Classical Yang-Baxter Equation Research Articles

Double Constructions of Compatible Associative Algebras

A compatible associative algebra is a pair of associative algebras satisfying that any linear combination of the two associative products is still an associative product. We construct a compatible associative algebra with a decomposition into the direct sum of the underlying vector space of another compatible associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant. This compatible associative algebra is equivalent to a certain bialgebra structure of compatible associative algebras, which is an analogue of a Lie bialgebra. Many properties of the bialgebra are presented. In particular, the coboundary bialgebra theory leads to the system of associative Yang–Baxter equations in compatible associative algebras, which is an analogue of the classical Yang–Baxter equation in a Lie algebra. Furthermore, the bialgebra can also be regarded as a “compatible version” of antisymmetric infinitesimal bialgebras, that is, a pair of antisymmetric infinitesimal bialgebras satisfying any linear combination of them is still an antisymmetric infinitesimal bialgebra.

Tri-vector deformations in d = 11 supergravity

We construct a d = 11 supergravity analogue of the open-closed string map in the context of SL(5) Exceptional Field Theory (ExFT). The deformation parameter tri-vector Ω generalizes the non-commutativity bi-vector parameter Θ of the open string. When applied to solutions in d = 11, this map provides an economical way of performing TsT deformations, and may be used to recover d = 10 Yang-Baxter deformations after dimensional reduction. We present a generalization of the Classical Yang-Baxter Equation (CYBE) for rank 3 objects, which emerges from d = 11 supergravity and the SL(5) ExFT. This equation is shown to reduce to the d = 10 CYBE upon dimensional reduction.

Marginal deformations of WZW models and the classical Yang–Baxter equation

We show how so-called Yang–Baxter (YB) deformations of sigma models, based on an R-matrix solving the classical Yang–Baxter equation (CYBE), give rise to marginal current–current deformations when applied to the Wess–Zumino–Witten (WZW) model. For non-compact groups these marginal deformations are more general than the ones usually considered, since they can involve a non-Abelian current subalgebra. We classify such deformations of the string.

Deformations and Their Controlling Cohomologies of $${\mathcal{O}}$$ O -Operators

$${\mathcal{O}}$$ -operators are important in broad areas of mathematics and physics, such as integrable systems, the classical Yang–Baxter equation, pre-Lie algebras and splitting of operads. In this paper, a deformation theory of $${\mathcal{O}}$$ -operators is established in consistence with the general principles of deformation theories. On the one hand, $${\mathcal{O}}$$ -operators are shown to be characterized as the Maurer–Cartan elements in a suitable graded Lie algebra. A given $${\mathcal{O}}$$ -operator gives rise to a differential graded Lie algebra whose Maurer–Cartan elements characterize deformations of the given $${\mathcal{O}}$$ -operator. On the other hand, a Lie algebra with a representation is identified from an $${\mathcal{O}}$$ -operator T such that the corresponding Chevalley–Eilenberg cohomology controls deformations of T, thus can be regarded as an analogue of the Andre–Quillen cohomology for the $${\mathcal{O}}$$ -operator. Thereafter, linear and formal deformations of $${\mathcal{O}}$$ -operators are studied. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order n deformations of an $${\mathcal{O}}$$ -operator are also characterized in terms of the new cohomology theory. Applications are given to deformations of Rota–Baxter operators of weight 0 and skew-symmetric r-matrices for the classical Yang–Baxter equation. For skew-symmetric r-matrices, there is an independent Maurer–Cartan characterization of the deformations as well as an analogue of the Andre–Quillen cohomology, which turn out to have an explicit relationship with the ones obtained as $${\mathcal{O}}$$ -operators associated to the coadjoint representations. Finally, linear deformations of skew-symmetric r-matrices and their corresponding triangular Lie bialgebras are studied.

