Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1807.07154

Pointed Hopf algebras: a guided tour to the liftings

Iván Angiono·Agustín García Iglesias
View original source ↗Read source document ↗Export metadata ↓

Abstract

This article serves a two-fold purpose. On the one hand, it is a survey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. On the other, we present a step-by-step guide to carry out the strategy developed to construct the liftings. As an example, we conclude the work with the classification of pointed Hopf algebras of type $B_2$.

Explore related subjects

math.QAmath.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Iván Angiono, Agustín García Iglesias. 2018-07-31. Pointed Hopf algebras: a guided tour to the liftings. https://arxiv.org/abs/1807.07154

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Classification of finite-dimensional Nichols algebras of rank two in twisted Yetter--Drinfeld categories

Let \(G\) be a finite non-abelian group, let \(Φ\in Z^3(G,\mathbb C^\times)\) be normalized, and let \(V,W\) be finite-dimensional simple objects of \({}_G^G\mathcal{YD}^Φ\). We classify, up to interchange, the braided-indecomposable pairs \((V,W)\) whose supports generate \(G\) and for which \(\mathcal B(V\oplus W)\) is finite-dimensional. The classification consists of eight cases with five possible support quandles. In every case the Cartan graph is standard of type \(A_2\), \(B_2\), or \(G_2\), and the dimension is determined explicitly. A new phenomenon occurs {in the \(Γ_2\) case}: the twisted setting admits a family of type \(G_2\) absent from the ordinary \(Γ_2\) classification and we construct an explicit example over a non-abelian group of order \(16\).

math.QA↗
arXiv2026

Facets of nondegeneracy: central type factor groups, Heisenberg representations and Hadamard matrices

Given a pair of complementary subgroups $P$ and $N$ of a finite group $G$, we first discuss how the same type of functions $\mathbb E\colon P\times N\to \mathbb T$, which we call bi-$1$-cocycles, appear naturally in the study of projective representations of $G$ of Heisenberg type and second cohomology of $G$ and of its dual quantum group $\widehat G$. We show then that such representations are irreducible and the corresponding ordinary and dual $2$-cocycles are nondegenerate if and only if the matrix $(\mathbb E(p,n))_{p,n}$ is invertible, if and only if it is Hadamard. We illustrate the result with examples arising from symplectic nilpotent Lie algebras over finite fields with two complementary Lagrangian subalgebras.

math.QA↗