Associative Yang-Baxter Equation Research Articles

On deformations of Rota–Baxter algebra morphisms

Rota–Baxter algebras are important in probability, combinatorics, associative Yang–Baxter equation and splitting of algebras. This paper studies the formal deformations of Rota–Baxter algebra morphisms. As a consequence, we develop a cohomology theory of Rota–Baxter algebra morphisms to interpret the lower degree cohomology groups as formal deformations. Finally, we prove the cohomology comparison theorem of Rota–Baxter algebra morphisms, i.e. the cohomology of a morphism of Rota–Baxter algebras is isomorphic to the cohomology of an auxiliary Rota–Baxter algebra.

Matching Rota–Baxter modules and matching 𝒪-operators of associative algebras

In this paper, firstly, we introduce the notion of matching Rota–Baxter modules, which generalizes the notions of matching Rota–Baxter algebras and Rota–Baxter modules, and give its characteristic and construction. Secondly, we give the relation between matching Rota–Baxter modules and other modules structure. Finally, we introduce the notion of matching [Formula: see text]-operators of associative algebras which also generalizes the notion of matching Rota–Baxter algebras and give the solution of polarized associative Yang–Baxter equation by using the matching [Formula: see text]-operators of associative algebras. In addition, we introduce the definitions of the compatible matching [Formula: see text]-operators of associative algebras, matching Nijenhui operators, and compatible matching dendriform algebras, and consider their connection.

Nijenhuis operators and associative D-bialgebras

Firstly, we prove that a quasitriangular associative D-bialgebra together with a coquasitriangular structure can induce a Nijenhuis algebra. This shows that Nijenhuis operators and associative D-bialgebras are closely related. Secondly, we investigate some properties of a representation of a Nijenhuis algebra and introduce the notion of a corepresentation of a Nijenhuis coalgebra, which also can be seen as an infinitesimal deformation of compatible bicomodule. Thirdly, a Nijenhuis associative D-bialgebra can be established by a Nijenhuis algebra and a Nijenhuis coalgebra satisfying some compatible conditions. Furthermore, we find that a Nijenhuis associative D-bialgebra is equivalent to a matched pair of Nijenhuis algebras or a double construction of a Nijenhuis Frobenius algebra. Lastly, we provide some methods to construct Nijenhuis associative D-bialgebras through admissible associative Yang-Baxter equations (aYBes for short) and O-operators on Nijenhuis algebras.

Higher-rank generalization of the 11-vertex rational $$R$$-matrix: IRF–vertex relations and the associative Yang–Baxter equation

Higher-rank generalization of the 11-vertex rational $$R$$-matrix: IRF–vertex relations and the associative Yang–Baxter equation

Higher-rank generalization of the 11-vertex rational $R$-matrix: IRF-vertex relations and the associative Yang-Baxter equation

Изучается $GL_N$ рациональная $R$-матрица, которая становится 11-вершинной $R$-матрицей в случае $N=2$. Описываются ее связи с динамическими и полудинамическими $R$-матрицами с использованием преобразования типа IRF-Vertex. Получена новая явная форма для $GL_N$ $R$-матрицы. Доказываются квантовое и ассоциативное уравнения Янга-Бакстера. Также доказан ряд других свойств и тождеств для данной $R$-матрицы.

Anisotropic Spin Generalization of Elliptic Macdonald–Ruijsenaars Operators and R-Matrix Identities

We propose commuting set of matrix-valued difference operators in terms of the elliptic Baxter-Belavin $R$-matrix in the fundamental representation of ${\rm GL}_M$. In the scalar case $M=1$ these operators are the elliptic Macdonald-Ruijsenaars operators, while in the general case they can be viewed as anisotropic versions of the quantum spin Ruijsenaars Hamiltonians. We show that commutativity of the operators for any $M$ is equivalent to a set of $R$-matrix identities. The proof of identities is based on the properties of elliptic $R$-matrix including the quantum and the associative Yang-Baxter equations. As an application of our results, we introduce elliptic generalization of q-deformed Haldane-Shastry model.

Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras

We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of O-operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras and the set of coherent derivations on an ASI bialgebra composes a Lie algebra which is the Lie algebra of the Lie group consisting of coherent automorphisms on this ASI bialgebra. Finally, we apply the study of differential ASI bialgebras to Poisson bialgebras, extending the construction of Poisson algebras from commutative differential algebras with two commuting derivations to the context of bialgebras, which is consistent with the well constructed theory of Poisson bialgebras. In particular, we construct Poisson bialgebras from differential Zinbiel algebras.

