Yang-Baxter Research Articles

A note on right-nil and strong-nil skew braces

Nilpotency concepts for skew braces are among the main tools with which we are nowadays classifying certain special solutions of the Yang–Baxter equation, a consistency equation that plays a relevant role in quantum statistical mechanics and in many areas of mathematics. In this context, two relevant questions have been raised in F. Cedó, A. Smoktunowicz and L. Vendramin (Skew left braces of nilpotent type. Proc. Lond. Math. Soc. (3) 118 (2019), 1367–1392) (see questions 2.34 and 2.35) concerning right- and central nilpotency. The aim of this short note is to give a negative answer to both questions: thus, we show that a finite strong-nil brace B need not be right-nilpotent. On a positive note, we show that there is one (and only one, by our examples) special case of the previous questions that actually holds. In fact, we show that if B is a skew brace of nilpotent type and $b\ \ast \ b=0$ for all $b\in B$ , then B is centrally nilpotent.

Tetrahedron equation and Schur functions

Abstract The tetrahedron equation introduced by Zamolodchikov is a three-dimensional generalization of the Yang–Baxter equation. Several types of solutions to the tetrahedron equation that have connections to quantum groups can be viewed as q-oscillator valued vertex models with matrix elements of the L-operators given by generators of the q-oscillator algebra acting on the Fock space. Using one of the q = 0-oscillator valued vertex models introduced by Bazhanov–Sergeev, we introduce a family of partition functions that admits an explicit algebraic presentation using Schur functions. Our construction is based on the three-dimensional realization of the Zamolodchikov–Faddeev algebra provided by Kuniba–Maruyama–Okado. Furthermore, we investigate an inhomogeneous generalization of the three-dimensional lattice model. We show that the inhomogeneous analog of (a certain subclass of) partition functions can be expressed as loop elementary symmetric functions.

Set-Theoretical Solutions for the Yang–Baxter Equation in GE-Algebras: Applications to Quantum Spin Systems

This manuscript presents set-theoretical solutions to the Yang–Baxter equation within the framework of GE-algebras by constructing mappings that satisfy the braid condition and exploring the algebraic properties of GE-algebras. Detailed proofs and the use of left and right translation operators are provided to analyze these algebraic interactions, while an algorithm is introduced to automate the verification process, facilitating broader applications in quantum mechanics and mathematical physics. Additionally, the Yang–Baxter equation is applied to spin transformations in quantum mechanical spin-12 systems, with transformations like rotations and reflections modeled using GE-algebras. A Cayley table is used to represent the algebraic structure of these transformations, and the proposed algorithm ensures that these solutions are consistent with the Yang–Baxter equation, offering new insights into the role of GE-algebras in quantum spin systems.

Entwining Yang–Baxter maps over Grassmann algebras

In this work we construct novel solutions to the set-theoretical entwining Yang–Baxter equation. These solutions are birational maps involving non-commutative dynamical variables which are elements of the Grassmann algebra of order n. The maps arise from refactorisation problems of Lax supermatrices associated to a nonlinear Schrödinger equation. In this non-commutative setting, we construct a spectral curve associated to each of the obtained maps using the characteristic function of its monodromy supermatrix. We find generating functions of invariants for the entwining Yang–Baxter maps from the moduli of the spectral curves. Moreover, we show that a hierarchy of birational entwining Yang–Baxter maps with commutative variables can be obtained by fixing the order n of the Grassmann algebra, and we present the cases n=1 (dual numbers) and n=2. Then we discuss the integrability properties, such as Lax matrices, invariants, and measure preservation, for the obtained discrete dynamical systems.

Bialgebras and related algebras for Novikov-Poisson algebras

In this paper, we develop the bialgebra theory for Novikov-Poisson algebras and establish the equivalence between matched pairs, Manin triples and Novikov-Poisson bialgebras. We study coboundary Novikov-Poisson bialgebras which leads to the introduction of the Novikov-Poisson Yang-Baxter equation (NPYBE). A skew-symmetric solution of the NPYBE naturally gives a coboundary Novikov-Poisson bialgebra. Moreover, O -operators and related algebras are considered.

Hybrid algorithm for the time-dependent Hartree–Fock method using the Yang–Baxter equation on quantum computers**The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (‘Argonne’). Argonne, a U.S. Department of Energy Office of Science laboratory, operated under Contract No. DE-AC02-06CH11357 and Pacific Northwest National Laboratory (PNNL), a U.S. Department of Energy Office of Science laboratory, operated

Hybrid algorithm for the time-dependent Hartree–Fock method using the Yang–Baxter equation on quantum computers**The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (‘Argonne’). Argonne, a U.S. Department of Energy Office of Science laboratory, operated under Contract No. DE-AC02-06CH11357 and Pacific Northwest National Laboratory (PNNL), a U.S. Department of Energy Office of Science laboratory, operated

