Classical Yang-Baxter Equation Research Articles

Integrability of classical strings dual for noncommutative gauge theories

We derive the gravity duals of noncommutative gauge theories from the Yang-Baxter sigma model description of the AdS5 × S5 superstring with classical r-matrices. The corresponding classical r-matrices are 1) solutions of the classical Yang-Baxter equation (CYBE), 2) skew-symmetric, 3) nilpotent and 4) abelian. Hence these should be called abelian Jordanian deformations. As a result, the gravity duals are shown to be integrable deformations of AdS5 × S5. Then, abelian twists of AdS5 are also investigated. These results provide a support for the gravity/CYBE correspondence proposed in arXiv:1404.1838.

A Jordanian deformation of AdS space in type IIB supergravity

We consider a Jordanian deformation of the AdS_5xS^5 superstring action by taking a simple R-operator which satisfies the classical Yang-Baxter equation. The metric and NS-NS two-form are explicitly derived with a coordinate system. Only the AdS part is deformed and the resulting geometry contains the 3D Schrodinger spacetime as a subspace. Then we present the full solution in type IIB supergravity by determining the other field components. In particular, the dilaton is constant and a R-R three-form field strength is turned on. The symmetry of the solution is [SL(2,R)xU(1)^2] x [SU(3)xU(1)] and contains an anisotropic scale symmetry.

A deformation of quantum affine algebra in squashed Wess-Zumino-Novikov-Witten models

We proceed to study infinite-dimensional symmetries in two-dimensional squashed Wess-Zumino-Novikov-Witten models at the classical level. The target space is given by squashed S3 and the isometry is SU(2)L × U(1)R. It is known that SU(2)L is enhanced to a couple of Yangians. We reveal here that an infinite-dimensional extension of U(1)R is a deformation of quantum affine algebra, where a new deformation parameter is provided with the coefficient of the Wess-Zumino term. Then we consider the relation between the deformed quantum affine algebra and the pair of Yangians from the viewpoint of the left-right duality of monodromy matrices. The integrable structure is also discussed by computing the r/s-matrices that satisfy the extended classical Yang-Baxter equation. Finally, two degenerate limits are discussed.

On the local structure of noncommutative deformations

Let (M,π,D) be a Poisson manifold endowed with a flat, torsion-free contravariant connection. We show that if D is an F-connection then there exists a tensor T such that DT is the metacurvature tensor introduced by E. Hawkins in his work on noncommutative deformations. We compute T and the metacurvature tensor in this case and show that if T=0 then near any regular point π and D are defined in a natural way by a Lie algebra action and a solution of the classical Yang–Baxter equation. Moreover, when D is the contravariant Levi-Civita connection associated to π and a Riemannian metric, the Lie algebra action can be chosen in such a way that it preserves the metric. This solves the inverse problem of a result of the second author.

Lattices in some symplectic or affine nilpotent Lie groups

The main aim of this paper is the description of a large class of lattices in some nilpotent Lie groups, sometimes filiforms, carrying a flat left invariant linear connection and often a left invariant symplectic form. As a consequence we obtain an infinity of, non homeomorphic, compact affine or symplectic nilmanifolds.We review some new facts about the geometry of compact symplectic nilmanifolds and we describe symplectic reduction for these manifolds. For the Heisenberg–Lie group, defined over a local associative and commutative finite dimensional real algebra, a necessary and sufficient condition for the existence of a left invariant symplectic form, is given. Finally in the symplectic case we show that a lattice in the group determines naturally lattices in the double Lie group corresponding to any solution of the classical Yang–Baxter equation.

Classical exchange algebra of the superstring on S5 with AdS-time

A classical exchange algebra of the superstring on S5 with AdS-time is shown on the light-like plane. To this end, we use the geometrical method of which consistency is guaranteed by the classical Yang–Baxter equation. The Dirac method does not work, since there are constraints in which first-class and second-class constraints are mixed and one can hardly disentangle with each other keeping the isometry.

Rota-Baxter operators on $\mathrm{sl(2,\mathbb {C})}$ sl (2,C) and solutions of the classical Yang-Baxter equation

We explicitly determine all Rota-Baxter operators (of weight zero) on $\mathrm{sl(2,\mathbb {C})}$ sl (2,C) under the Cartan-Weyl basis. For the skew-symmetric operators, we give the corresponding skew-symmetric solutions of the classical Yang-Baxter equation in $\mathrm{sl(2,\mathbb {C})}$ sl (2,C), confirming the related study by Semenov-Tian-Shansky. In general, these Rota-Baxter operators give a family of solutions of the classical Yang-Baxter equation in the six-dimensional Lie algebra $\mathrm{sl(2,\mathbb {C})}\ltimes _{{\rm ad}^{\ast }} \mathrm{sl(2,\mathbb {C})}^{\ast }$ sl (2,C)⋉ ad * sl (2,C)*. They also give rise to three-dimensional pre-Lie algebras which in turn yield solutions of the classical Yang-Baxter equation in other six-dimensional Lie algebras.

