Related papers: Integral operators on lattices
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 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…
All Rota-Baxter operators of weight zero on split octonion algebra over a~field of characteristic not 2 are classified up to conjugation by automorphisms and antiautomorphisms. Thus, the classification of Rota-Baxter operators on…
In this paper, we shall describe all the Rota-Baxter operators with any weight on split semi-quaternion algebra. Firstly, we give the matrix characterization of the Rota-Baxter operator on split semi-quaternion algebra. Then we give the…
In this paper, we study relative Rota-Baxter operators of weight $0$ on groups and give various examples. In particular, we propose different approaches to study Rota-Baxter operators of weight $0$ on groups and Lie groups. We establish…
In this paper, we study Rota-Baxter operators and super $\mathcal{O}$-operator of associative superalgebras, Lie superalgebras, pre-Lie superalgebras and $L$-dendriform superalgebras. Then we give some properties of pre-Lie superalgebras…
This article explores Rota-Baxter operators on finite-dimensional $\omega$-Lie algebras over a field of characteristic not 2. We provide several methods for constructing left-symmetric algebras, $\omega$-Lie algebras, and Hom-Lie algebras…
We classify all Rota-Baxter operators of nonzero weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which are not arisen from the decompositions of the entire algebra into a direct vector…
Rota-Baxter groups with weights $\pm 1$ have attracted quite much attention since their recent introduction, thanks to their connections with Rota-Baxter Lie algebras, factorizations of Lie groups, post- and pre-Lie algebras, braces and…
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…
We describe all Rota-Baxter operators $R$ of weight zero on the matrix algebra $M_3(F)$ over a quadratically closed field $F$ of characteristic not 2 or 3 such that $R(1)\neq0$. Thus, we get a partial classification of solutions to the…
Rota-Baxter operators and more generally $\mathcal{O}$-operators play a crucial role in broad areas of mathematics and physics, such as integrable systems, the Yang-Baxter equation and pre-Lie algebras. The main objects of study in the…
Rota-Baxter operators on the polynomial algebra have been actively studied since the work of S.H. Zheng, L. Guo, and M. Rosenkranz (2015). Monomial operators of an arbitrary weight (2016), as well as injective operators of weight zero on…
We describe all Rota-Baxter operators $R$ of weight zero on the algebra $U_3(F)$ of upper-triangular matrices of order three over a field of characteristic 0. For this, we apply the following three ingredients: properties of $R(1)$,…
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…
We describe Rota-Baxter operators on split octonions. It turns out that up to some transformations there exists exactly one such non-splitting operator over any field. We also obtain a description of all decompositions of split octonions…
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…
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…
We give the description of homogeneous Rota-Baxter operators, Reynolds operators, Nijenhuis operators, Average operators and differential operator of weight 1 of null-filiform associative algebras of arbitrary dimension.
The algebraic study of special integral operators led to the notions of Rota-Baxter operators and shuffle products which have found broad applications. This paper carries out an algebraic study of general integral operators and equations,…