Classical Yang-Baxter equation from \u03b2-supergravity

Yang-Baxter deformations of superstring σ-models have recently inspired a supergravity solution generating technique. Using the open/closed string map and a Killing bi-vector as a deformation parameter, new solutions can be built, such that the (generalised) supergravity field equations were conjectured to always reduce to the classical Yang-Baxter equation (CYBE) [1]. In this work we provide a proof of this conjecture, using a systematic approach based on the so-called β-supergravity, which is a dynamical theory for the field βmn instead of the NSNS 2-form bmn.

Supergravity backgrounds of the \u03b7-deformed AdS2 \xd7 S2 \xd7 T6 and AdS5 \xd7 S5 superstrings

We construct supergravity backgrounds for the integrable η-deformations of the AdS2 × S2 × T6 and AdS5 × S5 superstring sigma models. The η-deformation is governed by an R-matrix that solves the non-split modified classical Yang-Baxter equation on the superisometry algebra of the model. Such R-matrices include those of Drinfel’d-Jimbo type, which are constructed from a Dynkin diagram and the associated Cartan-Weyl basis. Drinfel’d-Jimbo R-matrices associated with inequivalent bases will typically lead to different deformed backgrounds. For the two models under consideration we find that the unimodularity condition, implying that there is no Weyl anomaly, is satisfied if and only if all the simple roots are fermionic. For AdS2 × S2 × T6 we construct backgrounds corresponding to the three Dynkin diagrams. When all the simple roots are fermionic we find a supergravity background previously obtained by directly solving the supergravity equations. For AdS5 × S5 we construct a supergravity background corresponding to the Dynkin diagram with all fermionic simple roots.

Yang-Baxter deformations of the AdS4\u2009\xd7\u2009\u2102\u21193 superstring sigma model

The gravity dual of β-deformed ABJM theory can be obtained by a TsT transformation of AdS4 × ℂℙ3. We present a supercoset construction of ℂℙ3 to obtain this gravity dual theory as a Yang-Baxter deformation. This is done by selecting a convenient combination of Cartan generators in order to get an Abelian r-matrix satisfying the classical Yang-Baxter equation. Our results provide another illustration of the relation between Abelian r-matrices and TsT transformations.

Embedding the modified CYBE in supergravity

It has recently been demonstrated that given a generic solution, the Classical Yang–Baxter Equation (CYBE) emerges from supergravity via the open-closed string map, thus providing tangible evidence for the conjectured equivalence between supergravity equations of motion and the homogeneous CYBE. To date, study of this equivalence has largely been confined to the NS sector. In this work, we make two extensions. First, we revisit the transformation of the RR sector and clarify its precise role in the emergence of the CYBE. Secondly, we identify direct products of coset geometries as the only setting where the transformation permits embeddings of the modified CYBE. We illustrate our solution generating technique with deformations of AdS_3 times S^3 times M_4, where M_4 = T^4 (K3) and S^3 times S^1, and explicitly construct one and two-parameter integrable q-deformations that are solutions to generalised supergravity.

Subprime solutions of the classical Yang–Baxter equation

We introduce a new family of classical r-matrices for the Lie algebra sln that lies in the Zariski boundary of the Belavin–Drinfeld space M of quasi-triangular solutions to the classical Yang–Baxter equation. In this setting M is a finite disjoint union of components; exactly ϕ(n) of these components are SLn-orbits of single points. These points are the generalized Cremmer–Gervais r-matrices ri,n which are naturally indexed by pairs of positive coprime integers, i and n, with i<n. A conjecture of Gerstenhaber and Giaquinto states that the boundaries of the Cremmer–Gervais components contain r-matrices having maximal parabolic subalgebras pi,n⊆sln as carriers. We prove this conjecture in the cases when n≡±1 (mod i). The subprime linear functionals f∈pi,n⁎ and the corresponding principal elements H∈pi,n play important roles in our proof. Since the subprime functionals are Frobenius precisely in the cases when n≡±1 (mod i), this partly explains our need to require these conditions on i and n. We conclude with a proof of the GG boundary conjecture in an unrelated case, namely when (i,n)=(5,12), where the subprime functional is no longer a Frobenius functional.