Crossing symmetry and the crossing map

We introduce and study the crossing map, a closed linear map acting on operators on the tensor square of a given Hilbert space that is inspired by the crossing property of quantum field theory. This map turns out to be closely connected to Tomita–Takesaki modular theory. In particular, crossing symmetric operators, namely those operators that are mapped to their adjoints by the crossing map, define endomorphisms of standard subspaces. Conversely, such endomorphisms can be integrated to crossing symmetric operators. We also investigate the relation between crossing symmetry and natural compatibility conditions with respect to unitary representations of certain symmetry groups, and furthermore introduce a generalized crossing map defined by a real object in an abstract [Formula: see text]-tensor category, not necessarily consisting of Hilbert spaces and linear maps. This latter crossing map turns out to be closely related to the (unshaded, finite-index) subfactor theoretical Fourier transform. Lastly, we provide families of solutions of the crossing symmetry equation, solving in addition the categorical Yang–Baxter equation, associated with an arbitrary Q-system.

Yang-Baxter Equations and Relative Rota-Baxter Operators for Left-Alia Algebras Associated to Invariant Theory

Left-Alia algebras are a class of algebras with symmetric Jacobi identities. They contain several typical types of algebras as subclasses, and are closely related to the invariant theory. In this paper, we study the construction theory of left-Alia bialgebras. We introduce the notion of the left-Alia Yang-Baxter equation. We show that an antisymmetric solution of the left-Alia Yang-Baxter equation gives rise to a left-Alia bialgebra that we call triangular. The notions of relative Rota-Baxter operators of left-Alia algebras and pre-left-Alia algebras are introduced to provide antisymmetric solutions of the left-Alia Yang-Baxter equation.

Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical $R$-Matrices for Superspin Chains from the Bethe/Gauge Correspondence

We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, classical Higgs branches of 3d $\mathcal N=2$ quiver gauge theories. The Bethe/gauge correspondence relates such a gauge theory to an anisotropic/elliptic superspin chain, and the stable envelopes compute the $R$-matrix that solves the dynamical Yang-Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE for the elliptic $\mathfrak{sl}(1|1)$ spin chain with fundamental representations using the corresponding 3d $\mathcal N=2$ SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on $I \times \mathbb E$ for an interval $I$ and an elliptic curve $\mathbb E$ compute the elliptic stable envelopes, and in turn the geometric elliptic $R$-matrix, of the anisotropic $\mathfrak{sl}(1|1)$ spin chain. Furthermore, we consider the 2d and 1d reductions of elliptic stable envelopes and the $R$-matrix. The reduction to 2d gives the K-theoretic stable envelopes and the trigonometric $R$-matrix, and a further reduction to 1d produces the cohomological stable envelopes and the rational $R$-matrix. The latter recovers Rimányi-Rozansky's results that appeared recently in the mathematical literature.

Cohomology and extensions of relative Rota–Baxter groups

Relative Rota–Baxter groups are generalisations of Rota–Baxter groups and recently shown to be intimately related to skew left braces, which are well-known to yield bijective non-degenerate solutions to the Yang–Baxter equation. In this paper, we develop an extension theory of relative Rota–Baxter groups and introduce their low dimensional cohomology groups, which are distinct from the ones known in the context of Rota–Baxter operators on Lie groups. We establish an explicit bijection between the set of equivalence classes of extensions of relative Rota–Baxter groups and their second cohomology. Further, we delve into the connections between this cohomology and the cohomology of associated skew left braces. We prove that for bijective relative Rota–Baxter groups, the two cohomologies are isomorphic in dimension two.

Some properties of relative Rota–Baxter operators on groups

We find connection between relative Rota–Baxter operators and usual Rota–Baxter operators. We prove that any relative Rota–Baxter operator on a group H with respect to ( G , Ψ ) defines a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G . On the other side, we give condition under which a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G defines a relative Rota–Baxter operator on H with respect to ( G , Ψ ) . We introduce homomorphic post-groups and find their connection with λ-homomorphic skew left braces. Further, we construct post-group on arbitrary group and a family of post-groups which depends on integer parameter on any two-step nilpotent group. We find all verbal solutions of the quantum Yang-Baxter equation on two-step nilpotent group.

The yang-baxter equation, quantum computing and quantum entanglement

We present a method to construct infinite families of entangling (and primitive) 2-qudit gates, and amongst them entangling (and primitive) 2-qudit gates which satisfy the Yang-Baxter equation. We show that, given 2-qudit gates c and d, if c or d is entangling, then their Tracy-Singh product c ⊠ d is also entangling and we can provide decomposable states which become entangled after the application of c ⊠ d.

An integrability illusion: Vanishing transfer matrices associated with generalised Gaudin superalgebras

The Lie superalgebra gl(m|n) admits irreducible, finite-dimensional representations that continuously depend on a free parameter. For each representation, we define a generalised Gaudin superalgebra through an associated solution of the classical Yang-Baxter equation. The universal enveloping superalgebra of the Gaudin superalgebra formally contains a commuting family of transfer matrices, given by a universal expression. It will be shown that in many instances these transfer matrices are identically zero in the universal enveloping superalgebra. We offer an alternative formulation for identifying transfer matrices.

