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2009
Shibukawa, Youichi (J-HOKKS) Dynamical Yang-Baxter maps with an invariance condition. (English summary)
Journal of Pure and Applied Algebra, 2013
In this article, we propose the concept of a dynamical brace, which is an algebraic system that corresponds to a certain class of dynamical Yang-Baxter maps. Moreover, we study combinatorial aspects of the dynamical brace.
Journal of Algebra, 2018
A dynamical Yang-Baxter map, introduced by Shibukawa, is a solution of the set-theoretical analogue of the dynamical Yang-Baxter equation. In this paper, we initiate a quiver-theoretical approach for the study of dynamical Yang-Baxter maps. Our key observation is that the category of dynamical sets over a set Λ, introduced by Shibukawa to establish a categorical framework to deal with dynamical Yang-Baxter maps, can be embedded into the category of quivers with vertices Λ. By using this embedding, we shed light on Shibukawa's classification result of a certain class of dynamical Yang-Baxter maps and extend his construction to obtain a new class of dynamical Yang-Baxter maps. We also discuss a relation between Shibukawa's bialgebroid associated to a dynamical Yang-Baxter map and Hayashi's weak bialgebra associated to a star-triangular face model.
2016
We propose a nonperturbative approach to nonabelian two-form gauge theory. We formulate the theory on a lattice in terms of plaquette as fundamental dynamical variable, and assign U(N) Chan-Paton colors at each boundary link. We show that, on hypercubic lattices, such colored plaquette variables constitute Yang-Baxter maps, where holonomy is characterized by certain dynamical deformation of quantum Yang-Baxter equations. Consistent dimensional reduction to Wilson's lattice gauge theory singles out unique compactness condition. We study a class of theories where the compactness condition is solved by Lax pair ansatz. We find that, in naive classical continuum limit, these theories recover Lorentz invariance but have degrees of freedom that scales as N 2 at large N. This implies that nontrivial quantum continuum limit must be sought for. We demonstrate that, after dimensional reduction, these theories are reduced to Wilson's lattice gauge theory. We also show that Wilson surfaces are well-defined physical observables without ordering ambiguity. Utilizing lattice strong coupling expansion, we compute partition function and correlation functions of the Wilson surfaces. We discover that, at large N limit, the character expansion coefficients exhibit large-order behavior growing faster than exponential, in striking contrast to Wilson's lattice gauge theory. This hints a hidden, weakly coupled theory dual to the proposed tensor gauge theory. We finally discuss relevance of our study to topological quantum order in strongly correlated systems. This paper is organized as follows. We begin in section 2 with etiology of Chan-Paton factors. We assign Chan-Paton factors to boundary links of an elementary plaquette so that it carries four 'color' indices. We take these objects as fundamental dynamical variables and construct in section 3 a nonabelian two-form tensor gauge theory defined on a d-dimensional hypercubic lattice. Expressing the plaquette variable as (N 2 × N 2) matrices of U(N) gauge group, we construct action for nonabelian tensor gauge theory. We then study possible compactness conditions. We show that consistent reduction to Wilson's lattice gauge theory [16] by a dimensional reduction and unitarity or reflection-positivity singles out a unique choice of the condition. In section 4, we study ground † In this paper, we study exclusively two-form gauge theory-as will become evident in foregoing discussions, the construction is extendible to higher p-form gauge theories straightforwardly. ‡ There has been in the past occasional attempt for constructing nonabelian p-form gauge theories. See [14] and also [15].
2015
In this paper, we construct a noncommutative extension of the Adler-Yamilov Yang-Baxter map which is related to the nonlinear Schr�dinger equation. Moreover, we show that this map is partially integrable.
Tokyo Journal of Mathematics, 2015
We prove that braided semigroups with suitable conditions can produce solutions to the quantum Yang-Baxter equation in every tensor category. As an application, some dynamical Yang-Baxter maps, set-theoretic solutions to a version of the quantum dynamical Yang-Baxter equation, are constructed.
2010
It is shown that square free set theoretic involutive non-degenerate solutions of the Yang-Baxter equation whose associated permutation group (referred to as an involutive Yang-Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also implies that the corresponding finitely presented abelian-by-finite groups (called the structure groups) are poly-Z groups. Secondly, an example of a solution with an abelian involutive Yang-Baxter group which is not a generalized twisted union is constructed. This answers in the negative another problem of Gateva-Ivanova. The constructed solution is of multipermutation level 3. Retractability of solutions is also proved in the case where the natural generators of the IYB group are cyclic permutations. Moreover, it is shown that such solutions are generalized twisted unions. * Research partially supported by grants of MICIN-FEDER (Spain) MTM2008-06201-C02-01, Generalitat de Catalunya 2005SGR00206, Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish-Polish bilateral agreement BIL2005/VUB/06 and MNiSW research grant N201 004 32/0088 (Poland).
