Classical Yang-Baxter Equation Research Articles

Yang-Baxter deformations of Minkowski spacetime

We study Yang-Baxter deformations of 4D Minkowski spacetime. The Yang-Baxter sigma model description was originally developed for principal chiral models based on a modified classical Yang-Baxter equation. It has been extended to coset curved spaces and models based on the usual classical Yang-Baxter equation. On the other hand, for flat space, there is the obvious problem that the standard bilinear form degenerates if we employ the familiar coset Poincare group/Lorentz group. Instead we consider a slice of AdS5 by embedding the 4D Poincare group into the 4D conformal group SO(2, 4) . With this procedure we obtain metrics and B-fields as Yang-Baxter deformations which correspond to well-known configurations such as T-duals of Melvin backgrounds, Hashimoto-Sethi and Spradlin-Takayanagi-Volovich backgrounds, the T-dual of Grant space, pp-waves, and T-duals of dS4 and AdS4. Finally we consider a deformation with a classical r-matrix of Drinfeld-Jimbo type and explicitly derive the associated metric and B-field which we conjecture to correspond to a new integrable system.

A New Approach to Leibniz Bialgebras

A study of Leibniz bialgebras arising naturally through the double of Leibniz algebras analogue to the classical Drinfeld’s double is presented. A key ingredient of our work is the fact that the underline vector space of a Leibniz algebra becomes a Lie algebra and also a commutative associative algebra, when provided with appropriate new products. A special class of them, the coboundary Leibniz bialgebras, gives us the natural framework for studying the Yang-Baxter equation (YBE) in our context, inspired in the classical Yang-Baxter equation as well as in the associative Yang-Baxter equation. Results of the existence of coboundary Leibniz bialgebra on a symmetric Leibniz algebra under certain conditions are obtained. Some interesting examples of coboundary Leibniz bialgebras are also included. The final part of the paper is dedicated to coboundary Leibniz bialgebra structures on quadratic Leibniz algebras.

Homogeneous averaging operators on simple finite conformal Lie algebras

We describe all homogeneous averaging operators on finite simple conformal Lie algebras. In the case of current algebras, these operators are closely related to rational solutions of the classical Yang–Baxter equation.

Compatible Lie Bialgebras* *Supported by National Natural Science Foundation of China under Grant Nos. 11271202, 11221091, 11425104 and Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20120031110022

A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie algebras as an analogue of a Lie bialgebra. They can also be regarded as a “compatible version” of Lie bialgebras, that is, a pair of Lie bialgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra. Many properties of compatible Lie bialgebras as the “compatible version” of the corresponding properties of Lie bialgebras are presented. In particular, there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang–Baxter equation in compatible Lie algebras as a combination of two classical Yang–Baxter equations in Lie algebras. Furthermore, a notion of compatible pre-Lie algebra is introduced with an interpretation of its close relation with the classical Yang–Baxter equation in compatible Lie algebras which leads to a construction of the solutions of the latter. As a byproduct, the compatible Lie bialgebras fit into the framework to construct non-constant solutions of the classical Yang–Baxter equation given by Golubchik and Sokolov.

THE CLASSICAL HOM-YANG-BAXTER EQUATION AND HOM-LIE BIALGEBRAS

Motivated by recent work on Hom-Lie algebras and the HomYang-Baxter equation, we introduce a twisted generalization of the classical Yang-Baxter equation (CYBE), called the classical Hom-Yang-Baxter equation (CHYBE). We show how an arbitrary solution of the CYBE induces multiple infinite families of solutions of the CHYBE. We also introduce the closely related structure of Hom-Lie bialgebras, which generalize Drinfel’d’s Lie bialgebras. In particular, we study the questions of duality and cobracket perturbation and the sub-classes of coboundary and quasi-triangular Hom-Lie bialgebras.

On classical Yang-Baxter based deformations of the AdS5 × S5 superstring

Interesting deformations of AdS5 × S5 such as the gravity dual of noncommutative SYM and Schödinger spacetimes have recently been shown to be integrable. We clarify questions regarding the reality and integrability properties of the associated construction based on R matrices that solve the classical Yang-Baxter equation, and present an overview of manifestly real R matrices associated to the various deformations. We also discuss when these R matrices should correspond to TsT transformations, which not all do, and briefly analyze the symmetries preserved by these deformations, for example finding Schrödinger superalgebras that were previously obtained as subalgebras of $$ \mathfrak{p}\mathfrak{s}\mathfrak{u} $$ (2, 2|4). Our results contain a (singular) generalization of an apparently non-TsT deformation of AdS5 × S5, whose status as a string background is an interesting open question.

