Related papers: Inhomogeneous linear equation in Rota-Baxter algeb…
The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.
A Rota-Baxter operator defined on the polynomial algebra is called monomial if it maps each monomial to a monomial with some coefficient. We classify monomial Rota-Baxter operators defined on the algebra of polynomials in one variable…
A Rota--Baxter operator is an algebraic abstraction of integration, which is the typical example of a weight zero Rota-Baxter operator. We show that studying the modules over the polynomial Rota--Baxter algebra $(k[x],P)$ is equivalent to…
Leibniz algebras are non-skewsymmetric analogue of Lie algebras. In this paper, we consider weighted relative Rota-Baxter operators on Leibniz algebras. We define cohomology of such operators and as an application, we study their…
Rota-Baxter operators have been paid much attention in the last few decades as they have many applications in mathematics and physics. In this paper, our object of study is modified Rota-Baxter operators on Leibniz algebras. We investigate…
Rota-Baxter operators are an algebraic abstraction of integration. Following this classical connection, we study the relationship between Rota-Baxter operators and integrals in the case of the polynomial algebra $\mathbf{k}[x]$. We consider…
In this paper we introduce the concepts of a Rota-Baxter operator and a differential operator with weights on an $n$-algebra. We then focus on Rota-Baxter 3-Lie algebras and show that they can be derived from Rota-Baxter Lie algebras and…
In this article we continue to explore the notion of Rota-Baxter algebras in the context of the Hopf algebraic approach to renormalization theory in perturbative quantum field theory. We show in very simple algebraic terms that the…
A Rota-Baxter Leibniz algebra is a Leibniz algebra $(\mathfrak{g},[~,~]_{\mathfrak{g}})$ equipped with a Rota-Baxter operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. We define representation and dual representation of Rota-Baxter…
We describe all Rota-Baxter operators of any weight on the space of matrices from $M_2(F)$ considered under the product $a\circ b = (ab + ba)/2$ and usually denoted as $M_2(F)^{(+)}$. This algebra is known to be a simple Jordan one. We…
A generalisation of the notion of a Rota-Baxter operator is proposed. This generalisation consists of two operators acting on an associative algebra and satisfying equations similar to the Rota-Baxter equation. Rota-Baxter operators of any…
We give a broad study of representation and module theory of Rota-Baxter algebras. Regular-singular decompositions of Rota-Baxter algebras and Rota-Baxter modules are obtained under the condition of quasi-idempotency. Representations of an…
In this paper, we firstly construct an $L_\infty[1]$-algebra via the method of higher derived brackets, whose Maurer-Cartan elements correspond to relative $\Omega$-family Rota-Baxter algebras structures of weight $\lambda$. For a relative…
We prove that all Rota-Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota-Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra…
The all Rota-Baxter algebra structures on the polynomial algebra $R={\bf k}[x]$ are well known. We study the finite dimensional modules of polynomial Rota-Baxter algebras $(\bfk[x],P)$ or $(x {\bf k} [x],P)$ of weight nonzero since some…
In this paper, we introduce the notion of modified Rota-Baxter operators of non-zero weight on $3$-Lie algebras and provide some examples. Next, we give various constructions of modified Rota-Baxter operators of non-zero weight according to…
Weighted Rota-Baxter Jacobi-Jordan algebras and their representations are studied. Moreover, we consider weighted Rota-Baxter paired operators that are related to weighted Rota-Baxter Jacobi-Jordan algebras together with their…
The purpose of the present paper is to investigate cohomologies and deformations of weighted Rota-Baxter Lie algebras as well as weighted Rota-Baxter associative algebras with derivations. First we introduce a notion of weighted Rota-Baxter…
We study possible connections between Rota-Baxter operators of non-zero weight and non-skew-symmetric solutions of the classical Yang-Baxter equation on finite-dimensional quadratic Lie algebras. The particular attention is made to the case…
Let K be a field of characteristic zero, and let m=(x_1,...,x_n)) be a maximal ideal of the polynomial ring K[x_1,...,x_n]. We classify all Rota--Baxter operators of weights zero and one on the truncated polynomial algebra…