Alexander Odesskii, Vladimir Sokolov
We study associative multiplications in semi-simple associative algebras over C compatible with the usual one. An interesting class of such multiplications is related to the affine Dynkin diagrams of A, D, E-type. In this paper we investigate in details the multiplications of the A-type and integrable matrix ODEs and PDEs generated by them.
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We study associative multiplications in semi-simple associative algebras over C compatible with the usual one or, in other words, linear deformations of semi-simple associative algebras over C. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures in the matrix case and PM-structures in the case of direct sums of several matrix algebras. We also investigate various properties of PM-structures, provide numerous examples and describe an important class of PM-structures. The classification of these PM…
Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such deformations and construct numerous examples. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures. We also describe an important class of M-structures related to the affine Dynkin diagrams of A, D, E-type. These M-structures and their representations are describe…
This paper studies the concepts of a totally compatible dialgebra and a totally compatible Lie dialgebra, defined to be a vector space with two binary operations that satisfy individual and mixed associativity conditions and Lie algebra conditions respectively. We show that totally compatible dialgebras are closely related to bimodule algebras and semi-homomorphisms. More significantly, Rota-Baxter operators on totally compatible dialgebras provide a uniform framework to generalize known results that Rota-Baxter related operators give tridendriform algebras. Free totally compatible dialgebras …
We formulate an analog of Inverse Scattering Method for integrable systems on noncommutative associative algebras. In particular we define Hamilton flows, Casimir elements and noncommutative analog of the Lax matrix. The noncommutative Lax element generates infinite family of commuting Hamilton flows on an associative algebra. The proposed approach to integrable systems on associative algebras satisfy certain universal property, in particular it incorporates both classical and quantum integrable systems as well as provides a basis for further generalization. We motivate our definition by expli…
The paper is devoted to the Poisson brackets compatible with multiplication in associative algebras. These brackets are shown to be quadratic and their relations with the classical Yang--Baxter equation are revealed. The paper also contains a description of Poisson Lie structures on Lie groups whose Lie algebras are adjacent to an associative structure.
We consider dynamical systems on the space of functions taking values in a free associative algebra. The system is said to be integrable if it possesses an infinite dimensional Lie algebra of commuting symmetries. In this paper we propose a new approach to the problem of quantisation of dynamical systems, introduce the concept of quantisation ideals and provide meaningful examples.
We extend integrable systems on quad-graphs, such as the Hirota equation and the cross-ratio equation, to the non-commutative context, when the fields take values in an arbitrary associative algebra. We demonstrate that the three-dimensional consistency property remains valid in this case. We derive the non-commutative zero curvature representations for these systems, based on the latter property. Quantum systems with their quantum zero curvature representations are particular cases of the general non-commutative ones.
Quadratic Poisson brackets on associative algebras are studied. Such a bracket compatible with the multiplication is related to a differentiation in tensor square of the underlying algebra. Jacobi identity means that this differentiation satisfies a classical Yang--Baxter equation. Corresponding Lie groups are canonically equipped with a Poisson Lie structure. A way to quantize such structures is suggested.
We introduce the concept of {sigma, tau}-Rota-Baxter operator, as a twisted version of a Rota-Baxter operator of weight zero. We show how to obtain a certain {sigma, tau}-Rota-Baxter operator from a solution of the associative (Bi)Hom-Yang-Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.
We find explicit (multisoliton) solutions for nonabelian integrable systems such as periodic Toda field equations, Langmuir equations, and Schrodinger equations for functions with values in any associative algebra. The solution for nonabelian Toda field equations for root systems of types A, B, C was expressed by the authors in a previous paper (q-alg/9701008) using quasideterminants introduced by the last two authors. To find multisoliton solutions of periodic Toda equations and other nonabelian systems we use a combination of these ideas with important lemmas which are due to Marchenko.
