Related papers: Relative Rota-Baxter operators and crossed homomor…
In this paper we determine all the Rota-Baxter operators of weight zero on semigroup algebras of order two and three with the help of computer algebra. We determine the matrices for these Rota-Baxter operators by directly solving the…
We introduce notions of ${\mathcal O}$-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of Rota-Baxter operators. The invertible $\mathcal O$-operators give a sufficient and…
Inspired by the work of Wang and Zhou [4] for Rota-Baxter algebras, we develop a cohomology theory of Rota-Baxter systems and justify it by interpreting the lower degree cohomology groups as formal deformations and as abelian extensions of…
As an abstraction and generalization of the integral operator in analysis, integral operators (known as Rota-Baxter operators of weight zero) on associative algebras and Lie algebras have played an important role in mathematics and physics.…
This paper studies Rota-Baxter, Reynolds, Nijenhuis, and Averaging operators on 2-dimensional pre-Lie algebras over $\mathbb{C}$. Using the classification of 2-dimensional pre-Lie algebras and computational tools like Mathematica or Maple,…
To study operator algebras with symmetries in a wide sense we introduce a notion of {\em relative convolution operators} induced by a Lie algebra. Relative convolutions recover many important classes of operators, which have been already…
Representations of Hom-Jacobi-Jordan algebras are studied. In particular, adjoint representations and trivial representations are studied in detail. Derivations and central extensions of Hom-Jacobi-Jordan algebras are also discussed as an…
Rota--Baxter operators $R$ of weight $1$ on $\mathfrak{n}$ are in bijective correspondence to post-Lie algebra structures on pairs $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is complete. We use such Rota--Baxter operators to study…
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…
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…
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…
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…
The intensive study of Rota-Baxter operators on the polynomial algebra $F[x]$ has been started with the work of S.H. Zheng, L. Guo, and M. Rosenkranz (2015). We deal with the case of two variables and monomial Rota-Baxter operators of…
Family algebraic structures indexed by a semigroup first appeared in the algebraic aspects of renormalizations in quantum field theory. The concept of the Rota-Baxter family and its relation with (tri)dendriform family algebras have been…
In this paper, we introduce the theory of Rota-Baxter operators on Clifford semigroups, useful tools for obtaining dual weak braces, i.e., triples $\left(S,+,\circ\right)$ where $\left(S,+\right)$ and $\left(S,\circ\right)$ are Clifford…
We study the Yang-Baxter operator for the vector representation $V_m$ of the quantum group $U_q(sl_m)$. We consider the one-term Yang-Baxter homology with coefficients in $V_m$-modules and provide a direct sum decomposition of the one term…
A commutative Rota-Baxter algebra can be regarded as a commutative algebra that carries an abstraction of the integral operator. With the motivation of generalizing the study of algebraic geometry to Rota-Baxter algebra, we extend the…
We will introduce an associative (or quantum) version of Poisson structure tensors. This object is defined as an operator satisfying a "generalized" Rota-Baxter identity of weight zero. Such operators are called generalized Rota-Baxter…
This paper examines the connections between (relative) Rota--Baxter groups, skew left braces, and enlargements of these structures on naturally associated semi-direct products. Given a skew left brace, we define a new skew left brace,…
The notion of matching Rota-Baxter algebras was recently introduced by Gao, Guo and Zhang [{\em J. Algebra} 552 (2020) 134-170] motivated by the study of algebraic renormalization of regularity structures. The concept of matching…