Approximation of graded bialgebras

TL;DR

This paper introduces a new categorical framework for approximating connected bialgebras using inverse systems, extending previous equivalences and demonstrating compatibility with cocycle twisting in Yetter-Drinfeld modules.

Contribution

It develops a novel approximation scheme for connected bialgebras, generalizes it beyond degree 1 generation, and links it to existing categorical equivalences.

Findings

Categories of connected bialgebras modulo d+1 form an inverse system.

The inverse limit of these categories is equivalent to the category of connected bialgebras.

Approximation is compatible with cocycle twisting in Yetter-Drinfeld modules.

Abstract

Motivated by an equivalence of categories established by Kapranov and Schechtman, we introduce, for each non-negative integer d, the category of connected bialgebras modulo d+1. We show that these categories fit into an inverse system of categories whose inverse limit category is equivalent to the category of connected bialgebras. In addition, we extend the notion of approximation of connected bialgebras to those that are not necessarily generated in degree 1 and show that, for connected bialgebras in the category of Yetter-Drinfeld modules over a Hopf algebra, approximation is compatible with cocycle twisting.