Yang-Baxter Research Articles

Tetrahedron equation and quantum cluster algebras

We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y-variables in terms q-Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev’s R matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when q is specialized to a root of unity.

Optimal Realization of Yang–Baxter Gate on Quantum Computers

AbstractQuantum computers provide a promising method to study the dynamics of many‐body systems beyond classical simulation. On the other hand, the analytical methods developed and results obtained from the integrable systems provide deep insights on the many‐body system. Quantum simulation of the integrable system not only provides a valid benchmark for quantum computers but is also the first step in studying integrable‐breaking systems. The building block for the simulation of an integrable system is the Yang–Baxter gate. It is vital to know how to optimally realize the Yang–Baxter gates on quantum computers. Based on the geometric picture of the Yang–Baxter gates, the optimal realizations of two types of Yang–Baxter gates with a minimal number of controlled NOT (CNOT) or gates are presented. It is also shown how to systematically realize the Yang–Baxter gates via the pulse control. The different realizations on IBM quantum computers are tested and compared. It is found that the pulse realizations of the Yang–Baxter gates always have a higher gate fidelity compared to the optimal CNOT or realizations. On the basis of the above optimal realizations, the simulation of the Yang–Baxter equation on quantum computers is demonstrated. These results provide a guideline and standard for further experimental studies based on the Yang–Baxter gate.

Dehornoy’s class and Sylows for set-theoretical solutions of the Yang–Baxter equation

We explain how the germ of the structure group of a cycle set decomposes as a product of its Sylow-subgroups, and how this process can be reversed to construct cycle sets from ones with coprime classes. We study Dehornoy’s class associated to a cycle set, and conjecture a bound that we prove in a specific case. We combine the use of braces and a monomial representation, in particular to answer a question by Dehornoy on retrieving the Garside structure without a theorem of Rump, while also retrieving said theorem.

Rota–Baxter mock-Lie bialgebras and related structures

The aim of this paper is to introduce the notion of Rota–Baxter mock-Lie bialgebras [Formula: see text] and their admissibility conditions in terms of dual representations [Formula: see text]. Next, we show that Rota–Baxter mock-Lie bialgebras are characterized by matched pairs and Manin triples of Rota–Baxter mock-Lie algebras. Furthermore, the coboundary case leads to the introduction of the admissible mock-Lie Yang–Baxter equation in Rota–Baxter mock-Lie algebras, whose skew-symmetric solutions are used to construct Rota–Baxter mock-Lie bialgebras.

Finite skew braces of square-free order and supersolubility

Abstract The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers and that in an arbitrary supersoluble skew brace B many relevant skew brace-theoretical properties are easier to identify: For example, a centrally nilpotent ideal of B is B-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, B has finite multipermutational level if and only if $(B,+)$ is nilpotent. Given a finite presentation of the structure skew brace $G(X,r)$ associated with a finite nondegenerate solution of the Yang–Baxter equation (YBE), there is an algorithm that decides if $G(X,r)$ is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on.

Classical Yang–Baxter Equation, Lagrangian Multiforms and Ultralocal Integrable Hierarchies

We cast the classical Yang–Baxter equation (CYBE) in a variational context for the first time, by relating it to the theory of Lagrangian multiforms, a framework designed to capture integrability in a variational fashion. This provides a significant connection between Lagrangian multiforms and the CYBE, one of the most fundamental concepts of integrable systems. This is achieved by introducing a generating Lagrangian multiform which depends on a skew-symmetric classical r-matrix with spectral parameters. The multiform Euler–Lagrange equations produce a generating Lax equation which yields a generating zero curvature equation. The CYBE plays a role at three levels: (1) it ensures the commutativity of the flows of the generating Lax equation; (2) it ensures that the generating zero curvature equation holds; (3) it implies the closure relation for the generating Lagrangian multiform. The specification of an integrable hierarchy is achieved by fixing certain data: a finite set S⊂CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S\\subset \\mathbb {C}P^1$$\\end{document}, a Lie algebra g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}$$\\end{document}, a g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}$$\\end{document}-valued rational function with poles in S and an r-matrix. We show how our framework is able to generate a large class of ultralocal integrable hierarchies by providing several known and new examples pertaining to the rational or trigonometric class. These include the Ablowitz–Kaup–Newell–Segur hierarchy, the sine-Gordon (sG) hierarchy and various hierarchies related to Zakharov–Mikhailov type models which contain the Faddeev–Reshetikhin (FR) model and recently introduced deformed Gross–Neveu models as particular cases. The versatility of our method is illustrated by showing how to couple integrable hierarchies together to create new examples of integrable field theories and their hierarchies. We provide two examples: the coupling of the nonlinear Schrödinger system to the FR model and the coupling of sG with the anisotropic FR model.