Physics Letters A, 1990
The possibility of characterization of nonlinear quantum dynamical maps in terms of complete positivity is discussed. The example of the Hartree type equation illustrates the difficulties associated with such an attempt.
The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter's work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative geometry, etc. Many scientists have found solutions for the Yang-Baxter equation, obtaining qualitative results (using the axioms of various algebraic structures) or quantitative results (usually using computer calculations). However, the full classification of its solutions remains an open problem. In this paper, we present the (set-theoretical) Yang-Baxter equation, we sketch the proof of a new theorem, we state some problems, and discuss about directions for future research.
2010
We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation.
Physics Letters A, 1992
A new Yang-Baxter structure for time-discrete integrable models is introduced. It contains the time-dependent part of the Lax representation and can be applied in investigations on integrable quantum mappings. As a particular class of examples we present the application to quantum mappings associated with the lattice Gerfand-Dikii hierarchy.
Journal of Physics A: Mathematical and Theoretical, 2021
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 approxima...
2001
We construct Drinfel'd twists that define deformed Hopf structures. In particular, we obtain deformed double Yangians and dynamical double Yangians.
Journal of Physics A: Mathematical and Theoretical, 2019
We construct birational maps that satisfy the parametric set-theoretical Yang-Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable Nonlinear Schrödinger type equations and study the refactorisation problems of the product of their associated Darboux matrices. Additionally, we study various algebraic properties of the derived maps, such as invariants and associated symplectic or Poisson structures, and we prove their complete integrability in the Liouville sense.
Journal of Physics A: Mathematical and Theoretical, 2016
In this paper we show that there are explicit Yang-Baxter maps with Darboux-Lax representation between Grassmann extensions of algebraic varieties. Motivated by some recent results on noncommutative extensions of Darboux transformations, we first derive a Darboux matrix associated with the Grassmann-extended derivative Nonlinear Schrödinger (DNLS) equation, and then we deduce novel endomorphisms of Grassmann varieties, which possess the Yang-Baxter property. In particular, we present ten-dimensional maps which can be restricted to eight-dimensional Yang-Baxter maps on invariant leaves, related to the Grassmann-extended NLS and DNLS equations. We consider their vector generalisations.
arXiv: Operator Algebras, 2019
Every unitary solution of the Yang-Baxter equation (R-matrix) in dimension $d$ can be viewed as a unitary element of the Cuntz algebra ${\mathcal O}_d$ and as such defines an endomorphism of ${\mathcal O}_d$. These Yang-Baxter endomorphisms restrict and extend to endomorphisms of several other $C^*$- and von Neumann algebras and furthermore define a II$_1$ factor associated with an extremal character of the infinite braid group. This paper is devoted to a detailed study of such Yang-Baxter endomorphisms. Among the topics discussed are characterizations of Yang-Baxter endomorphisms and the relative commutants of the various subfactors they induce, an endomorphism perspective on algebraic operations on R-matrices such as tensor products and cabling powers, and properties of characters of the infinite braid group defined by R-matrices. In particular, it is proven that the partial trace of an R-matrix is an invariant for its character by a commuting square argument. Yang-Baxter endomorp...
2000
Sufficient conditions for an invertible two-tensor F to relate two solutions to the Yang-Baxter equation via the transformation R → F −1 21 RF are formulated. Those conditions include relations arising from twisting of certain quasitriangular bialgebras.
2008
We present solutions for the (constant and spectral-parameter) Yang-Baxter equations and Yang-Baxter systems, arising from algebra structures. We discuss about their applications in theoretical physics.
Communications in Algebra, 2005
It is shown that a Yang-Baxter system can be constructed from any entwining structure. It is also shown that, conversely, Yang-Baxter systems of certain type lead to entwining structures. Examples of Yang-Baxter systems associated to entwining structures are given, and a Yang-Baxter operator of Hecke type is defined for any bijective entwining map.
Letters in Mathematical Physics, 2005
star-products and dynamical twists very explicitly (cf. ). We also propose a method that allows one (under certain conditions) to obtain non-dynamical twists from dynamical ones. This is a quantization of the "classical" result obtained in Appendix B].
Journal of Physics A: Mathematical and Theoretical, 2013
In this paper we construct Yang-Baxter (YB) maps using Darboux matrices which are invariant under the action of finite reduction groups. We present 6-dimensional YB maps corresponding to Darboux transformations for the Nonlinear Schrödinger (NLS) equation and the derivative Nonlinear Schrödinger (DNLS) equation. These YB maps can be restricted to 4−dimensional YB maps on invariant leaves. The former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries [14]. We give a 6− dimensional YB-map corresponding to the Darboux transformation for a deformation of the DNLS equation. We also consider vector generalisations of the YB maps corresponding to the NLS and DNLS equation.
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