Pointed Hopf algebras revisited, with a view from tensor categories
Iván Angiono
TL;DR
The article surveys the classification program for finite-dimensional pointed Hopf algebras through the Lifting Method, emphasizing the categorical perspective provided by Yetter–Drinfeld modules, Nichols algebras, and cocycle deformations. It details how one fixes the coradical $H_0$, studies finite-dimensional Nichols algebras $\mathcal{B}(V)$ in ${}^{H_0}_{H_0}\mathcal{YD}$, and then classifies coradically graded connected Hopf algebras $R$ with $R^1=V$ to recover $\operatorname{gr} H\simeq R\# H_0$ and lift to $H$. The Weyl groupoid and root-system machinery play a central role in determining finite-dimensional Nichols algebras, especially for diagonal (abelian) and certain non-abelian cases, with definitive classifications in several abelian scenarios (Cartan, super, modular, unidentified families) and substantial progress for non-abelian groups via Fomin–Kirillov algebras, HV-type results, and almost-diagonal twists. The survey also connects these classifications to related problems in tensor categories, Hopf algebras without the Chevalley property, and module categories, illustrating how cocycle deformations and de-equivariantization inform the structure and interrelations of these broader contexts. Overall, the work outlines a cohesive framework for constructing and classifying pointed Hopf algebras and their categorical equivalents, highlighting key open questions and ongoing developments.
Abstract
Hopf algebras appear in connection with various problems in Pure Mathematics and Theoretical Physics, mainly through their categoriesof representations, which are examples of tensor categories. In recent years, there have been major advances in the classification of finite-dimensional Hopf algebras over the complex numbers, especially when restricted to the pointed case. In this survey, we aim to review the main results in this direction, stating the classification theorems recently obtained and the problems still open, and describing the tools needed to solve the problem by means of the so-called Lifting Method of Andruskiewitsch and Schneider. We will highlight a more categorical point of view related to these classification results, which offers for a better description of the so-called liftings up to involving cocycle deformations, and which in turn allows certain problems to be reduced to the associated graded pointed Hopf algebras. We will emphasise this point of view with applications to other contexts related to the aforementioned pointed Hopf algebras, such as finite pointed tensor categories, Hopf algebras that do not satisfy the Chevalley property and their finite-dimensional Nichols algebras, and finally module categories over the categories of representations of pointed Hopf algebras.