Hopf heap modules, Rota-Baxter operators, and their structure theorems

TL;DR

This work develops a unified heap-based framework for Hopf algebraic structures by systematically relating Hopf heaps, Hopf trusses, and Hopf braces, and by introducing Hopf heap modules with structure theorems. It shows how Hopf heap theory interplays with classical Hopf algebras via the construction $H_x(H)$ and establishes equivalences between categories of Hopf heap modules and Hopf modules, yielding explicit decompositions $M\cong H\otimes M^{coH}_x$ and $M\cong Hp(H,[-,-,-])\otimes M^{coH}_x$. The authors then extend the theory to Rota-Baxter operators on Hopf heaps and on Hopf heap modules, developing descendent Hopf heaps and associated co-brace structures, and proving structure theorems that express Rota-Baxter Hopf heap modules as $M_x^{coH}\otimes H$-type objects. Overall, the results provide a cohesive algebraic toolkit for studying heap-based generalizations of Hopf theory and their Rota-Baxter deformations, with potential applications in quantum algebra and renormalization contexts.

Abstract

This paper is primarily devoted to the study of Hopf heaps and Hopf heap modules. We redefine the structure of Hopf trusses by means of Hopf heaps, establish the connection between Hopf trusses and Hopf braces, and provide a series of examples of Hopf truss structures from the perspective of Hopf heaps. Most importantly, we introduce the conception of Hopf heap modules, and present its structure theorem. Finally, we introduce the notions of Rota-Baxter operators on Hopf heaps and Hopf heap modules, and present the structure theorem for Rota-Baxter Hopf heap modules.

Theorems & Definitions (86)

• Definition 2.1
• Remark 2.2
• Example 2.3
• Example 2.4
• Remark 2.5
• Remark 2.6
• Definition 2.7
• Remark 2.8
• Proposition 2.9
• proof