Schrödinger geometries arising from Yang-Baxter deformations

We present further examples of the correspondence between solutions of type IIB supergravity and classical $r$-matrices satisfying the classical Yang-Baxter equation (CYBE). In the previous works, classical $r$-matrices have been composed of generators of only one of either $\mathfrak{so}(2,4)$ or $\mathfrak{so}(6)$. In this paper, we consider some examples of $r$-matrices with both of them. The $r$-matrices of this kind contain (generalized) Schr\"odinger spacetimes and gravity duals of dipole theories. It is known that the generalized Schr\"odinger spacetimes can also be obtained via a certain class of TsT transformations called null Melvin twists. The metric and NS-NS two-form are reproduced by following the Yang-Baxter sigma-model description.

Vector bundles on plane cubic curves and the classical Yang–Baxter equation

In this article, we develop a geometric method to construct solutions of the classical Yang-Baxter equation, attaching to the Weierstrass family of plane cubic curves and a pair of coprime positive integers, a family of classical r-matrices. It turns out that all elliptic r-matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin's classification and prove that they are degenerations of the corresponding elliptic solutions.

Yang-Baxter deformations and string dualities

We further study integrable deformations of the AdS$_5\times$S$^5$ superstring by following the Yang-Baxter sigma model approach with classical $r$-matrices satisfying the classical Yang-Baxter equation (CYBE). Deformed string backgrounds specified by $r$-matrices are considered as solutions of type IIB supergravity, and therefore the relation between gravitational solutions and $r$-matrices may be called the gravity/CYBE correspondence. In this paper, we present a family of string backgrounds associated with a classical $r$-matrices carrying two parameters and its three-parameter generalization. The two-parameter case leads to the metric and NS-NS two-form of a solution found by Hubeny-Rangamani-Ross [hep-th/0504034] and another solution in [arXiv:1402.6147]. For all of the backgrounds associated with the three-parameter case, the metric and NS-NS two-form are reproduced by performing TsT transformations and S-dualities for the undeformed AdS$_5\times$S$^5$ background. As a result, one can anticipate the R-R sector that should be reproduced via a supercoset construction.

Yang–Baxter sigma models based on the CYBE

It is known that Yang–Baxter sigma models provide a systematic way to study integrable deformations of both principal chiral models and symmetric coset sigma models. In the original proposal and its subsequent development, the deformations have been characterized by classical r-matrices satisfying the modified classical Yang–Baxter equation (mCYBE). In this article, we propose the Yang–Baxter sigma models based on the classical Yang–Baxter equations (CYBE) rather than the mCYBE. This generalization enables us to utilize various kinds of solutions of the CYBE to classify integrable deformations. In particular, it is straightforward to realize partial deformations of the target space without loss of the integrability of the parent theory.

Deformations of T 1,1 as Yang-Baxter sigma models

We consider a family of deformations of T 1,1 in the Yang-Baxter sigma model approach. We first discuss a supercoset description of T 1,1, which makes manifest the full symmetry of the space and leads to the standard Sasaki-Einstein metric. Next, we consider three-parameter deformations of T 1,1 by using classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). The resulting metric and NS-NS two-form agree exactly with the ones obtained via TsT transformations, and contain the Lunin-Maldacena background as a special case. It is worth noting that for AdS5 × T 1,1 , classical integrability for the full sector has been argued to be lost. Hence our result indicates that the Yang-Baxter sigma model approach is applicable even for non-integrable cosets. This observation suggests that the gravity/CYBE correspondence can be extended beyond integrable cases.

Integrable deformations of the AdS5×S5 superstring and the classical Yang-Baxter equation - Towards the gravity/CYBE correspondence -

Based on the formulation of Yang-Baxter sigma models developed by Klimcik and Delduc-Magro-Vicedo, we explain that various deformations of type IIB superstring on AdS5 × S5 can be charactered by classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). The relation may be referred to as the gravity/CYBE correspondence. We present non-trivial examples of the correspondence including Lunin-Maldacena backgrounds for β-deformations of the N = 4 super Yang-Mills theory and the gravity duals for non-commutative gauge theories. We also discuss non-integrable backgrounds such as AdS5 × T1,1 as a generalization.

Note on the 3-Graded Modified Classical Yang-Baxter Equations and Integrable Systems

The 6=3×2 huge Lie algebra Ξ of all local and non-local differential operators on a circle is applied to the standard Adler-Kostant-Symes (AKS) R-bracket scheme. It is shown in particular that there exist three additional Lie structures, associated to three graded modified classical Yang-Baxter (GMCYB) equations. As we know from the standard case, these structures can be used to classify in a more consistent way a wide class of integrable systems. Other algebraic properties are also presented.

Exotic cluster structures on SL5

A conjecture by Gekhtman, Shapiro and Vainshtein suggests a correspondence between the Belavin–Drinfeld (BD) classification of solutions of the classical Yang–Baxter equation and cluster structures on simple Lie groups. This paper confirms the conjecture for SL5. Given a BD class, we construct the corresponding cluster structure in , and show that it satisfies all parts of the conjecture.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Cluster algebras in mathematical physics’.