Corrigendum and addendum to "Structure monoids of set-theoretic solutions of the Yang-Baxter equation”

Corrigendum and addendum to "Structure monoids of set-theoretic solutions of the Yang-Baxter equation”

On bosonic Thirring model in Minkowski signature

We present the way to continue the bosonic Thirring model or βγ-system with quartic interaction to Minkowski signature, based on the symmetries of this model. It is shown that the considered Minkowski version of the model is one-loop renormalizable. Based on this, we find the amplitudes of scattering of the excitations corresponding to the γ and γ¯ fields up to the one-loop order. In particular, it was computed that the 2→2 amplitudes of these excitations possess property of reflectionless scattering and thus the corresponding S-matrix of such excitations satisfies the Yang-Baxter equation. The obtained S-matrix elements for γ and γ¯ are shown to coincide with the corresponding S-matrix elements of the solitons in the complex sine-Gordon model proposed by Dorey and Hollowood.

Bialgebras, the Yang–Baxter equation and Manin triples for mock-Lie algebras

The aim of this paper is to introduce the notion of a mock-Lie bialgebra which is equivalent to a Manin triple of mock-Lie algebras. The study of a special case called coboundary mock-Lie bialgebra leads to introducing the mock-Lie Yang–Baxter equation on a mock-Lie algebra which is an analogue of the classical Yang–Baxter equation on a Lie algebra. Note that a skew-symmetric solution of mock-Lie Yang–Baxter equation gives a mock-Lie bialgebra. Finally, O-operators are studied to construct a skew-symmetric solution of a mock-Lie Yang–Baxter equation.

A Remark on the q-Hypergeometric Integral Bailey Pair and the Solution to the Star-Triangle Equation

A Remark on the q-Hypergeometric Integral Bailey Pair and the Solution to the Star-Triangle Equation

A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie Poisson bialgebras

The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang–Baxter equation and [Formula: see text]-operators are studied.

Free fermionic Schur functions

We introduce a new family of Schur functions sλ/μ;a,b(x/y) that depend on two sets of variables and two sequences of parameters. These free fermionic Schur functions generalize and unify double, supersymmetric, and dual Schur functions from literature. These functions have a hidden symmetry that is manifested in the supersymmetric Cauchy identity∑λsλ;a,b(x/y)sˆλ;a,b(z/w)=∏i,j1+yizj1−xizj1+xiwj1−yiwj, where sˆλ;a,b(z/w)=sλ′;b′,a′(w/z) are the dual functions.Our approach is based on the integrable six vertex model with free fermionic Boltzmann weights. We show that these weights satisfy the refined Yang-Baxter equations, which allows us to prove the generalizations of well-known properties of the Schur functions. We emphasize that some of our results and proofs are novel even in special cases.

The structure skew brace associated with a finite non-degenerate solution of the Yang-Baxter equation is finitely presented

The aim of this paper is to show that the structure skew brace associated with a finite non-degenerate solution of the Yang-Baxter equation is finitely presented.

(In)decomposability of finite solutions of the Yang-Baxter equation

(In)decomposability of finite solutions of the Yang-Baxter equation

On integrability of the one-dimensional Hubbard model

We find a family of solutions to Zamolodchikov's tetrahedral algebra corresponding to the fermionic R-operator for the free fermion model of the difference type in one of the spectral parameters, construct an extension of the R-operator for a system of two spins satisfying the Yang-Baxter equation, and find the local charges. We also construct a twisted monodromy operator, which leads to the one-dimensional Hubbard model.