Rota–Baxter operators on 4-dimensional complex simple associative algebras

Rota–Baxter operators or relations were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics. In this paper, we commence to study the Rota–Baxter operators of weight zero on a 4-dimensional complex simple associative algebra, which is isomorphic to the algebra of all 2×2 matrices over the field of complex numbers. Such operators satisfy (the operator form of) the classical Yang–Baxter equation on the general linear Lie algebra gl2(C). We provide two different solving methods: one by hand to show some tricks for solving a large nonlinear system, and another one by Computer Algebra to show the possibility of solving higher dimensional cases.

Infinite-dimensional Lie algebras, classical r-matrices, and Lax operators: Two approaches

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 solutions r(u, v) of the generalized classical Yang-Baxter equation, we construct infinite-dimensional Lie algebras \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}$\end{document}g-valued meromorphic functions. We outline two ways of embedding of the Lie algebra \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− into a larger Lie algebra with Kostant-Adler-Symmes decomposition. The first of them is an embedding of \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− into Lie algebra \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}(u^{-1},u))$\end{document}g̃(u−1,u)) of formal Laurent power series. The second is an embedding of \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− as a quasigraded Lie subalgebra into a quasigraded Lie algebra \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}_r$\end{document}g̃r: \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}_r=\widetilde{\mathfrak {g}}^{-}_r+\widetilde{\mathfrak {g}}^{+}_r$\end{document}g̃r=g̃r−+g̃r+, such that the Kostant-Adler-Symmes decomposition is consistent with a chosen quasigrading. We construct dual spaces \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^*_r$\end{document}g̃r*, \documentclass[12pt]{minimal}\begin{document}$(\widetilde{\mathfrak {g}}^{\pm }_r)^*$\end{document}(g̃r±)* and explicit form of the Lax operators L(u), L±(u) as elements of these spaces. We develop a theory of integrable finite-dimensional hamiltonian systems and soliton hierarchies based on Lie algebras \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}_r$\end{document}g̃r, \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{\pm }_r$\end{document}g̃r±. We consider examples of such systems and soliton equations and obtain the most general form of integrable tops, Kirchhoff-type integrable systems, and integrable Landau-Lifshitz-type equations corresponding to the Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}$\end{document}g.

Restoring unitarity in the q-deformed world-sheet S-matrix

The world-sheet S-matrix of the string in AdS5 x S5 has been shown to admit a q-deformation that relates it to the S-matrix of a generalization of the sine-Gordon theory, which arises as the Pohlmeyer reduction of the superstring. Whilst this is a fascinating development the resulting S-matrix is not explicitly unitary. The problem has been known for a long time in the context of S-matrices related to quantum groups. A braiding relation often called "unitarity" actually only corresponds to quantum field theory unitarity when the S-matrix is Hermitian analytic and quantum group S-matrices manifestly violate this. On the other hand, overall consistency of the S-matrix under the bootstrap requires that the deformation parameter is a root of unity and consequently one is forced to perform the "vertex" to IRF, or SOS, transformation on the states to truncate the spectrum consistently. In the IRF formulation unitarity is now manifest and the string S-matrix and the S-matrix of the generalised sine-Gordon theory are recovered in two different limits. In the latter case, expanding the Yang-Baxter equation we find that the tree-level S-matrix of the Pohlmeyer-reduced string should satisfy a modified classical Yang-Baxter equation explaining the apparent anomaly in the perturbative computation. We show that the IRF form of the S-matrix meshes perfectly with the bootstrap equations.

Classification of constant solutions of the associative Yang-Baxter equation on Mat3

We find all non-equivalent constant solutions for classical associative Yang-Baxter equation for $gl(3)$. New examples found in the classification yield the corresponding quadratic trace Poisson brackets, double Poisson structures on free associative algebra with three generators and anti-Frobenius associative algebras.

GAUGE FIXING AND QUANTUM GROUP SYMMETRIES IN (2+1)-GRAVITY

We summarize the results obtained by applying Dirac's gauge fixing formalism to the combinatorial description of the Chern–Simons formulation of (2+1)-gravity and their implications for the symmetries of the quantum theory. While the combinatorial description of the phase space exhibits standard Poisson–Lie symmetries, every gauge fixing condition based on two point particles yields a Poisson structure determined by a dynamical classical r-matrix. By considering transformations between different gauge fixing conditions, it is possible to classify all gauge fixed Poisson structures in terms of two standard solutions of the dynamical classical Yang–Baxter equation. We discuss the conclusions that can be drawn from this about the symmetries of (2+1)-dimensional quantum gravity.

RELATIVE ROTA–BAXTER OPERATORS AND TRIDENDRIFORM ALGEBRAS

A relative Rota–Baxter operator is a relative generalization of a Rota–Baxter operator on an associative algebra. In the Lie algebra context, it is called an [Formula: see text]-operator, originated from the operator form of the classical Yang–Baxter equation. We generalize the well-known construction of dendriform and tridendriform algebras from Rota–Baxter algebras to a construction from relative Rota–Baxter operators. In fact we give two such generalizations, on the domain and range of the operator respectively. We show that each of these generalized constructions recovers all dendriform and tridendriform algebras. Furthermore the construction on the range induces bijections between certain equivalence classes of invertible relative Rota–Baxter operators and tridendriform algebras.

