Normalized quadratic extensions of pointed Hopf algebras
Abstract
Let L be a finite-dimensional pointed Hopf algebra over an algebraically closed field. We establish an intrinsic characterization of index-two extensions: every Hopf algebra H containing L as a Hopf subalgebra and satisfying H0=L0 and dim H=2dim L is a normalized quadratic extension of L. For a fixed support (g,h), such extensions are described by a coalgebra Hochschild $2$-cocycle together with Ore-type data $(σ,δ,u,v)$ satisfying explicit compatibility conditions. Their isomorphism classes are parameterized by the orbits of gauge transformations and Hopf automorphisms of L. As an application, we classify non-connected pointed Hopf algebras of dimension $16$ in characteristic $2$ with one-dimensional infinitesimal braiding whose diagram is not a Nichols algebra.
Related
- Pre-Nichols algebras and generation of pointed Hopf algebras in positive characteristic
- Freeness and divisibility for right H-simple left H-comodule algebras over a pointed Hopf algebra H
- A q-Weyl Freeness Principle for Nichols Algebras and Pointed Hopf Algebras of Square-Free Dimension
- Exact Factorizations of Rank-One Pointed Hopf Algebras in Positive Characteristic, I
- Dihedral pointed fusion categories: diagrammatics, 3-cocycle representatives, and planar algebras
BibTeX
@misc{xiong2026normalized,
title = {{Normalized quadratic extensions of pointed Hopf algebras}},
author = {Rongchuan Xiong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11622},
eprint = {2609.11622},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11622},
}
RIS
TY - UNPB
TI - Normalized quadratic extensions of pointed Hopf algebras
AU - Rongchuan Xiong
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11622
UR - https://arxiv.org/abs/2609.11622
AB - Let L be a finite-dimensional pointed Hopf algebra over an algebraically closed field. We establish an intrinsic characterization of index-two extensions: every Hopf algebra H containing L as a Hopf subalgebra and satisfying H_0=L_0 and dim H=2dim L is a normalized quadratic extension of L. For a fixed support (g,h), such extensions are described by a coalgebra Hochschild $2$-cocycle together with Ore-type data $(σ,δ,u,v)$ satisfying explicit compatibility conditions. Their isomorphism classes are parameterized by the orbits of gauge transformations and Hopf automorphisms of L. As an application, we classify non-connected pointed Hopf algebras of dimension $16$ in characteristic $2$ with one-dimensional infinitesimal braiding whose diagram is not a Nichols algebra.
ER -