Related papers: Rota-Baxter Algebra. The Combinatorial Structure o…
In recent years, algebraic studies of the differential calculus and integral calculus in the forms of differential algebra and Rota-Baxter algebra have been merged together to reflect the close relationship between the two calculi through…
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…
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…
We count the number of all Rota-Baxter operators on a finite direct sum $A = F\oplus F\oplus \ldots \oplus F$ of fields and count all of them up to conjugation with an automorphism. We also study Rota-Baxter operators on $A$ corresponding…
Many years ago, Rota proposed a program on determining algebraic identities that can be satisfied by linear operators. After an extended period of dormant, progress on this program picked up speed in recent years, thanks to perspectives…
Rota-Baxter systems of T. Brzezi\'{n}ski are a generalization of Rota-Baxter operators that are related to dendriform structures, associative Yang-Baxter pairs and covariant bialgebras. In this paper, we consider Rota-Baxter systems in the…
We introduce the notion of a matching Rota-Baxter algebra motivated by the recent work on multiple pre-Lie algebras arising from the study of algebraic renormalization of regularity structures~[10,18]. This notion is also related to…
In this paper, we establish a bialgebra theory for Reynolds Lie algebras. First we introduce the notion of a quadratic Reynolds Lie algebra and show that it induces an isomorphism from the adjoint representation to the coadjoint…
This paper establishes a uniform procedure to split the operations in any algebraic operad, generalizing previous known notions of splitting algebraic structures from the dendriform algebra of Loday that splits the associative operation to…
In this paper, we prove results on enumerations of sets of Rota-Baxter words in a finite number of generators and a finite number of unary operators. Rota-Baxter words are words formed by concatenating generators and images of words under…
An operator deformed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its $q \to 1$ limit appear to be the most general Yang-Baxter algebra underlying…
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…
Left-Alia algebras are a class of algebras with symmetric Jacobi identities. They contain several typical types of algebras as subclasses, and are closely related to the invariant theory. In this paper, we study the construction theory of…
A Rota-Baxter operator on a Lie group $ G $ is a smooth map $ B : G \to G $ such that $ B(g)B(h) = B(gB(g)hB(g)^{-1}) $ for all $ g, h \in G $. This concept was introduced in 2021 by Guo, Lang and Sheng as a Lie group analogue 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 first introduce the notion of an extending datum of a Rota-Baxter Lie algebra through a vector space. We then construct a unified product for the Rota-Baxter Lie algebra with a vector space as a main ingredient in our…
Yang-Baxter operators from algebra structures appeared for the first time in [16], [17] and [8]. Later, Yang-Baxter systems from entwining structures were constructed in [5]. In this paper we show that an algebra factorisation can be…
Rota-Baxter operators and bialgebras are closely connected in several applications, such as the Connes-Kreimer renormalization framework and the operator approach to the classical Yang-Baxter equation. The concept of a Rota-Baxter system…
Combining the notions of braces and relative Rota-Baxter operators on groups in connection with the Yang-Baxter equation and a factorization theorem of Lie groups from integrable systems, relative Rota-Baxter operators on braces and…
The notion of Rota-Baxter groups was recently introduced by Guo, Lang and Sheng [{\em Adv. Math.} 387 (2021), 107834, 34 pp.] in the geometric study of Rota-Baxter Lie algebras. They are closely related to skew braces as observed by…