We define quantum determinants in Quantum Matrix Algebras, related to couples of compatible braidings following the scheme from [G]. We establish relations between these determinants and the so-called column-(row-)determinants, often used in the theory of integrable systems. Also, we generalize the quantum integrable spin systems from [CFRS] by using generalized Yangians, related to couples of compatible braidings. We demonstrate that such quantum integrable spin systems are not uniquely determined by the "quantum coordinate ring" of the basic space V. For instance, the "quantum plane" xy=qyx …
Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that is compatible with the Rota-Baxter operator, called the Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the context of Rota-Baxter algebras. The study of the coboundary case leads to an enrichment of the associative Yang-B…
After giving some definitions for vertex operator SUPERalgebras and their modules, we construct an associative algebra corresponding to any vertex operator superalgebra, such that the representations of the vertex operator algebra are in one-to-one correspondence with those of the corresponding associative algebra. A way is presented to decribe the fusion rules for the vertex operator superalgebras via modules of the associative algebra. The above are generalizations of Zhu's constructions for vertex operator algebras. Then we deal in detail with vertex operator superalgebras corresponding to …
In this paper we propose versions of the associative Yang-Baxter equation and higher order $R$-matrix identities which can be applied to quantum dynamical $R$-matrices. As is known quantum non-dynamical $R$-matrices of Baxter-Belavin type satisfy this equation. Together with unitarity condition and skew-symmetry it provides the quantum Yang-Baxter equation and a set of identities useful for different applications in integrable systems. The dynamical $R$-matrices satisfy the Gervais-Neveu-Felder (or dynamical Yang-Baxter) equation. Relation between the dynamical and non-dynamical cases is descr…
This paper studies operator forms of the nonhomogeneous associative classical Yang-Baxter equation (nhacYBe), extending and generalizing such studies for the classical Yang-Baxter equation and associative Yang-Baxter equation that can be tracked back to the works of Semonov-Tian-Shansky and Kupershmidt on Rota-Baxter Lie algebras and $\mathcal{O}$-operators. In general, solutions of the nhacYBe are characterized in terms of generalized $\mathcal{O}$-operators. The characterization can be given by the classical $\mathcal{O}$-operators precisely when the solutions satisfy an invariant condition.…
The survey is devoted to associative $\Z_{\ge0}$-graded algebras presented by n generators and n(n-1)/2 quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). We consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve and a point on this curve) which are flat deformations of the polynomial ring in n variables. Diverse properties of these algebras are described, together with their relations to integrable systems, deformation quantization, moduli spaces and other directions of modern investigations.
We consider an unified model, called ancestor model, associated with twisted trigonometric $R$ matrix which model leads to several descendant integrable lattice models related to the U$_{q^{1/2}}(\hat{sl_2})$ symmetry. Boundary operators compatible with integrability are introduced to this model. Reflection and dual reflection equations to ensure integrability of the system are shown to be same as the untwisted case. It follows that underlying symmetry of the ancestor model with integrable boundaries is identified with the q-deformed analogue of Onsager's symmetry. The transfer matrix and its …
This paper gives a systematic study of matching dialgebras corresponding to the operad $As^{(2)}$ in \cite{Zi} as the only Koszul self dual operad there other than the operads of associative algebras and Poisson algebras. The close relationship of matching dialgebras with semi-homomorphisms and matched pairs of associative algebras are established. By anti-symmetrizing, matching dialgerbas are also shown to give compatible Lie algebras, pre-Lie algebras and PostLie algebras. By the rewriting method, the operad of matching dialgebras is shown to be Koszul and the free objects are constructed in…
Anti-dendriform algebras, new splitting of operations and Novikov type algebras
本文引入反树形代数作为分裂结合性的新途径，通过反$\mathcal O$-算子和反Rota-Baxter算子给出解释，并借助$q$-代数建立其与树形代数的对应，进而提出Novikov型树形代数及一般类比框架。
Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algebras, later called Rota's program on algebraic operators. Recent progresses on this program have been achieved in the contexts of operated algebra, rewriting systems and Groebner-Shirshov bases. These developments also suggest that Rota's insight can be applied to determine operator identities on Lie algebras, and thus to put the various linear operators on Lie algebras in a uniform perspective. This paper carries out…
Module Monoidal Categories as Categorification of Associative Algebras
本文引入非单位模幺半范畴的概念以范畴化结合代数的刻画，并证明在有单位情况下该定义与先前模张量范畴定义的2-范畴等价性。
In this paper we study (associative) Nijenhuis algebras, with emphasis on the relationship between the category of Nijenhuis algebras and the categories of NS algebras. This is in analogy to the well-known theory of the adjoint functor from the category of Lie algebras to that of associative algebras, and the more recent results on the adjoint functor from the categories of dendriform and tridendriform algebras to that of Rota-Baxter algebras. We first give an explicit construction of free Nijenhuis algebras and then apply it to obtain the universal enveloping Nijenhuis algebra of an NS algebr…
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