Classical Yang-Baxter Equation Research Articles

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 [10–12] and explicitly construct the corresponding abcd-tensors — both classical and quantum. Small n examples are also considered in some details.

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.

Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras

It is known that the operads of perm algebras and pre-Lie algebras are the Koszul dual each other and hence there is a Lie algebra structure on the tensor product of a perm algebra and a pre-Lie algebra. Conversely, we construct a special perm algebra structure and a special pre-Lie algebra structure on the vector space of Laurent polynomials such that the tensor product with a pre-Lie algebra and a perm algebra being a Lie algebra structure characterizes the pre-Lie algebra and the perm algebra respectively. This is called the affinization of a pre-Lie algebra and a perm algebra respectively. Furthermore we extend such correspondences to the context of bialgebras, that is, there is a bialgebra structure for a perm algebra or a pre-Lie algebra which could be characterized by the fact that its affinization by a quadratic pre-Lie algebra or a quadratic perm algebra respectively gives an infinite-dimensional Lie bialgebra. In the case of perm algebras, the corresponding bialgebra structure is called a perm bialgebra, which can be independently characterized by a Manin triple of perm algebras as well as a matched pair of perm algebras. The notion of the perm Yang-Baxter equation is introduced, whose symmetric solutions give rise to perm bialgebras. There is a correspondence between symmetric solutions of the perm Yang-Baxter equation in perm algebras and certain skew-symmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras induced from the perm algebras. In the case of pre-Lie algebras, the corresponding bialgebra structure is a pre-Lie bialgebra which is well-constructed. The similar correspondences for the related structures are given.

Heterotic integrable deformation of the principal chiral model

A novel classically integrable model is proposed. It is a deformation of the two-dimensional principal chiral model, embedded into a heterotic σ model by a particular heterotic gauge field. This is inspired by the bosonic part of the heterotic σ model and its recent Hamiltonian formulation in terms of O(d,d+n)-generalized geometry in Hatsuda [Gauged double field theory, current algebras and heterotic sigma models, ]. Classical integrability is shown by the construction of a Lax pair and a classical R matrix. The latter is almost of the canonical form with a twist function and solves the classical Yang-Baxter equation. Published by the American Physical Society 2024

An Elliptic Solution of the Classical Yang–Baxter Equation Associated with the Queer Lie Superalgebra

A solution of the classical Yang–Baxter equation associated with the queer Lie superalgebra is constructed in terms of Hermite theta functions.

Deformations of modified $r$-matrices and cohomologies of related algebraic structures

Modified r -matrices are solutions of the modified classical Yang–Baxter equation, introduced by Semenov-Tian-Shansky, and play important roles in mathematical physics. In this paper, first we introduce a cohomology theory for modified r -matrices. Then we study three kinds of deformations of modified r -matrices using the established cohomology theory, including algebraic deformations, geometric deformations and linear deformations. We give the differential graded Lie algebra that governs algebraic deformations of modified r -matrices. For geometric deformations, we prove the rigidity theorem and study when is a neighborhood of a modified r -matrix smooth in the space of all modified r -matrix structures. In the study of trivial linear deformations, we introduce the notion of a Nijenhuis element for a modified r -matrix. Finally, applications are given to study deformations of the complement of the diagonal Lie algebra and compatible Poisson structures.

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.

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 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.

Chaotic string dynamics in bosonic η-deformed AdS5×T1,1 background

In this paper, we investigate a new class of [Formula: see text]-deformed [Formula: see text] backgrounds produced by classical [Formula: see text]-matrices that satisfy the modified classical Yang–Baxter equation [JHEP 03, 094 (2022)]. We consider the string sigma model on [Formula: see text]-deformed [Formula: see text] background. We numerically examined the classical phase space dynamics of these (semi)classical strings over this deformed background and computed several chaos indicators. These involve figuring out the Poincaré section and computing the Lyapunov exponents. In the (semi)classical limit, we discover evidence that supports a non-integrable phase space dynamics.

