Baxter Equation Research Articles

Segre products and Segre morphisms in a class of Yang–Baxter algebras

Let (X,r_X) and (Y,r_Y) be finite nondegenerate involutive set-theoretic solutions of the Yang–Baxter equation, and let A_X = mathcal {A}({{textbf {k}}}, X, r_X) and A_Y= mathcal {A}({{textbf {k}}}, Y, r_Y) be their quadratic Yang–Baxter algebras over a field {{textbf {k}}}. We find an explicit presentation of the Segre product A_Xcirc A_Y in terms of one-generators and quadratic relations. We introduce analogues of Segre maps in the class of Yang–Baxter algebras and find their images and their kernels. The results agree with their classical analogues in the commutative case.

Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation

Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation

Birational solutions to the set-theoretical 4-simplex equation

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 Schrödinger type parametric 4-simplex map which is the first parametric 4-simplex map in the literature.

Fusion hierarchies, T-systems and Y-systems for the dilute loop models on a strip

We study the dilute loop models on the geometry of a strip of width N. Two families of boundary conditions are known to satisfy the boundary Yang–Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as its T- and Y-systems, for these four boundary conditions and with a generic crossing parameter λ. For rational and thus a root of unity, we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large-N expansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for small N. The present work complements a previous study (Morin-Duchesne and Pearce 2019 J. Stat. Mech. 1909) that investigated the dilute loop models with periodic boundary conditions.

Rota–Baxter operators on groups

The theory of Rota–Baxter operators on rings and algebras has been developed since 1960. In 2020, the notion of Rota–Baxter operator on a group was defined. Further, it was proved that one may define a skew left brace on any group endowed with a Rota–Baxter operator. Thus, a group endowed with a Rota–Baxter operator gives rise to a set-theoretical solution to the Yang–Baxter equation. We provide some general constructions of Rota–Baxter operators on a group. Given a map on a group, we study its extensions to a Rota–Baxter operator. We state the connection between Rota–Baxter operators on a group and Rota–Baxter operators on an associated Lie ring. We describe Rota–Baxter operators on sporadic simple groups.

Solutions of the Yang–Baxter Equation and Automaticity Related to Kronecker Modules

The Kronecker algebra K is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over K are said to be Kronecker modules. The classification of these modules can be obtained by solving a well-known tame matrix problem. Such a classification deals with solving systems of differential equations of the form Ax=Bx′, where A and B are m×n, F-matrices with F an algebraically closed field. On the other hand, researching the Yang–Baxter equation (YBE) is a topic of great interest in several science fields. It has allowed advances in physics, knot theory, quantum computing, cryptography, quantum groups, non-associative algebras, Hopf algebras, etc. It is worth noting that giving a complete classification of the YBE solutions is still an open problem. This paper proves that some indecomposable modules over K called pre-injective Kronecker modules give rise to some algebraic structures called skew braces which allow the solutions of the YBE. Since preprojective Kronecker modules categorize some integer sequences via some appropriated snake graphs, we prove that such modules are automatic and that they induce the automatic sequences of continued fractions.

Lie Bialgebras on the Rank Two Heisenberg–Virasoro Algebra

The rank two Heisenberg–Virasoro algebra can be viewed as a generalization of the twisted Heisenberg–Virasoro algebra. Lie bialgebras play an important role in searching for solutions of quantum Yang–Baxter equations. It is interesting to study the Lie bialgebra structures on the rank two Heisenberg–Virasoro algebra. Since the Lie brackets of rank two Heisenberg–Virasoro algebra are different from that of the twisted Heisenberg–Virasoro algebra and Virasoro-like algebras, and there are inner derivations (from itself to its tensor space) which are hidden more deeply in its interior algebraic structure, some new techniques and strategies are employed in this paper. It is proved that every Lie bialgebra structure on the rank two Heisenberg–Virasoro algebra is triangular coboundary.

