Year: 2026
Type: preprint
Abstract: For a module \(M\) over a cocommutative Hopf algebra \(\mathscr A\), we define extensive amenability by amenability of its symmetric algebra as a module over \(\operatorname{Sym}(M)\#\mathscr A\). Using the author's coalgebraic rounding and quotient theorems, we prove that nonzero extensively amenable modules are amenable, that every module over an amenable Hopf algebra is extensively amenable, and that extensive amenability is preserved and reflected by short exact sequences. For permutation modules this definition agrees, over every field, with extensive amenability of group actions. A final section gives the modifications for exterior algebras and establishes the comparison between the two definitions in several cases.
Source: arXiv (Cornell University)
Author: Bartholdi, Laurent
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215796802