Classification of D-bialgebra structures on power series algebras

In this paper, we use algebro-geometric methods in order to derive classification results for so-called D-bialgebra structures on the power series algebra A〚z〛 for certain central simple non-associative algebras A. These structures are closely related to a version of the classical Yang-Baxter equation (CYBE) over A.If A is a Lie algebra, we obtain new proofs for pivotal steps in the known classification of non-degenerate topological Lie bialgebra structures on A〚z〛 as well as of non-degenerate solutions of the usual CYBE.If A is associative, we achieve the classification of non-triangular topological balanced infinitesimal bialgebra structures on A〚z〛 as well as of all non-degenerate solutions of an associative version of the CYBE.

Stokes phenomena, Poisson–Lie groups and quantum groups

Let g be a complex semisimple Lie algebra, b±⊂g a pair of opposite Borel subalgebras, and r∈b−⊗b+ the corresponding solution of the classical Yang–Baxter equations. Let G be the simply–connected Poisson–Lie group corresponding to (g,r), H⊂B±⊂G the subgroups with Lie algebras h=b−∩b+ and b±, and G⁎=B+×HB− the Poisson–Lie group dual of G. G–valued Stokes phenomena were used by Boalch [3,4] to give a canonical, analytic linearisation of the Poisson–Lie group structure on G⁎. Ug–valued Stokes phenomena were used by the first author to construct a twist killing the KZ associator, and therefore give a transcendental construction of the Drinfeld–Jimbo quantum group Uħg[23]. In the present paper, we show that the former construction can be obtained as semiclassical limit of the latter. Along the way, we also show that the R–matrix of Uħg is a Stokes matrix for the dynamical KZ equations.

All Jordanian deformations of the $AdS_5 \times S^5$ superstring

We explicitly construct and classify all Jordanian solutions of the classical Yang-Baxter equation on \mathfrak{psu}(2,2|4)𝔭𝔰𝔲(2,2|4), corresponding to Jordanian Yang-Baxter deformations of the AdS_5\times S^5AdS5 × S5 superstring. Such deformations preserve the classical integrability of the underlying sigma-model and thus are a subclass of all possible integrable deformations. The deformations that we consider are divided into two families, unimodular and non-unimodular ones. The former ensure that the deformed backgrounds are still solutions of the type IIB supergravity equations. For the simplest unimodular solutions, we find that the corresponding backgrounds preserve a number N < 32 of supercharges that can be N=12,8,6,4,0.

Infinite-Dimensional Lie Bialgebras via Affinization of Novikov Bialgebras and Koszul Duality

Balinsky and Novikov showed that the affinization of a Novikov algebra naturally defines a Lie algebra, a property that in fact characterizes the Novikov algebra. It is also an instance of the operadic Koszul duality. In this paper, we develop a bialgebra theory for the Novikov algebra, namely the Novikov bialgebra, which is characterized by the fact that its affinization (by a quadratic right Novikov algebra) gives an infinite-dimensional Lie bialgebra, suggesting a Koszul duality for properads. A Novikov bialgebra is also characterized as a Manin triple of Novikov algebras. The notion of Novikov Yang-Baxter equation is introduced, whose skewsymmetric solutions can be used to produce Novikov bialgebras and hence Lie bialgebras. Moreover, these solutions also give rise to skewsymmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras from the Novikov algebras.

Topological Manin pairs and (n,s)-type series

Lie subalgebras of L = {mathfrak {g}}(!(x)!) times {mathfrak {g}}[x]/x^n{mathfrak {g}}[x] , complementary to the diagonal embedding Delta of {mathfrak {g}}[![x]!] and Lagrangian with respect to some particular form, are in bijection with formal classical r-matrices and topological Lie bialgebra structures on the Lie algebra of formal power series {mathfrak {g}}[![x]!] . In this work we consider arbitrary subspaces of L complementary to Delta and associate them with so-called series of type (n, s) . We prove that Lagrangian subspaces are in bijection with skew-symmetric (n, s) -type series and topological quasi-Lie bialgebra structures on {mathfrak {g}}[![x]!] . Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type (n, s) , solving the generalized classical Yang-Baxter equation, correspond to subalgebras of L. We discuss their possible utility in the theory of integrable systems.

