large. This model includes the dimer, ice and zero-field Ising, F and KDP models as special cases. In general the free energy has a branch point singularity at a phase transition, with an irrational exponent.
In this paper we present a systematic method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred. It is argued that these difference-difference equations can be regarded as arising from Bianchi identities expressing the commutativity of Bäcklund transformations. Applying an appropriate continuum limit we first obtain integrable nonlinear differential-difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations. As special cases we treat the difference-difference versions and the differential-difference versions of the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the nonlinear Schrödinger equation, the isotropic classical Heisenberg spin chain, and the complex and real sine-Gordon equation.
In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A striking instance of such a procedure discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrödinger operator are integrals of the Korteweg-de Vries equation.
In Section 2 we prove the simplest case of a conjecture of Kruskal and Zabusky concerning the existence of double wave solutions of the Korteweg-de Vries equation, i.e., of solutions which for |I| large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of remarkable series of integrals discovered by Kruskal and Zabusky.
Nonlinear evolutions
- Maillet
The algebraic structure of d-simplex equations, CERN preprint
- J M Maillet
- F W Nijhoff
Gauge theories and integrable systems in d ⩾ 3; Gauging the quantum groups, CERN preprints
- J M Maillet
- F W Nijhoff