Related papers: Rota-Baxter operators on dihedral and alternating …
This paper first introduces the notion of a Rota-Baxter operator (of weight $1$) on a Lie group so that its differentiation gives a Rota-Baxter operator on the corresponding Lie algebra. Direct products of Lie groups, including the…
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…
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…
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 the paper, we introduce the notion of a Rota-Baxter operator of a non-scalar weight. As a motivation, we show that there is a natural connection between Rota-Baxter operators of this type and structures of quasitriangular Lie bialgebras…
A Rota-Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota-Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota-Baxter…
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 generalize the notion of a Rota-Baxter operator on groups and the notion of a Rota-Baxter operator of weight 1 on Lie algebras and define and study the notion of a Rota-Baxter operator on a cocommutative Hopf algebra $H$. If $H=F[G]$ is…
Rota-Baxter operators present a natural generalisation of integration by parts formula for the integral operator. In 2015, Zheng, Guo, and Rosenkranz conjectured that every injective Rota-Baxter operator of weight zero on the polynomial…
In the present article we define and investigate relative Rota--Baxter operators and relative averaging operators on racks and rack algebras. Also, if B is a Rota--Baxter or averaging operator on a rack X, then we can extend B by linearity…
Gian-Carlo Rota suggested in one of his last articles the problem of developing a theory around the notion of integration algebras, complementary to the already existing theory of differential algebras. This idea was mainly motivated by…
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…
Rota--Baxter operators over groups have been recently defined in \cite{LHY2021}, and they share a close connection with skew braces, as demonstrated in \cite{VV2022}. In this paper, we classify all Rota--Baxter operators of weight 1 over…
In this paper, we construct a categorical solution $(\huaC, R)$ of the Yang-Baxter equation, i.e. $\huaC$ is a small category and $R: \huaC\times\huaC\lon\huaC\times\huaC$ is an invertible functor satisfying $$…
We introduce the concept of an extended O-operator that generalizes the well-known concept of a Rota-Baxter operator. We study the associative products coming from these operators and establish the relationship between extended O-operators…
The purpose of this paper is to determine all Rota-Baxter operators on dual quaternion algebra $\mathcal{H}_d$ over the reals.
A long standing problem of Gian-Carlo Rota for associative algebras is the classification of all linear operators that can be defined on them. In the 1970s, there were only a few known operators, for example, the derivative operator, the…
In this paper, we introduce the notion of a relative Rota-Baxter operator of weight $\lambda$ on a Lie triple system with respect to an action on another Lie triple system, which can be characterized by the graph of their semidirect…
This paper introduces the notion of Rota-Baxter $C^{\ast}$-algebras. Here a Rota-Baxter $C^{\ast}$-algebra is a $C^{\ast}$-algebra with a Rota-Baxter operator. Symmetric Rota-Baxter operators, as special cases of Rota-Baxter operators on…
A Rota-Baxter operator of weight $\lambda$ is an abstraction of both the integral operator (when $\lambda=0$) and the summation operator (when $\lambda=1$). We similarly define a differential operator of weight $\lambda$ that includes both…