Tri-vector deformations on compact isometries

Classical Yang–Baxter equation governing bi-vector deformations of 10d supergravity is known to have no solutions along non-abelian compact isometries. By providing explicit examples we show that this is in contrast to generalized Yang–Baxter equation governing tri-vector deformations of 11d supergravity. We present deformations of the AdS{}_7times mathbb {S}^4 and flat backgrounds with isometries generated by Killing vectors of a sphere. Isometries of the AdS space-time are preserved by such deformations.

Yang-Baxter deformation of WZW model based on Lie supergroups: The cases of GL(1|1) and (C3 + A)

We proceed to generalize the Yang-Baxter (YB) deformation of Wess-Zumino-Witten (WZW) model to the Lie supergroups case. This generalization enables us to utilize various kinds of solutions of the (modified) graded classical Yang-Baxter equation ((m)GCYBE) to classify the YB deformations of WZW models based on the Lie supergroups. We obtain the inequivalent solutions (classical r-matrices) of the (m)GCYBE for the gl(1|1) and (C3+A) Lie superalgebras in the non-standard basis, in such a way that the corresponding automorphism transformations are employed. Then, the YB deformations of the WZW models based on the GL(1|1) and (C3+A) Lie supergroups are specified by skew-supersymmetric classical r-matrices satisfying (m)GCYBE. In some cases for both families of deformed models, the metrics remain invariant under the deformation, while the components of B-fields are changed. After checking the conformal invariance of the models up to one-loop order, it is concluded that the GL(1|1) and (C3+A) WZW models are conformal theories within the classes of the YB deformations preserving the conformal invariance. However, our results are interesting in themselves, but at a constructive level, may prompt many new insights into (generalized) supergravity solutions.

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.

Lax pairs for new [formula omitted]-symmetric coset σ-models and their Yang-Baxter deformations

Two-dimensional σ-models with ZN-symmetric homogeneous target spaces have been shown to be classically integrable when introducing WZ-terms in a particular way. This article continues the search for new models of this type now allowing some kinetic terms to be absent – analogously to the Green-Schwarz superstring σ-model on Z4-symmetric homogeneous spaces. A list of such integrable ZN-symmetric (super)coset σ-models for N≤6 and their Lax pairs is presented. For arbitrary N a big class of integrable models is constructed that includes both the known pure spinor and Green-Schwarz superstring on Z4-symmetric cosets.Integrable Yang-Baxter deformations of this class of ZN-symmetric (super)coset σ-models can be constructed in same way as in the known Z2- or Z4-cases. Deformations based on solutions of the modified classical Yang-Baxter equation, the so-called η-deformation, require deformation of the constants defining the Lagrangian and the corresponding Lax pair. Homogeneous Yang-Baxter deformations (i.e. those based on solutions to the classical Yang-Baxter equation) leave the equations of motion and consequently the Lax pair invariant and are expected to be classically equivalent to the undeformed model.As an example, the relationship between Z3-symmetric homogeneous spaces and nearly (para-)Kähler geometries is revisited. Confirming existing literature it is shown that the integrable choice of WZ-term in the Z3-symmetric coset σ-model associated to a nearly Kähler background gives an imaginary contribution to the action.

Discrete integrable equations on face-centred cubics: consistency and Lax pairs of corner equations

A new set of discrete integrable equations, called face-centred quad equations, was recently obtained using new types of interaction-round-a-face solutions of the classical Yang–Baxter equation. These equations satisfy a new formulation of multidimensional consistency, known as consistency-around-a-face-centred-cube (CAFCC), which requires consistency of an overdetermined system of 14 five-point equations on the face-centred cubic unit cell. In this paper, a new formulation of CAFCC is introduced where so-called type C equations are centred at faces of the face-centred cubic unit cell, whereas previously they were only centred at corners. This allows type C equations to be regarded as independent multidimensionally consistent integrable systems on higher dimensional lattices and is used to establish their Lax pairs.