Related papers: Rota-Baxter systems and skew trusses
This paper studies formal deformations and homotopy theory of Rota-Baxter algebras of any weight. We define an $L_\infty$-algebra, which controls simultaneous deformations of associative products and Rota-Baxter operators. As a consequence,…
Skew bracoids have been shown to have applications in Hopf-Galois theory. We show that a certain family of skew bracoids correspond bijectively with left cancellative semibraces. A consequence of this correspondence is that skew bracoids in…
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…
We study Rota--Baxter operators on vertex algebras using the integrated $\lambda$-bracket formalism. A Rota--Baxter operator produces a deformed vertex algebra structure, and the difference between the deformed and original brackets yields…
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 $$…
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…
In this paper, we first introduce the notion of a Zinbiel bialgebra and show that Zinbiel bialgebras, matched pairs of Zinbiel algebras and Manin triples of Zinbiel algebras are equivalent. Then we study the coboundary Zinbiel bialgebras,…
The Rota-Baxter operator and the modified Rota-Baxter operator on various algebras are both important in mathematics and mathematical physics. The former is originated from the integration-by-parts formula and probability with applications…
Reductions for systems of ODEs integrable via the standard factorization method (the Adler-Kostant-Symes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such…
We study simple set-theoretic solutions of the Yang-Baxter equation that are finite and non-degenerate. Such retractable solutions are fully described and to investigate the irretracble solutions we give a new algebraic method. Our approach…
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 give a construction of Rota-Baxter coalgebras from Hopf module coalgebras and also derive the structures of the pre-Lie coalgebras via Rota-Baxter coalgebras of different weight. Finally, the notion of Rota-Baxter bialgebra is introduced…
In this paper, first we construct two subcategories (using symmetric representations and antisymmetric representations) of the category of relative Rota-Baxter operators on Leibniz algebras, and establish the relations with the categories…
We introduce the notion of Rota-Baxter coalgebra which can be viewed as the dual notion of Rota-Baxter algebra. We provide some concrete examples and establish various properties of this new object. We also consider comodules over…
We introduce the concept of an extension of a semilattice of groups $A$ by a group $G$ and describe all the extensions of this type which are equivalent to the crossed products $A*_\Theta G$ by twisted partial actions $\Theta$ of $G$ on…
A skew brace is a ring-like and group-like algebraic structure that was introduced in the study of set-theoretic solutions to the Yang-Baxter equation. In this survey paper, we shall consider the left series, right series, socle series, and…
The notion of a geometric crystal was introduced by A.Berenstein and D.Kazhdan, motivated by the needs of representation theory of p-adic groups. It was shown by A.Braverman, A.Berenstein, and D.Kazhdan that some particular geometric…
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…
This article gives a brief introduction to some recent work on deformation and homotopy theories of Rota-Baxter operators and more generally $\mathcal{O}$-operators on Lie algebras, by means of the differential graded Lie algebra approach.…
In this paper we extend to left skew trusses $(T,+,\circ,\sigma)$ previous work on left skew rings. We had presented a left skew ring as a group $(N,+)$ with two binary operations $\circ$ and $\cdot$ with $\circ$ associative, $\cdot$ left…