Academic literature on the topic 'Rota-Baxter operator'

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.

In this paper, we first introduce the concept of a Rota–Baxter operator on a cocommutative weak Hopf algebra H and give some examples. We then construct Rota–Baxter operators from the normalized integral, antipode, and target map of H. Moreover, we construct a new multiplication “∗” and an antipode SB from a Rota–Baxter operator B on H such that HB=(H,∗,η,Δ,ε,SB) becomes a new weak Hopf algebra. Finally, all Rota–Baxter operators on a weak Hopf algebra of a matrix algebra are given.

<abstract><p>Uchino initiated the investigation of twisted Rota-Baxter operators on associative algebras. Relevant studies have been extensive in recent times. In this paper, we introduce the notion of a twisted Rota-Baxter operator on a Hom-Lie algebra. By utilizing higher derived brackets, we establish an explicit $ L_{\infty} $-algebra whose Maurer-Cartan elements are precisely twisted Rota-Baxter operators on Hom-Lie algebra s. Additionally, we employ Getzler's technique of twisting $ L_\infty $-algebras to establish the cohomology of twisted Rota-Baxter operators. We demonstrate that this cohomology can be regarded as the Chevalley-Eilenberg cohomology of a specific Hom-Lie algebra with coefficients in an appropriate representation. Finally, we study the linear and formal deformations of twisted Rota-Baxter operators by using the cohomology defined above. We also show that the rigidity of a twisted Rota-Baxter operator can be derived from Nijenhuis elements associated with a Hom-Lie algebra.</p></abstract>

In this paper, we establish the cohomology of relative Rota–Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then, we use this type of cohomology to characterize deformations of relative Rota–Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota–Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order n deformation of a relative Rota–Baxter operator can be extended to an order n+1 deformation if and only if the obstruction class in the second cohomology group is trivial.

In this paper, we first introduce the notion of Hom–Leibniz bialgebras, which is equivalent to matched pairs of Hom–Leibniz algebras and Manin triples of Hom–Leibniz algebras. Additionally, we extend the notion of relative Rota–Baxter operators to Hom–Leibniz algebras and prove that there is a Hom–pre-Leibniz algebra structure on Hom–Leibniz algebras that have a relative Rota–Baxter operator. Finally, we study the classical Hom–Leibniz Yang–Baxter equation on Hom–Leibniz algebras and present its connection with the relative Rota–Baxter operator.

In this paper, we introduce the cohomology theory of relative Rota–Baxter operators on Leibniz algebras. We use the cohomological approach to study linear and formal deformations of relative Rota–Baxter operators. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order [Formula: see text] deformations of a relative Rota–Baxter operator are also characterized in terms of the cohomology theory.

A dendriform algebra defined by Loday has two binary operations that give a two-part splitting of the associativity in the sense that their sum is associative. Similar dendriform type algebras with three-part and four-part splitting of the associativity were later obtained. These structures can also be derived from actions of suitable linear operators, such as a Rota-Baxter operator or TD operator, on an associative algebra. Motivated by finding a five-part splitting of the associativity, we consider the Rota-Baxter TD (RBTD) operator, an operator combining the Rota-Baxter operator and TD operator, and coming from a recent study of Rota’s problem concerning linear operators on associative algebras. Free RBTD algebras on rooted forests are constructed. We then introduce the concept of a quinquedendriform algebra and show that its defining relations are characterized by the action of an RBTD operator, similar to the cases of dendriform and tridendriform algebras.

AbstractWe introduce the concept of a $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$ -Rota–Baxter operator, as a twisted version of a Rota–Baxter operator of weight zero. We show how to obtain a certain $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$ -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.

A relative Rota–Baxter algebra is a triple ( A, M, T) consisting of an algebra A, an A-bimodule M, and a relative Rota–Baxter operator T. Using Voronov’s derived bracket and a recent work of Lazarev, Sheng, and Tang, we construct an L∞[1]-algebra whose Maurer–Cartan elements are precisely relative Rota–Baxter algebras. By a standard twisting, we define a new L∞[1]-algebra that controls Maurer–Cartan deformations of a relative Rota–Baxter algebra ( A, M, T). We introduce the cohomology of a relative Rota–Baxter algebra ( A, M, T) and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of triangular skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota–Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras.

We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota–Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this allows us to build an operator ring that acts naturally on the given Rota–Baxter hierarchy. We conjecture that the operator relations are a noncommutative Gröbner–Shirshov basis for the ideal they generate.

L'objectif de cette thèse est d'étudier les opérateurs de Rota-Baxter relatifs sur les algèbres ternaires de type Lie et de type Jordan. L'étude porte sur leur structure, leur cohomologie, leurs déformations et leur lien avec les équations de Yang-Baxter. Ce travail est divisé en trois parties. La première partie est consacrée à l'étude de l'algèbre de contrôle des systèmes triples de Lie, et à son application à la théorie existante de la cohomologie. De plus, nous introduisons la notion d'opérateur de Rota-Baxter relatif sur les systèmes triples de Lie et construisons une 3-algèbre de Lie comme cas spécial des L∞-algèbres dont les éléments de Maurer-Cartan sont des opérateurs de Rota-Baxter relatifs. Dans la deuxième partie, nous introduisons la notion d'opérateur de Rota-Baxter relatif twisté sur les algèbres 3-Lie et construisons une L∞-algèbre dont les éléments de Maurer-Cartan sont des opérateurs de Rota-Baxter relatifs twistés. Cela nous permet de définir la cohomologie de Chevalley-Eilenberg d'un opérateur de Rota-Baxter relatif twisté. Dans la dernière partie, nous étudions la représentation des algèbres ternaires de Jordan, ce qui nous permet d'introduire la notion d'algèbres ternaires de Jordan cohérentes. Ensuite, les opérateurs de Rota-Baxter relatifs des algèbres ternaires de Jordan sont introduits et les solutions de l'équation de Yang-Baxter de Jordan ternaire sont discutées en impliquant des opérateurs de Rota-Baxter relatifs
The goal of this thesis is to explore relative Rota-Baxter operators in the context of ternary algebras of both Lie and Jordan types. We mainly consider Lie triple systems, 3-Lie algebras and ternary Jordan algebras. The study covers their structure, cohomology, deformations, and their connection with the Yang-Baxter equations. The work is divided into three main parts. The first part aims first to introduce and study a graded Lie algebra whose Maurer-Cartan elements are Lie triple systems. It turns out to be the controlling algebra of Lie triple systems deformations and fits with the adjoint cohomology theory of Lie triple systems introduced by Yamaguti. In addition, we introduce the notion of relative Rota-Baxter operators on Lie triple systems and construct a Lie 3-algebra as a special case of L∞-algebras, where the Maurer-Cartan elements correspond to relative Rota-Baxter operators. In the second part, we introduce the concept of twisted relative Rota-Baxter operators on 3-Lie algebras and construct an L∞-algebra, where the Maurer-Cartan elements are twisted relative Rota-Baxter operators. This allows us to define the Chevalley-Eilenberg cohomology of a twisted relative Rota-Baxter operator. In the last part, we deal with a representation theory of ternary Jordan algebras. In particular, we introduce and discuss the concept of coherent ternary Jordan algebras. We then define relative Rota-Baxter operators for ternary Jordan algebras and discuss solutions ofthe ternary Jordan Yang-Baxter equation involving relative Rota-Baxter operators. Moreover, we investigate ternary pre-Jordan algebras as the underlying algebraic structure of relative Rota-Baxter operators