Indecomposable involutive set-theoretical solutions to the Yang–Baxter equation of size p 2

This article focuses on indecomposable involutive non-degenerate set-theoretical solutions to the Yang–Baxter equation. More specifically, we give a full classification of those solutions which are of size p 2 , for p a prime. We do this through a thorough analysis of their associated permutation braces and using the language of cycle sets.

On a new class of non-dynamical ABCD algebras for classical and quantum integrable systems

We consider classical and quantum non-dynamical quadratic abcd Lax algebras with classical and quantum gl(n)⊗gl(n)-valued abcd-tensors satisfying a set of quadratic non-dynamical Yang-Baxter-type equations generalizing those of Fredel and Maillet [1]. We establish a relation of some of these equations with the so-called “semi-dynamical” Yang-Baxter equations of [2]. We show that the linearization of the corresponding quadratic structures lead to linear tensor structures with the classical gl(n)⊗gl(n)-valued r-matrices satisfying usual “permuted” classical Yang-Baxter equations [1,3–5]. We consider example of our construction associated with the deformed Zn-graded r-matrix of [9–11] and explicitly construct the corresponding abcd-tensors — both classical and quantum. Small n examples are also considered in some details.

Novel non‐involutive solutions of the Yang–Baxter equation from (skew) braces

AbstractWe produce novel non‐involutive solutions of the Yang–Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from braces and skew braces, and surprisingly in the case of braces, they are not necessarily involutive. In the case of two‐sided (skew) braces, one can assign such solutions to every element of the set. Novel bijective maps associated to the inverse solutions are also introduced. Moreover, we show that the recently derived Drinfeld twists of the involutive case are still admissible in the non‐involutive frame, and we identify the twisted ‐matrices and twisted coproducts. We observe, as in the involutive case, that the underlying quantum algebra is not a quasi‐triangular bialgebra, as one would expect, but a quasi‐triangular quasi‐bialgebra. The same applies to the quantum algebra of the twisted ‐matrices, contrary to the involutive case.

Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices

AbstractWe present connections between left non-degenerate solutions of the set-theoretic braid equation and left shelves using Drinfel’d homomorphisms. We generalize the notion of affine quandle, by using heap endomorphisms and metahomomorphisms, and identify the underlying Yang–Baxter algebra for solutions of the braid equation associated to a given quandle. We introduce the notion of the pre-Lie skew brace and identify certain affine quandles that give rise to pre-Lie skew braces. Generalisations of the braiding of a group, associated to set-theoretic solutions of the braid equation are also presented. These generalized structures encode part of the underlying Hopf algebra. We then introduce the quasi-triangular (quasi) Hopf algebras and universalR-matrices for rack and set-theoretic algebras. Generic set-theoretic solutions coming from heap endomorphisms are also identified.

Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras

Bethe Ansatz was discovered in 1932. Half a century later its algebraic structure was unearthed: Yang-Baxter equation was discovered, as well as its multidimensional generalizations [tetrahedron equation and d-simplex equations]. Here we describe a universal method to solve these equations using Clifford algebras. The Yang-Baxter equation (d=2), Zamolodchikov's tetrahedron equation (d=3) and the Bazhanov-Stroganov equation (d=4) are special cases. Our solutions form a linear space. This helps us to include spectral parameters. Potential applications are discussed.

Colored vertex models and Iwahori Whittaker functions

We give a recursive method for computing all values of a basis of Whittaker functions for unramified principal series invariant under an Iwahori or parahoric subgroup of a split reductive group G over a nonarchimedean local field F. Structures in the proof have surprising analogies to features of certain solvable lattice models. In the case G=GLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=\ extrm{GL}_r$$\\end{document} we show that there exist solvable lattice models whose partition functions give precisely all of these values. Here ‘solvable’ means that the models have a family of Yang–Baxter equations which imply, among other things, that their partition functions satisfy the same recursions as those for Iwahori or parahoric Whittaker functions. The R-matrices for these Yang–Baxter equations come from a Drinfeld twist of the quantum group Uq(gl^(r|1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_q(\\widehat{\\mathfrak {gl}}(r|1))$$\\end{document}, which we then connect to the standard intertwining operators on the unramified principal series. We use our results to connect Iwahori and parahoric Whittaker functions to variations of Macdonald polynomials.

Novel ASEP-inspired solutions of the Yang-Baxter equation

Abstract We explore the algebraic structure of a particular ansatz of the Yang-Baxter equation (YBE), which is inspired by the Bethe Ansatz treatment of the asymmetric simple exclusion process spin-model. Various classes of Hamiltonian density arriving from the two types of R-matrices are found, which also appear as solutions of the constant YBE. We identify the idempotent and nilpotent categories of such constant R-matrices and perform a rank-1 numerical search for the lowest dimension. A summary of the final results reveals general non-Hermitian spin-1/2 chain models.