Rota-Baxter Operators Research Articles

Yang-Baxter Equations and Relative Rota-Baxter Operators for Left-Alia Algebras Associated to Invariant Theory

Left-Alia algebras are a class of algebras with symmetric Jacobi identities. They contain several typical types of algebras as subclasses, and are closely related to the invariant theory. In this paper, we study the construction theory of left-Alia bialgebras. We introduce the notion of the left-Alia Yang-Baxter equation. We show that an antisymmetric solution of the left-Alia Yang-Baxter equation gives rise to a left-Alia bialgebra that we call triangular. The notions of relative Rota-Baxter operators of left-Alia algebras and pre-left-Alia algebras are introduced to provide antisymmetric solutions of the left-Alia Yang-Baxter equation.

Lie Rota–Baxter operators on the Sweedler algebra H4

If [Formula: see text] is an associative algebra, then we can define the adjoint Lie algebra [Formula: see text] and Jordan algebra [Formula: see text]. It is easy to see that any associative Rota–Baxter (RB) operator on [Formula: see text] induces a Lie and Jordan RB operator on [Formula: see text] and [Formula: see text], respectively. Are there Lie (Jordan) RB operators, which are not associative RB operators? In this paper, we explore these questions for the Sweedler algebra [Formula: see text], which is a 4-dimensional non-commutative Hopf algebra. More precisely, we describe the RB operators on the adjoint Lie algebra [Formula: see text].

Some constructions of 3-pre-Lie-Rinehart algebras

3-Lie-Rinehart algebras are an algebraic axiomatization of the basic properties of Filippov algebroids (for n = 3). The purpose of this paper is to introduce the notion of relative Rota-Baxter operators on a 3-Lie-Rinehart algebra and give some classical characterizations. Next, we define 3-pre-Lie-Rinehart algebras which can be seen as a dendrification of 3-Lie-Rinehart algebras by means of relative Rota-Baxter operators. Moreover, we introduce the notion of trace functions on Lie-Rinehart algebra and pre-Lie-Rinehart algebra to construct their induced 3-Lie-Rinehart algebra and 3-pre-Lie-Rinehart algebra, respectively.

Some properties of relative Rota–Baxter operators on groups

We find connection between relative Rota–Baxter operators and usual Rota–Baxter operators. We prove that any relative Rota–Baxter operator on a group H with respect to ( G , Ψ ) defines a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G . On the other side, we give condition under which a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G defines a relative Rota–Baxter operator on H with respect to ( G , Ψ ) . We introduce homomorphic post-groups and find their connection with λ-homomorphic skew left braces. Further, we construct post-group on arbitrary group and a family of post-groups which depends on integer parameter on any two-step nilpotent group. We find all verbal solutions of the quantum Yang-Baxter equation on two-step nilpotent group.

Rota–Baxter Operators of Weight Zero on Upper-Triangular Matrices of Order Three

Rota–Baxter Operators of Weight Zero on Upper-Triangular Matrices of Order Three

Modified double brackets and a conjecture of S. Arthamonov

Around 20 years ago, M. Van den Bergh introduced double Poisson brackets as operations on associative algebras inducing Poisson brackets under the representation functor. Weaker versions of these operations, called modified double Poisson brackets, were later introduced by S. Arthamonov in order to induce a Poisson bracket on moduli spaces of representations of the corresponding associative algebras. Moreover, he defined two operations that he conjectured to be modified double Poisson brackets. The first case of this conjecture was recently proved by M. Goncharov and V. Gubarev motivated by the theory of Rota-Baxter operators of nonzero weight. We settle the conjecture by realising the second case as part of a new family of modified double Poisson brackets. These are obtained from mixed double Poisson algebras, a new class of algebraic structures that are introduced and studied in the present work.

Rota–baxter operators and skew left brace structures over heisenberg group

Rota–baxter operators and skew left brace structures over heisenberg group

Quasi-triangular, factorizable Leibniz bialgebras and relative Rota–Baxter operators

Abstract We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang–Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota–Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota–Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota–Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight 0, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.

Generalized NS-algebras

We generalize the notion of an NS-algebra, which was previously only considered for associative, Lie and Leibniz algebras, to arbitrary categories of binary algebras with one operation. We do this by defining these algebras using a bimodule property, as we did in our earlier work for defining the notions of a dendriform and tridendriform algebra for such categories of algebras. We show that several types of operators lead to NS-algebras: Nijenhuis operators, twisted Rota-Baxter operators and relative Rota-Baxter operators of arbitrary weight. We thus provide a general framework in which several known results and constructions for associative, Lie and Leibniz-NS-algebras are unified, along with some new examples and constructions that we also present.

Cohomologies and deformations of relative Rota-Baxter operators on n-Lie superalgebras

In this paper, we characterize a relative Rota-Baxter operator on an n-Lie superalgebra with a representation (called an n-LSRep pair) as a Maurer-Cartan element of a certain Lie n-algebra. Then, we introduce the notions of n-pre-Lie superalgebras and symplectic n-Lie superalgebras and give some related results based on relative Rota-Baxter operators. In the next part, we give the cohomologies of relative Rota-Baxter operators on n-LSRep pairs and study infinitesimal deformations and extensions of finite order deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Finally, we build the relationship between the cohomology groups of relative Rota-Baxter operators on n-LSRep pairs and those on ( n + 1 ) -LSRep pairs by certain linear functions.

