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Twisted Post-Hopf Algebras, Twisted Relative Rota-Baxter Operators and Hopf Trusses

José Manuel Fernández Vilaboa, Ramón González Rodríguez, Brais Ramos Pérez

TL;DR

This work unifies Hopf trusses, weak twisted post-Hopf algebras, and twisted relative Rota--Baxter operators within braided monoidal categories by establishing categorical equivalences under a relaxed cocommutativity framework. It introduces weak twisted post-Hopf algebras and their cocycle data, proving an isomorphism with Hopf trusses and deriving new Hopf-algebra structures on cocycle images when split idempotents exist. By defining weak twisted relative Rota--Baxter operators and constructing adjoint functors between the associated categories, the paper extends Li–Sheng–Tang’s correspondences to the twisted, braided setting and, under cocommutativity-class conditions, yields full equivalences among the relevant categories, including their cocommutative subcategories. These results provide a coherent, generalized framework for generating and translating Hopf-truss structures from twisted Rota--Baxter data, with explicit constructions and examples from idempotent endomorphisms and semidirect product decompositions.

Abstract

The present article is devoted to studying the categorical relationships between the categories of Hopf trusses, weak twisted post-Hopf algebras, introduced by Wang (2023), and weak twisted relative Rota-Baxter operators. The latter objects are a generalisation of the relative Rota-Baxter operators defined by Li-Sheng-Tang (2024), where the Rota-Baxter condition is modified through a cocycle. Under certain conditions, this work shows that the three aforementioned categories are equivalent.

Twisted Post-Hopf Algebras, Twisted Relative Rota-Baxter Operators and Hopf Trusses

TL;DR

This work unifies Hopf trusses, weak twisted post-Hopf algebras, and twisted relative Rota--Baxter operators within braided monoidal categories by establishing categorical equivalences under a relaxed cocommutativity framework. It introduces weak twisted post-Hopf algebras and their cocycle data, proving an isomorphism with Hopf trusses and deriving new Hopf-algebra structures on cocycle images when split idempotents exist. By defining weak twisted relative Rota--Baxter operators and constructing adjoint functors between the associated categories, the paper extends Li–Sheng–Tang’s correspondences to the twisted, braided setting and, under cocommutativity-class conditions, yields full equivalences among the relevant categories, including their cocommutative subcategories. These results provide a coherent, generalized framework for generating and translating Hopf-truss structures from twisted Rota--Baxter data, with explicit constructions and examples from idempotent endomorphisms and semidirect product decompositions.

Abstract

The present article is devoted to studying the categorical relationships between the categories of Hopf trusses, weak twisted post-Hopf algebras, introduced by Wang (2023), and weak twisted relative Rota-Baxter operators. The latter objects are a generalisation of the relative Rota-Baxter operators defined by Li-Sheng-Tang (2024), where the Rota-Baxter condition is modified through a cocycle. Under certain conditions, this work shows that the three aforementioned categories are equivalent.
Paper Structure (4 sections, 28 theorems, 99 equations)

This paper contains 4 sections, 28 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. (Weak) twisted post-Hopf algebras and Hopf trusses
  4. Twisted relative Rota--Baxter operators and Hopf trusses

Key Result

Theorem 2.10

Let $X=(X,\eta_{X},\mu_{X},\varepsilon_{X},\delta_{X},\lambda_{X})$ and $H=(H,\eta_{H},\mu_{H},\varepsilon_{H},\delta_{H},\lambda_{H})$ be Hopf algebras in $\sf{C}$ such that there exists a morphism $\varphi_{H}\colon X\otimes H\rightarrow H$ satisfying the following conditions: Then, $\varphi_{H}\circ(X\otimes\eta_{H})=\varepsilon_{X}\otimes\eta_{H}$ holds.

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 44 more