Approximation of graded bialgebras
Giovanna Carnovale, Francesco Esposito, Lleonard Rubio y Degrassi
TL;DR
This work develops a braided-category framework for approximating connected bialgebras by introducing $\mathcal{CB}^{\le d}(\mathcal{V})$, the category of connected bialgebras modulo $d+1$, and two fundamental functors: the truncation $\Theta_d$ and the extension $\Psi_{d!}$. The $d$-th approximation functor $F_d=\Psi_{d!}\circ\Theta_d$ is shown to be left-adjoint to a dual construction, and an inverse-limit result demonstrates that $\varprojlim_d\mathcal{CB}^{\le d}(\mathcal{V})$ is equivalent to the full category $\mathcal{CB}(\mathcal{V})$, unifying finite-stage truncations with the entire theory. The framework is shown to be compatible with category equivalences, notably cocycle twists in Yetter-Drinfeld settings, ensuring that twist-equivalent algebras have corresponding equivalent approximations; this yields consequences for Nichols algebras under twists. Collectively, the results provide a robust, generalizable approach to understand finite-presentability and approximation phenomena for connected bialgebras in braided categories, with geometric translations anticipated via a forthcoming link to perverse sheaves on $\mathrm{Sym}(\mathbb{C})$.
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.