Gauge Equivalence Between 1 + 1 Rational Calogero–Moser Field Theory and Higher Rank Landau–Lifshitz Equation

In this paper we study 1 + 1 field generalization of the rational N-body Calogero–Moser model. We show that this model is gauge equivalent to some special higher rank matrix Landau–Lifshitz equation. The latter equation is described in terms of {text{G}}{{{text{L}}}_{N}} rational R-matrix, which turns into the 11-vertex R-matrix in the N = 2 case. The rational R-matrix satisfies the associative Yang–Baxter equation, which underlies construction of the Lax pair for the Zakharov–Shabat equation. The field analogue of the IRF-Vertex transformation is proposed. It allows to compute explicit change of variables between the field Calogero–Moser model and the Landau–Lifshitz equation.

Lax equations for relativistic Gaudin models on elliptic curve

We describe the most general GL NM classical elliptic finite-dimensional integrable system, which Lax matrix has n simple poles on elliptic curve. For M = 1 it reproduces the classical inhomogeneous spin chain, for N = 1 it is the Gaudin type (multispin) extension of the spin Ruijsenaars–Schneider model, and for n = 1 the model of M interacting relativistic GL N tops emerges in some particular case. In this way we present a classification for relativistic Gaudin models on GL-bundles over elliptic curve. As a by-product we describe the inhomogeneous Ruijsenaars chain. We show that this model can be considered as a particular case of multispin Ruijsenaars–Schneider model when residues of the Lax matrix are of rank one. An explicit parametrization of the classical spin variables through the canonical variables is obtained for this model. Finally, the most general GL NM model is also described through R-matrices satisfying associative Yang–Baxter equation. This description provides the trigonometric and rational analogues of GL NM models.

Higher Rank 1 + 1 Integrable Landau–Lifshitz Field Theories from the Associative Yang–Baxter Equation

We propose a construction of 1+1 integrable Heisenberg-Landau-Lifshitz type equations in the ${\rm gl}_N$ case. The dynamical variables are matrix elements of $N\times N$ matrix $S$ with the property $S^2={\rm const}\cdot S$. The Lax pair with spectral parameter is constructed by means of a quantum $R$-matrix satisfying the associative Yang-Baxter equation. Equations of motion for ${\rm gl}_N$ Landau-Lifshitz model are derived from the Zakharov-Shabat equations. The model is simplified when ${\rm rank}(S)=1$. In this case the Hamiltonian description is suggested. The described family of models includes the elliptic model coming from ${\rm GL}_N$ Baxter-Belavin elliptic $R$-matrix. In $N=2$ case the widely known Sklyanin's elliptic Lax pair for XYZ Landau-Lifshitz equation is reproduced. Our construction is also valid for trigonometric and rational degenerations of the elliptic $R$-matrix.

On elliptic solutions of the associative Yang–Baxter equation

We give a direct proof of the fact that elliptic solutions of the associative Yang–Baxter equation arise from appropriate spherical orders on an elliptic curve.

Admissible Poisson bialgebras

An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommutative and one anti-cocommutative comultiplications generalizes and illustrates the polarization–depolarization process in the context of bialgebras. The study of a special class of adm-Poisson bialgebras which include the known coboundary Poisson bialgebras in [27] as a proper subclass in general, illustrating an advantage in terms of the presentation with one operation, leads to the introduction of adm-Poisson Yang–Baxter equation in an adm-Poisson algebra. It is an unexpected consequence that both the adm-Poisson Yang–Baxter equation and the associative Yang–Baxter equation have the same form and thus it motivates and simplifies the involved study from the study of the associative Yang–Baxter equation, which is another advantage in terms of the presentation with one operation. A skew-symmetric solution of adm-Poisson Yang–Baxter equation gives an adm-Poisson bialgebra. Finally, the notions of an [Formula: see text]-operator of an adm-Poisson algebra and a pre-adm-Poisson algebra are introduced to construct skew-symmetric solutions of adm-Poisson Yang–Baxter equation and hence adm-Poisson bialgebras. Note that a pre-adm-Poisson algebra gives an equivalent presentation for a pre-Poisson algebra introduced by Aguiar.