Non-abelian T-duality and Yang-Baxter deformations of Green-Schwarz strings

We perform non-abelian T-duality for a generic Green-Schwarz string with respect to an isometry (super)group G, and we derive the transformation rules for the supergravity background fields. Specializing to G bosonic, or G fermionic but abelian, our results reproduce those available in the literature. We discuss also continuous deformations of the T-dual models, obtained by adding a closed B-field before the dualization. This idea can also be used to generate deformations of the original (un-dualized) model, when the 2-cocycle identified from the closed B is invertible. The latter construction is the natural generalization of the so-called Yang-Baxter deformations, based on solutions of the classical Yang-Baxter equation on the Lie algebra of G and originally constructed for group manifolds and (super)coset sigma models. We find that the deformed metric and B-field are obtained through a generalization of the map between open and closed strings that was used also in the discussion by Seiberg and Witten of non-commutative field theories. When applied to integrable sigma models these deformations preserve the integrability.

Classical Yang-Baxter equation from supergravity

We promote the open-closed string map, originally formulated by Seiberg \& Witten, to a solution generating prescription in generalized supergravity. The approach hinges on a knowledge of an antisymmetric bivector $\Theta$, built from antisymmetric products of Killing vectors, which is specified by the equations of motion. In the cases we study, the equations of motion reproduce the Classical Yang-Baxter equation (CYBE) and $\Theta$ is the most general $r$-matrix solution. Our work generalizes Yang-Baxter deformations to non-coset spaces and unlocks gravity as a means to classify $r$-matrix solutions to the CYBE.

Modified Rota–Baxter Algebras, Shuffle Products and Hopf Algebras

In this paper, we begin a systematic study of modified Rota–Baxter algebras, as an associative analogue of the modified classical Yang–Baxter equation. We construct free commutative modified Rota–Baxter algebras by a variation of the shuffle product and describe the structure both recursively and explicitly. We then provide these algebras with a Hopf algebra structure by applying a Hochschild cocycle.

Torsion Free Sheaves on Weierstrass Cubic Curves and the Classical Yang–Baxter Equation

This work deals with an algebro–geometric theory of solutions of the classical Yang–Baxter equation based on torsion free coherent sheaves of Lie algebras on Weierstras cubic curves.

Yang-Baxter deformations beyond coset spaces (a slick way to do TsT)

Yang-Baxter string sigma-models provide a systematic way to deform coset geometries, such as AdSp × Sp, while retaining the σ-model integrability. It has been shown that the Yang-Baxter deformation in target space is simply an open-closed string map that can be defined for any geometry, not just coset spaces. Given a geometry with an isometry group and a bivector that is assumed to be a linear combination of antisymmetric products of Killing vectors, we show the equations of motion of (generalized) supergravity reduce to the Classical Yang-Baxter Equation associated with the isometry group, proving the statement made in [1]. These results bring us closer to the proof of the “YB solution generating technique” for (generalized) supergravity advertised in [1] and in particular provide an economical way to perform TsT transformations.

Conformal twists, Yang–Baxter σ-models & holographic noncommutativity

Expanding upon earlier results (Araujo et al 2017 Phys. Rev. D 95 105006), we present a compendium of σ-models associated with integrable deformations of AdS5 generated by solutions to homogenous classical Yang–Baxter equation. Each example we study from four viewpoints: conformal (Drinfeld) twists, closed string gravity backgrounds, open string parameters and proposed dual noncommutative (NC) gauge theory. Irrespective of whether the deformed background is a solution to supergravity or generalized supergravity, we show that the open string metric associated with each gravity background is undeformed AdS5 with constant open string coupling and the NC structure Θ is directly related to the conformal twist. One novel feature is that Θ exhibits ‘holographic noncommutativity’: while it may exhibit non-trivial dependence on the holographic direction, its value everywhere in the bulk is uniquely determined by its value at the boundary, thus facilitating introduction of a dual NC gauge theory. We show that the divergence of the NC structure Θ is directly related to the unimodularity of the twist. We discuss the implementation of an outer automorphism of the conformal algebra as a coordinate transformation in the AdS bulk and discuss its implications for Yang–Baxter σ-models and self-T-duality based on fermionic T-duality. Finally, we comment on implications of our results for the integrability of associated open strings and planar integrability of dual NC gauge theories.