The classical Yang–Baxter equation and the associated Yangian symmetry of gauged WZW-type theories

We construct the Lax-pair, the classical monodromy matrix and the corresponding solution of the Yang–Baxter equation, for a two-parameter deformation of the Principal chiral model for a simple group. This deformation includes as a one-parameter subset, a class of integrable gauged WZW-type theories interpolating between the WZW model and the non-Abelian T-dual of the principal chiral model. We derive in full detail the Yangian algebra using two independent methods: by computing the algebra of the non-local charges and alternatively through an expansion of the Maillet brackets for the monodromy matrix. As a byproduct, we also provide a detailed general proof of the Serre relations for the Yangian symmetry.

Parameter-dependent associative Yang–Baxter equations and Poisson brackets

We discuss associative analogues of classical Yang–Baxter equation (CYBE) meromorphically dependent on parameters. We discover that such equations enter in a description of a general class of parameter-dependent Poisson structures and double Lie and Poisson structures in sense of Van den Bergh. We propose a classification of all solutions for one-dimensional associative Yang–Baxter equations (AYBE).

Planck constant as spectral parameter in integrable systems and KZB equations

We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard (KZB) equations with $\tilde N$ punctures by deformation of the corresponding quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is $\tau$. At the level of classical mechanics the deformation parameter $\tau$ allows to extend the previously obtained modified Gaudin models to the modified Schlesinger systems. Next, we notice that the identities underlying generic (elliptic) KZB equations follow from some additional relations for the properly normalized $R$-matrices. The relations are noncommutative analogues of identities for (scalar) elliptic functions. The simplest one is the unitarity condition. The quadratic (in $R$ matrices) relations are generated by noncommutative Fay identities. In particular, one can derive the quantum Yang-Baxter equations from the Fay identities. The cubic relations provide identities for the KZB equations as well as quadratic relations for the classical $r$-matrices which can be halves of the classical Yang-Baxter equation. At last we discuss the $R$-matrix valued linear problems which provide ${\rm gl}_{\tilde N}$ Calogero-Moser (CM) models and Painleve equations via the above mentioned identities. The role of the spectral parameter plays the Planck constant of the quantum $R$-matrix. When the quantum ${\rm gl}_N$ $R$-matrix is scalar ($N=1$) the linear problem reproduces the Krichever's ansatz for the Lax matrices with spectral parameter for the ${\rm gl}_{\tilde N}$ CM models. The linear problems for the quantum CM models generalize the KZ equations in the same way as the Lax pairs with spectral parameter generalize those without it.

On left-symmetric conformal bialgebras

The notion of left-symmetric bialgebra was introduced in [C. M. Bai, Left-symmetric bialgebra and an analogue of the classical Yang–Baxter equation, Commun. Contemp. Math. 10(2) (2008) 221–260] which is equivalent to a parakähler Lie algebra which is the Lie algebra of a Lie group G with a G-invariant parakähler structure. In this paper, we study a conformal analog of left-symmetric bialgebras. The notions of left-symmetric conformal coalgebra and bialgebra are introduced. Moreover, the constructions of matched pairs of Lie conformal algebras and left-symmetric conformal algebras are presented. We show that a finite left-symmetric conformal bialgebra which is free as a ℂ[∂]-module is equivalent to a parakähler Lie conformal algebra. We also obtain a conformal analog of the S-equation (see [C. M. Bai, Left-symmetric bialgebras and an analogue of the classical Yang–Baxter equation, Commun. Contemp. Math. 10(2) (2008) 221–260]), and give a construction of the conformal symplectic double.

“Many-poled” r-matrix Lie algebras, Lax operators, and integrable systems

For each finite-dimensional simple Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}$\end{document}g, starting from a general \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}\otimes \mathfrak {g}$\end{document}g⊗g-valued solution r(u, v) of the generalized non-dynamical classical Yang-Baxter equation, we construct “N-poled” infinite-dimensional Lie algebras \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-,N}_r$\end{document}g̃r−,N of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}$\end{document}g-valued meromorphic functions with the poles in a fixed set of points ν1, …,νN. We apply the constructed algebras to the theory of finite-dimensional integrable systems and theory of soliton equations and obtain with their help the most general form of anisotropic chiral field-type equations as well as the most general form of integrable N-top systems and integrable cases of N interacting Kirchhoff systems.

Lunin-Maldacena backgrounds from the classical Yang-Baxter equation — towards the gravity/CYBE correspondence

We consider \gamma-deformations of the AdS_5xS^5 superstring as Yang-Baxter sigma models with classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). An essential point is that the classical r-matrices are composed of Cartan generators only and then generate abelian twists. We present examples of the r-matrices that lead to real \gamma-deformations of the AdS_5xS^5 superstring. Finally we discuss a possible classification of integrable deformations and the corresponding gravity solution in terms of solutions of CYBE. This classification may be called the gravity/CYBE correspondence.