Relative Rota–Baxter groups and skew left braces

Abstract Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and have been introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota–Baxter groups with skew left braces, which are well known to give bijective non-degenerate set-theoretical solutions of the Yang–Baxter equation. We prove that every relative Rota–Baxter group gives rise to a skew left brace, and conversely, every skew left brace arises from a relative Rota–Baxter group. It turns out that there is an isomorphism between the two categories under some mild restrictions. We propose an efficient GAP algorithm, which would enable the computation of relative Rota–Baxter operators on finite groups. In the end, we introduce the notion of isoclinism of relative Rota–Baxter groups and prove that an isoclinism of these objects induces an isoclinism of corresponding skew left braces.

On Hom–Nambu–Poisson superalgebras structures and their representations

The purpose of this paper is to study some results of constructions on Hom–Poisson superalgebras. We use the representations and Rota–Baxter operators. We introduce the notion of [Formula: see text]-ary Hom–Nambu–Poisson superalgebras and their representations and we study relationships between Hom–Poisson superalgebras and its induced [Formula: see text]-ary Hom–Nambu–Poisson superalgebras. The same concept is applied for the representations of [Formula: see text]-ary Hom–Nambu–Poisson superalgebras.

3-pre-Leibniz Algebras, Deformations and Cohomologies of Relative Rota-Baxter Operators on 3-Leibniz Algebras

In this paper, first we introduce the notions of 3-pre-Leibniz algebras and relative Rota-Baxter operators on 3-Leibniz algebras. We show that a 3-pre-Leibniz algebra gives rise to a 3-Leibniz algebra and a representation such that the identity map is a relative Rota-Baxter operator. Conversely, a relative Rota-Baxter operator naturally induces a 3-pre-Leibniz algebra. Then we construct a Lie 3-algebra, and characterize relative Rota-Baxter operators as its Maurer-Cartan elements. Consequently, we obtain the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty$$\\end{document}-algebra that controls deformations of relative Rota-Baxter operators on 3-Leibniz algebras. Next we define the cohomology of relative Rota-Baxter operators on 3-Leibniz algebras and show that infinitesimal deformations of a relative Rota-Baxter operator are classified by the second cohomology group. Finally, we construct an L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty$$\\end{document}-algebra whose Maurer-Cartan elements are relative Rota-Baxter 3-Leibniz algebra structures, and define the cohomology of relative Rota-Baxter 3-Leibniz algebras.

Rota–Baxter Operators on Skew Braces

In this paper, we introduce the concept of Rota–Baxter skew braces, and provide classifications of Rota–Baxter operators on various skew braces, such as (Z,+,∘) and (Z/(4),+,∘). We also present a necessary and sufficient condition for a skew brace to be a co-inverse skew brace. Additionally, we describe some constructions of Rota–Baxter quasiskew braces, and demonstrate that every Rota–Baxter skew brace can induce a quasigroup and a Rota–Baxter quasiskew brace.

Some Results on Zinbiel Algebras and Rota–Baxter Operators

Rota–Baxter operators (RBOs) play a substantial role in many subfields of mathematics, especially in mathematical physics. In the article, RBOs on Zinbiel algebras (ZAs) and their sub-adjacent algebras are first investigated. Moreover, all the RBOs on two and three-dimensional ZAs are presented. Finally, ZAs are also realized in low dimensions of the RBOs of commutative associative algebras. It was found that not all ZAs can be attained in this way.

Twisted Lie algebras by invertible derivations

In this paper, we introduce an algebra structure denoted by InvDer algebra in which we twist an algebra with an invertible derivation whose inverse is also a derivation. We define InvDer Lie algebras, InvDer associative algebras, InvDer Zinbiel algebras and InvDer dendriforme algebras. We also study the relations between these structures by using Rota–Baxter operators and endomorphism operators.

Twisted relative Rota-Baxter operators on Leibniz conformal algebras

In this paper, we first investigate some properties of relative Rota-Baxter operators on Leibniz conformal algebras with respect to representations and their connections with Leibniz dendriform conformal algebras. Next, we introduce the notion of a twisted relative Rota-Baxter operator and construct a conformal NS-Leibniz algebra structure related to twisted relative Rota-Baxter operators. Furthermore, we define the cohomology of a twisted relative Rota-Baxter operator with coefficients in a suitable representation. Finally, we consider formal deformations of twisted relative Rota-Baxter operators from cohomological points of view.

L ∞ -structures and cohomology theory of compatible O-operators and compatible dendriform algebras

The notion of O-operator is a generalization of the Rota–Baxter operator in the presence of a bimodule over an associative algebra. A compatible O-operator is a pair consisting of two O-operators satisfying a compatibility relation. A compatible O-operator algebra is an algebra together with a bimodule and a compatible O-operator. In this paper, we construct a graded Lie algebra and an L∞-algebra that respectively characterize compatible O-operators and compatible O-operator algebras as Maurer–Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible O-operators and compatible O-operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible O-operators.

Pre-Leibniz algebras

The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classify some special types of homotopy pre-Leibniz algebras.

Lie theory and cohomology of relative Rota–Baxter operators

AbstractIn this paper, we establish a local Lie theory for relative Rota–Baxter operators of weight 1. First we recall the category of relative Rota–Baxter operators of weight 1 on Lie algebras and construct a cohomology theory for them. We use the second cohomology group to study infinitesimal deformations of relative Rota–Baxter operators and modified ‐matrices. Then we introduce a cohomology theory of relative Rota–Baxter operators on a Lie group. We construct the differentiation functor from the category of relative Rota–Baxter operators on Lie groups to that on Lie algebras, and extend it to the cohomology level by proving the Van Est theorem between the two cohomology theories. We integrate a relative Rota–Baxter operator of weight 1 on a Lie algebra to a local relative Rota–Baxter operator on the corresponding Lie group, and show that the local integration and differentiation are adjoint to each other. Finally, we give two applications of our integration of Rota–Baxter operators: one is to give an explicit formula for the factorization problem, and the other is to provide an integration for matched pairs.