Bailey pairs for the q-hypergeometric integral pentagon identity

In this work, we construct new Bailey pairs for the integral pentagon identity in terms of q-hypergeometric functions. The pentagon identity considered here represents the equality of the partition functions of certain three-dimensional supersymmetric dual theories. It can be also interpreted as the star-triangle relation for the Ising-type integrable lattice model.

Integrable Quantum Circuits from the Star-Triangle Relation

The star-triangle relation plays an important role in the realm of exactly solvable models, offering exact results for classical two-dimensional statistical mechanical models. In this article, we construct integrable quantum circuits using the star-triangle relation. Our construction relies on families of mutually commuting two-parameter transfer matrices for statistical mechanical models solved by the star-triangle relation, and differs from previously known constructions based on Yang-Baxter integrable vertex models. At special value of the spectral parameter, the transfer matrices are mapped into integrable quantum circuits, for which infinite families of local conserved charges can be derived. We demonstrate the construction by giving two examples of circuits acting on a chain of Q−state qudits: Q-state Potts circuits, whose integrability has been conjectured recently by Lotkov et al., and ZQ circuits, which are novel to our knowledge. In the first example, we present for Q=3 a connection to the Zamolodchikov-Fateev 19-vertex model.

Matching Rota–Baxter modules and matching 𝒪-operators of associative algebras

In this paper, firstly, we introduce the notion of matching Rota–Baxter modules, which generalizes the notions of matching Rota–Baxter algebras and Rota–Baxter modules, and give its characteristic and construction. Secondly, we give the relation between matching Rota–Baxter modules and other modules structure. Finally, we introduce the notion of matching [Formula: see text]-operators of associative algebras which also generalizes the notion of matching Rota–Baxter algebras and give the solution of polarized associative Yang–Baxter equation by using the matching [Formula: see text]-operators of associative algebras. In addition, we introduce the definitions of the compatible matching [Formula: see text]-operators of associative algebras, matching Nijenhui operators, and compatible matching dendriform algebras, and consider their connection.

Malcev Yang-Baxter equation, weighted $\mathcal{O}$-operators on Malcev algebras and post-Malcev algebras

The purpose of this paper is to study the $\mathcal{O}$-operators on Malcev algebras and discuss the solutions of Malcev Yang-Baxter equation by $\mathcal{O}$-operators. Furthermore we introduce the notion of weighted $\mathcal{O}$-operators on Malcev algebras, which can be characterized by graphs of the semi-direct product Malcev algebra. Then we introduce a new algebraic structure called post-Malcev algebras. Therefore, post-Malcev algebras can be viewed as the underlying algebraic structures of weighted $\mathcal{O}$-operators on Malcev algebras. A post-Malcev algebra also gives rise to a new Malcev algebra. Post-Malcev algebras are analogues for Malcev algebras of post-Lie algebras and fit into a bigger framework with a close relationship with post-alternative algebras.

Nijenhuis operators and associative D-bialgebras

Firstly, we prove that a quasitriangular associative D-bialgebra together with a coquasitriangular structure can induce a Nijenhuis algebra. This shows that Nijenhuis operators and associative D-bialgebras are closely related. Secondly, we investigate some properties of a representation of a Nijenhuis algebra and introduce the notion of a corepresentation of a Nijenhuis coalgebra, which also can be seen as an infinitesimal deformation of compatible bicomodule. Thirdly, a Nijenhuis associative D-bialgebra can be established by a Nijenhuis algebra and a Nijenhuis coalgebra satisfying some compatible conditions. Furthermore, we find that a Nijenhuis associative D-bialgebra is equivalent to a matched pair of Nijenhuis algebras or a double construction of a Nijenhuis Frobenius algebra. Lastly, we provide some methods to construct Nijenhuis associative D-bialgebras through admissible associative Yang-Baxter equations (aYBes for short) and O-operators on Nijenhuis algebras.