Rota–Baxter operators on Clifford semigroups and the Yang–Baxter equation

In this paper, we introduce the theory of Rota–Baxter operators on Clifford semigroups, useful tools for obtaining dual weak braces, i.e., triples (S,+,∘) where (S,+) and (S,∘) are Clifford semigroups such that a∘(b+c)=a∘b−a+a∘c and a∘a−=−a+a, for all a,b,c∈S. To each algebraic structure is associated a set-theoretic solution of the Yang–Baxter equation that has a behaviour near to the bijectivity and non-degeneracy. Drawing from the theory of Clifford semigroups, we provide methods for constructing dual weak braces and deepen some structural aspects, including the notion of ideal.

On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang–Baxter equation

Given a finite bijective non-degenerate set-theoretic solution ( X , r ) of the Yang–Baxter equation we characterize when its structure monoid M ( X , r ) is Malcev nilpotent. Applying this characterization to solutions coming from racks, we rediscover some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of finite abelian racks and quandles. We also investigate bijective non-degenerate multipermutation (not necessarily finite) solutions ( X , r ) and show, for example, that this property is equivalent to the solution associated to the structure monoid M ( X , r ) (respectively structure group G ( X , r ) ) being a multipermuation solution and that G ( X , r ) is solvable of derived length not exceeding the multipermutation level of ( X , r ) enlarged by one, generalizing results of Gateva-Ivanova and Cameron obtained in the square-free involutive case. Moreover, we also prove that if X is finite and G = G ( X , r ) is nilpotent, then the torsion part of the group G is finite, it coincides with the commutator subgroup [ G , G ] + of the additive structure of the skew left brace G and G / [ G , G ] + is a trivial left brace.

On the enumeration of finite 𝐿-algebras

We use Constraint Satisfaction Methods to construct and enumerate finite L L -algebras up to isomorphism. These objects were recently introduced by Rump and appear in Garside theory, algebraic logic, and the study of the combinatorial Yang–Baxter equation. There are 377,322,225 isomorphism classes of L L -algebras of size eight. The database constructed suggests the existence of bijections between certain classes of L L -algebras and well-known combinatorial objects. We prove that Bell numbers enumerate isomorphism classes of finite linear L L -algebras. We also prove that finite regular L L -algebras are in bijective correspondence with infinite-dimensional Young diagrams.

Rota---Baxter operators on $Cur(sl_2(\mathbb{C}))$

We classify all Rota---Baxter operators on the simple Lie conformal algebra $\Cur(\sl_2(\mathbb{C}))$ and clarify which of them arise from the solutions to the conformal classical Yang---Baxter equation due to the connection discovered by Y. Hong and C. Bai in 2020.

Conformal Yang–Baxter equation on Cur(sl2(C))

In 2008, Liberati [J. Algebra 319, 2295–2318 (2008)] defined what a conformal Lie bialgebra is and introduced the conformal classical Yang–Baxter equation (CCYBE). An L-invariant solution to the weak version of CCYBE provides a conformal Lie bialgebra structure. We describe all solutions to the CCYBE on the current Lie conformal algebra Cur(sl2(C)) and to the weak version of it.

"My life can be said to form a circle" — An interview with Nobel laureate Professor Chen-Ning Yang

<abstract> <p><bold>Editor's note:</bold> Chen-Ning Yang (杨振宁, 1 October 1922 –), or C.N. Yang or by the English name Frank Yang received the 1957 Nobel Prize in Physics with Tsung-Dao Lee for their work on parity non-conservation of weak interaction. Yang and Robert Mills also proposed the non-abelian gauge theory, widely known as the Yang–Mills theory. Yang and Baxter found the Yang–Baxter equation that plays an important role in integrable models and has influenced several branches of physics and mathematics. Professor C.N. Yang is not only a great physicist, but also an outstanding educator and a thoughtful philosopher. This edited interview was conducted in 2014 by the author. Parts of the Chinese version were published in a few Chinese newspapers and magazines at the time. This English version, translated by Tyler Ross from New York, USA, provides readers worldwide with an opportunity to share and understand Professor Yang's long-lasting successful journey and thoughts on scientific preparation and innovation, cross-disciplinary integration, and effective collaboration, in particular how Yang's in-depth understanding and appreciation of mathematics in his successes in physics. This interview is also to celebrate Professor Yang's 101st birthday and his public speech at Zhejiang University in China in 2014.</p> </abstract>

