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Classification of finite-dimensional Nichols algebras of rank two in twisted Yetter--Drinfeld categories
Abstract
Let G be a finite non-abelian group, let Φ∈ Z3(G,ℂ×) be normalized, and let V,W be finite-dimensional simple objects of GG𝒴𝒟Φ. We classify, up to interchange, the braided-indecomposable pairs (V,W) whose supports generate G and for which ℬ(V⊕ 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 A2, B2, or G2, and the dimension is determined explicitly. A new phenomenon occurs {in the Γ2 case}: the twisted setting admits a family of type G2 absent from the ordinary Γ2 classification and we construct an explicit example over a non-abelian group of order 16.
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BibTeX
@misc{li2026classification,
title = {{Classification of finite-dimensional Nichols algebras of rank two in twisted Yetter--Drinfeld categories}},
author = {Bowen Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.09812},
eprint = {2609.09812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09812},
}
RIS
TY - UNPB
TI - Classification of finite-dimensional Nichols algebras of rank two in twisted Yetter--Drinfeld categories
AU - Bowen Li
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2609.09812
UR - https://arxiv.org/abs/2609.09812
AB - Let G be a finite non-abelian group, let Φ∈ Z^3(G,ℂ^×) be normalized, and let V,W be finite-dimensional simple objects of _G^G𝒴𝒟^Φ. We classify, up to interchange, the braided-indecomposable pairs (V,W) whose supports generate G and for which ℬ(V⊕ 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.
ER -