Anti-flexible bialgebras

We establish a bialgebra theory for anti-flexible algebras in this paper. We introduce the notion of an anti-flexible bialgebra which is equivalent to a Manin triple of anti-flexible algebras. The study of a special case of anti-flexible bialgebras leads to the introduction of anti-flexible Yang–Baxter equation in an anti-flexible algebra which is an analogue of the classical Yang–Baxter equation in a Lie algebra or the associative Yang–Baxter equation in an associative algebra. It is unexpected consequence that both the anti-flexible Yang–Baxter equation and the associative Yang–Baxter equation have the same form. A skew-symmetric solution of anti-flexible Yang–Baxter equation gives an anti-flexible bialgebra. Finally the notions of an [Formula: see text]-operator of an anti-flexible algebra and a pre-anti-flexible algebra are introduced to construct skew-symmetric solutions of anti-flexible Yang–Baxter equation.

Rota–Baxter Operators on Unital Algebras

We state that all Rota-Baxter operators of nonzero weight on Grassmann algebra over a field of characteristic zero are projections on a subalgebra along another one. We show the one-to-one correspondence between the solutions of associative Yang-Baxter equation and Rota-Baxter operators of weight zero on the matrix algebra $M_n(F)$ (joint with P. Kolesnikov). We prove that all Rota-Baxter operators of weight zero on a unital associative (alternative, Jordan) algebraic algebra over a field of characteristic zero are nilpotent. For an algebra $A$, we introduce its new invariant the rb-index $\mathrm{rb}(A)$ as the nilpotency index for Rota-Baxter operators of weight zero on $A$. We show that $\mathrm{rb}(M_n(F)) = 2n-1$ provided that characteristic of $F$ is zero.

Odd supersymmetric Kronecker elliptic function and Yang–Baxter equations

We introduce an odd supersymmetric version of the Kronecker elliptic function. It satisfies the genus one Fay identity and supersymmetric version of the heat equation. As an application, we construct odd supersymmetric extensions of the elliptic R-matrices, which satisfy the classical and the associative Yang–Baxter equations.

Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras

The concept of weighted infinitesimal unitary bialgebra is an algebraic meaning of the nonhomogenous associative Yang–Baxter equation. In this paper, we equip the space of decorated planar rooted forests with a coproduct which makes it a weighted infinitesimal unitary bialgebra. Further, we construct an infinitesimal unitary Hopf algebra on decorated planar rooted forests in the sense of Loday and Ronco. We then introduce the concept of symmetric 1-cocycle condition, which is derived from the dual of the Hochschild cohomology. We study the universal properties of the space of decorated planar rooted forests with the symmetric 1-cocycle, leading to the notation of a weighted $$\Omega $$ -cocycle infinitesimal unitary bialgebra. As an application, we obtain the initial object in the category of free cocycle infinitesimal unitary bialgebras on the undecorated planar rooted forests, which is the object studied in the well-known noncommutative Connes–Kreimer Hopf algebra. Finally, we construct a pre-Lie algebra on decorated planar rooted forests.

Deformations of associative Rota-Baxter operators

Rota-Baxter operators and more generally O-operators on associative algebras are important in probability, combinatorics, associative Yang-Baxter equation and splitting of algebras. Using a method of Uchino, we construct an explicit graded Lie algebra whose Maurer-Cartan elements are given by O-operators. This allows us to construct cohomology for an O-operator. This cohomology can also be seen as the Hochschild cohomology of a certain algebra with coefficients in a suitable bimodule. Next, we study linear and formal deformations of an O-operator which are governed by the above-defined cohomology. We introduce Nijenhuis elements associated with an O-operator which give rise to trivial deformations. As applications, we conclude deformations of weight zero Rota-Baxter operators, associative r-matrices and averaging operators.

Odd supersymmetrization of elliptic R-matrices * * To the 80th anniversary of Andrei Slavnov.

We study a general ansatz for an odd supersymmetric version of the Kronecker elliptic function, which satisfies the genus one Fay identity. The obtained result is used for construction of the odd supersymmetric analogue for the classical and quantum elliptic R-matrices. They are shown to satisfy the classical Yang–Baxter equation and the associative Yang–Baxter equation. The quantum Yang–Baxter equation is discussed as well. It acquires additional term in the case of supersymmetric R-matrices.

Generalized model of interacting integrable tops

We introduce a family of classical integrable systems describing dynamics of M interacting glN integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the GLNR-matrix satisfying the associative Yang-Baxter equation. The obtained systems can be considered as extensions of the spin type Calogero-Moser models with (the classical analogues of) anisotropic spin exchange operators given in terms of the R-matrix data. In N = 1 case the spin Calogero-Moser model is reproduced. Explicit expressions for glNM -valued Lax pair with spectral parameter and its classical dynamical r-matrix are obtained. Possible applications are briefly discussed.

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.