Solutions of the Yang–Baxter Equation Arising from Brauer Configuration Algebras

Currently, researching the Yang–Baxter equation (YBE) is a subject of great interest among scientists of diverse areas in mathematics and other sciences. One of the fundamental open problems is to find all of its solutions. The investigation deals with developing theories such as knot theory, Hopf algebras, quandles, Lie and Jordan (super) algebras, and quantum computing. One of the most successful techniques to obtain solutions of the YBE was given by Rump, who introduced an algebraic structure called the brace, which allows giving non-degenerate involutive set-theoretical solutions. This paper introduces Brauer configuration algebras, which, after appropriate specializations, give rise to braces associated with Thompson’s group F. The dimensions of these algebras and their centers are also given.

Stokes Phenomenon and Reflection Equations

In this paper, we study the Stokes phenomenon of the cyclotomic Knizhnik–Zamolodchikov equation associated to the symmetric pair \((\mathfrak {gl}_n,\mathfrak {so}_n)\), and prove that its two types of Stokes matrices satisfy the Yang–Baxter and reflection equations respectively. We briefly discuss its isomonodromy deformation, and its relations with cyclotomic associators, twists and quantum symmetric pairs.

On bi-skew braces and brace blocks

L. N. Childs defined a bi-skew brace to be a skew brace such that if we swap the role of the two operations, then we find again a skew brace.In this paper, we give a systematic analysis of bi-skew braces. We study nilpotency and solubility, and connections between bi-skew braces and set-theoretic solutions of the Yang–Baxter equation. Further, we deal with Byott's conjecture in the case of bi-skew braces, and we use bi-skew braces as a tool to solve a classification problem proposed by L. Vendramin.In the final part, we investigate brace blocks, defined by A. Koch to be families of group operations on a given set such that any two of them yield a bi-skew brace. We provide a characterisation of brace blocks, illustrate how all known constructions in literature follow in a natural way from our characterisation, and give several new examples.

Mini-Workshop: Three Facets of R-Matrices

By definition, an R–matrix with spectral parameter is a solution to the Yang–Baxter equation, introduced in the 1970’s by C.N. Yang and R.J. Baxter. Such a matrix encodes the Boltzmann weights of a lattice model of statistical mechanics, and the Yang–Baxter equation appears naturally as a sufficient condition for its solvability. In the last decade, several mathematical and physical theories have led to seemingly different constructions of R–matrices. The theme of this workshop was the interaction of three such approaches, each of which has independently proven to be valuable: the geometric, analytic and gauge–theoretic constructions of R-matrices. Its aim was to bring together leading experts and researchers from each school of thought, whose recent works have given novel interpretations to this nearly classical topic.

Fusion for the Yang–Baxter equation and the braid group

Fusion for the Yang–Baxter equation and the braid group

Lax pair formulation for the open boundary Osp(1∣2) spin chain

Based on the Lax pair formulation, we study the integrable conditions of the Osp(1∣2) spin chain with open boundaries. We consider both the non-graded and graded versions of the Osp(1∣2) chain. The Lax pair operators M ± for the boundaries can be induced by the Lax operator M j for the bulk of the system. They correspond to the reflection equations (RE) and the Yang–Baxter equation, respectively. We further calculate the boundary K-matrices for both the non-graded and graded versions of the model with open boundaries. The double row monodromy matrix and transfer matrix of the spin chain have also been constructed. The K-matrices obtained from the Lax pair formulation are consistent with those from Sklyanin’s RE. This construction is another way to prove the quantum integrability of the Osp(1∣2) chain. We find that the Lax pair formulation has advantages in dealing with the boundary terms of the supersymmetric model.

Nilpotency of skew braces and multipermutation solutions of the Yang–Baxter equation

We study relations between different notions of nilpotency in the context of skew braces and applications to the structure of solutions to the Yang–Baxter equation. In particular, we consider annihilator nilpotent skew braces, an important class that turns out to be a brace-theoretic analog to the class of nilpotent groups. In this vein, several well-known theorems in group theory are proved in the more general setting of skew braces.