On string theory on [formula omitted] with mixed 3-form flux: Tree-level S-matrix

We consider superstring theory on AdS3×S3×T4 supported by a combination of RR and NSNS 3-form fluxes (with parameter of the NSNS 3-form q). This theory interpolates between the pure RR flux model (q=0) whose spectrum is expected to be described by a (thermodynamic) Bethe ansatz and the pure NSNS flux model (q=1) which is described by the supersymmetric extension of the SL(2,R)×SU(2) WZW model. As a first step towards the solution of this integrable theory for generic value of q we compute the corresponding tree-level S-matrix for massive BMN-type excitations. We find that this S-matrix has a surprisingly simple dependence on q: the diagonal amplitudes have exactly the same structure as in the q=0 case but with the BMN dispersion relation e2=p2+1 replaced by the one with shifted momentum and mass, e2=(p±q)2+1−q2. The off-diagonal amplitudes are then determined from the classical Yang–Baxter equation. We also construct the Pohlmeyer-reduced model corresponding to this superstring theory and find that it depends on q only through the rescaled mass parameter, μ→1−q2μ, implying that its relativistic S-matrix is q-independent.

Consistently constrained SL(N) WZWN models and classical exchange algebra

Currents of the SL(N) WZWN model are constrained so that the remaining symmetry is a symmetry of constrained currents as well. Such consistency enables us to study the Poisson structure of constrained SL(N) WZWN models properly. We establish the Poisson brackets which satisfy the Jacobi identities owing to the classical Yang-Baxter equation. The Virasoro algebra is shown by using them. An SL(N) conformal primary is constructed. It satisfies a quadratic algebra, which might become an exchange algebra by its quantum deformation.

Pre-jordan Algebras

The purpose of this paper is to introduce and study a notion of pre-Jordan algebra. Pre-Jordan algebras are regarded as the underlying algebraic structures of the Jordan algebras with a nondegenerate symplectic form. They are the algebraic structures behind the Jordan Yang-Baxter equation and Rota-Baxter operators in terms of $\mathcal{O}$-operators of Jordan algebras introduced in this paper. Pre-Jordan algebras are analogues for Jordan algebras of pre-Lie algebras and fit into a bigger framework with a close relationship with dendriform algebras. The anticommutator of a pre-Jordan algebra is a Jordan algebra and the left multiplication operators give a representation of the Jordan algebra, which is the beauty of such a structure. Furthermore, we introduce a notion of $\mathcal{O}$-operator of a pre-Jordan algebra which gives an analogue of the classical Yang-Baxter equation in a pre-Jordan algebra.

Poisson bialgebras

We introduce a notion of Poisson bialgebra as an analogue of a Lie bialgebra of Drinfeld. Poisson bialgebras exhibit many familiar properties of Lie bialgebras. In particular, they can be constructed from a combination of the classical Yang-Baxter equation and the associative Yang-Baxter equation and there exists a natural Drinfeld classical double. Moreover, Poisson bialgebras are related to certain algebraic structures and they fit naturally into a framework to construct compatible Poisson brackets in integrable systems.

ON SOME LEFT-SYMMETRIC SUPERALGEBRAS

In this paper, we classify the left-symmetric superalgebras in dimensions two and three over the field of complex numbers. As a direct consequence, we obtain some explicit solutions of the graded classical Yang–Baxter equation in certain Lie superalgebras and their corresponding Lie bisuperalgebras.

𝒪-operators on associative algebras and associative Yang–Baxter equations

An O-operator on an associative algebra is a generalization of a Rota-Baxter operator that plays an important role in the Hopf algebra approach of Connes and Kreimer to the renormalization of quantum field theory. It is also the associative analog of an O-operator on a Lie algebra in the study of the classical Yang-Baxter equation. We introduce the concept of an extended O-operator on an associative algebra whose Lie algebra analog has been applied to generalized Lax pairs and PostLie algebras. We study algebraic structures coming from extended O-operators. Continuing the work of Aguiar deriving Rota-Baxter operators from the associative Yang-Baxter equation, we show that its solutions correspond to extended O-operators through a duality. We also establish a relationship of extended O-operators with the generalized associative Yang-Baxter equation.

Octo-bialgebras

The notion of octo-algebra was introduced by Leroux as a Loday algebra with 8 operations. In this paper, we introduce a notion of octo-bialgebra as a bialgebra theory of octo-algebras, which is equivalent to a double construction of a quadri-algebra with a nondegenerate 2-cocycle or a double construction of an octo-algebra with a nondegenerate invariant bilinear form. Some properties of octo-bialgebras are given, including the study of the coboundary cases which leads to a construction from an analogue of the classical Yang-Baxter equation in an octo-algebra.