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 RotaBaxter bialgebra is introduced and some examples are provided.Key words: Rota-Baxter bialgebras, Radford biproduct, Rota-Baxter coalgebras. Mathematics Subject Classification: 16T05, 16W991 Introduction A Rota-Baxter algebra, also called a Baxter algebra, is an associative algebra with a linear operator which generalizes the algebra of continuous functions with the integral operator. More precisely, for a given field K and λ ∈ K, a Rota-Baxter K-algebra (of weight λ) is a K-algebra R together with a K-linear map P : R −→ R such that P (x)P (y) = P (xP (y)) + P (P (x)y) + λP (xy)for all x, y ∈ R. Such a linear operator is called a Rota-Baxter operator (of weight λ). Rota-Baxter algebras were introduced in [21] in the context of differential operators on commutative Banach algebras and since [4] intensively studied in probability and combinatorics, and more recently in mathematical physics, such as dendriform algebras * Corresponding author 1 (see [3,6,7,11]), free Rota-Baxter algebras (see [5,8,9,13,14]), Lie algebras (see [2,18]), multiple zeta values (see [8,15]) and Connes-Kreimer renormalization theory in quantum field theory (see [10]), etc. One can refer to the book [12] for the detailed theory of Rota-Baxter algebras. In 2014, Jian used Yetter-Drinfeld module algebras to construct Hopf module algebras, and then the corresponding Rota-Baxter algebras were derived (see [16]). Based on the dual method in the Hopf algebra theory, Jian and Zhang in [17] defined the notion of Rota-Baxter coalgebras and also provided various examples of the new object, including constructions by group-like elements and by smash coproduct.In this paper, we will continue to investigate the properties of Rota-Baxter coalgebras. 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 and some examples are provided. PreliminariesThroughout this paper, we assume that all vector spaces, algebras, coalgebras and tensor products are defined over a field K. An algebra is always assumed to be associative, but not necessarily unital. A coalgebra is always assumed to be coassociative, but not necessarily counital. Now, let C be a coalgebra. We use Sweedler's notation for the comultiplication (see [19]): ∆(c) = c 1 ⊗ c 2 for any c ∈ C. Denote the category of leftIn what follows, we recall some useful definitions and results which will be used later (see [12,16,17,19]).Hopf module (see [19]): A right (resp. left) H-Hopf module is a vector space M equipped simultaneously with a right (resp. left) H-module structure and a right (resp. left) H-comodule structure ρ R (resp. ρ L ) such that(resp. ρ L (h · m) = ∆(h)ρ L (m)) for all h ∈ H and m ∈ M .Remark ...
Abstract. Let (A, α) and (B, β) be two Hom-Hopf algebras. In this paper, we construct a class of new Hom-Hopf algebras: R-smash product (A♮ R B, α ⊗ β). Moreover, necessary and sufficient conditions for (A♮ R B, α ⊗ β) to be a cobraided Hom-Hopf algebra are given.
We introduce and discuss the notions of Rota–Baxter bialgebra equation systems and Rota–Baxter Hopf algebras. Then we construct a lot of examples based on Hopf quasigroups.
Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that is compatible with the Rota-Baxter operator, called the Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the context of Rota-Baxter algebras. The study of the coboundary case leads to an enrichment of the associative Yang-Baxter equation (AYBE) to Rota-Baxter algebras. Antisymmetric solutions of the equation are used to construct Rota-Baxter ASI bialgebras. The notions of an O-operator on a Rota-Baxter algebra and a Rota-Baxter dendriform algebra are also introduced to produce solutions of the AYBE in Rota-Baxter algebras and thus to provide Rota-Baxter ASI bialgebras. An unexpected byproduct is that a Rota-Baxter ASI bialgebra of weight zero gives rise to a quadri-bialgebra instead of bialgebra constructions for the dendriform algebra.
In this paper, we present a dual version of T. Brzeziński’s results about Rota–Baxter systems which appeared in [Rota–Baxter systems, dendriform algebras and covariant bialgebras, J. Algebra 460 (2016) 1–25]. Then as a generalization to bialgebras, we introduce the notion of Rota–Baxter bisystem and construct various examples of Rota–Baxter bialgebras and bisystems in dimensions 2, 3 and 4. On the other hand, we introduce a new type of bialgebras (named mixed bialgebras) which consist of an associative algebra and a coassociative coalgebra satisfying the compatible condition determined by two coderivations. We investigate coquasitriangular mixed bialgebras and the particular case of coquasitriangular infinitesimal bialgebras, where we give the double construction. Also, we show in some cases that Rota–Baxter cosystems can be obtained from a coquasitriangular mixed bialgebras.
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