Invariant Poisson–Nijenhuis structures on Lie groups and classification

We study right-invariant (respectively, left-invariant) Poisson–Nijenhuis structures ([Formula: see text]-[Formula: see text]) on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra [Formula: see text]. We show that [Formula: see text]-[Formula: see text] structures can be used to find compatible solutions of the classical Yang–Baxter equation (CYBE). Conversely, two compatible [Formula: see text]-matrices from which one is invertible determine an [Formula: see text]-[Formula: see text] structure. We classify, up to a natural equivalence, all [Formula: see text]-matrices and all [Formula: see text]-[Formula: see text] structures with invertible [Formula: see text] on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by [Formula: see text]-matrices on a phase space whose symmetry group is Lie group a [Formula: see text], can be specifically determined.

Classical r-matrices for the generalised Chern–Simons formulation of 3d gravity

We study the conditions for classical r-matrices to be compatible with the generalised Chern–Simons action for 3d gravity. Compatibility means solving the classical Yang–Baxter equations with a prescribed symmetric part for each of the real Lie algebras and bilinear pairings arising in the generalised Chern–Simons action. We give a new construction of r-matrices via a generalised complexification and derive a non-linear set of matrix equations determining the most general compatible r-matrix. We exhibit new families of solutions and show that they contain some known r-matrices for special parameter values.

Yang-Baxter deformations of supercoset sigma models with ℤ4m grading**Supported by National Natural Science Foundation of China (11375141, 11425522, 11547050), Natural Science Foundation of Shaanxi Province (2013JQ1011, 2017ZDJC-32, 2016JM1027), Special Foundation for Basic Scientific Research of Central Colleges (310812152001, 310812172001, 2013G1121082, CHD2012JC019), Scientific Research Program Funded by Shaanxi Provincial Education Department (2013JK0628), Xi’an Shiyou University Science and Technology Foundation (2010QN018) and

We have studied Yang-Baxter deformations of supercoset sigma models with ℤ4m grading. The deformations are specified by a skew-symmetric classical r-matrix satisfying the classical Yang-Baxter equations. The deformed action is constructed and the Lax pair is also presented. When m=1, our results reduce to those of the type IIB Green-Schwarz superstring on AdS5×S5 background recently given by Kawaguchi, Matsumoto and Yoshida.

Simple vector bundles on a nodal Weierstrass cubic and quasi-trigonometric solutions of the classical Yang–Baxter equation**Dedicated to the memory of Petr Petrovich Kulish.

In this paper we study the combinatorics of quasi-trigonometric solutions of the classical Yang–Baxter equation, arising from simple vector bundles on a nodal Weierstraß cubic.

3-Lie bialgebras and 3-Lie classical Yang–Baxter equations in low dimensions

In this paper, we give low-dimensional examples of local cocycle 3-Lie bialgebras and double construction 3-Lie bialgebras which were introduced in the study of the classical Yang–Baxter equation and Manin triples for 3-Lie algebras. We give an explicit and practical formula to compute the skew-symmetric solutions of the 3-Lie classical Yang–Baxter equation (CYBE). As an illustration, we obtain all skew-symmetric solutions of the 3-Lie CYBE in complex 3-Lie algebras of dimensions 3 and 4 and then the induced local cocycle 3-Lie bialgebras. On the other hand, we classify the double construction 3-Lie bialgebras for complex 3-Lie algebras in dimensions 3 and 4 and then give the corresponding eight-dimensional pseudo-metric 3-Lie algebras.