Construction of Rota-Baxter algebras via Hopf module algebras

Abstract

We propose the notion of Hopf module algebras and show that the projection onto the subspace of coinvariants is an idempotent Rota-Baxter operator of weight -1. We also provide a construction of Hopf module algebras by using Yetter-Drinfeld module algebras. As an application, we prove that the positive part of a quantum group admits idempotent Rota-Baxter algebra structures.Comment: 8 pages, some typos are corrected and some statements are improved; accepted for publication in Sci China Mat

Rota–Baxter coalgebras and Rota–Baxter bialgebras

“…In what follows, we recall some useful definitions and results which will be used later (see [12,16,17,19]). …”

“…Hopf module algebra (see [16]): A right H-Hopf module algebra is a right HHopf module M together with an associative multiplication µ : M ⊗ M −→ M (write µ(m ⊗ m ′ ) = mm ′ ) such that for any h ∈ H and m, m ′ ∈ M ,…”

“…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.…”

“…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]). …”

Rota–Baxter Operators on Cocommutative Weak Hopf Algebras

“…Recently, the relationships between Rota-Baxter operators and Hopf algebras have attracted many researchers, such as Jian [9], Yu, Guo and Thibon [10], and Zheng, Guo, and Zhang [11]. In 2021, Guo, Lang, and Sheng gave the notion of a Rota-Baxter operator on a group [12], moreover, based on the above notion, Goncharov introduced the definition of a Rota-Baxter operator on a cocommutative Hopf algebra and proved that the Rota-Baxter operator on the universal enveloping algebra U(L) of a Lie algebra L is one to one corresponding to the Rota-Baxter operator on L [13].…”