On stable twist 2-groups of quasi-Hopf algebras

@misc{ng2026stable,
  title = {{On stable twist 2-groups of quasi-Hopf algebras}},
  author = {Siu-Hung Ng and Xingting Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40228},
  eprint = {2610.40228},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40228},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - On stable twist 2-groups of quasi-Hopf algebras
AU  - Siu-Hung Ng
AU  - Xingting Wang
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40228
UR  - https://arxiv.org/abs/2610.40228
AB  - For any finite-dimensional quasi-Hopf algebra A, we define the notion of a stable twist of A. We show that stable twists form a strict 2-group 2ust(A), which is equivalent to OAut(CA)^rev. Given a group homomorphism ρ: G → ost(A), we determine the obstruction in H^3(G,Z_G(A)) to lift ρ to a G-action on CA. We also establish a bijection between such liftings and cocentral cleft extensions of k[G] by A. The main application of 2ust(A) is the computation of OAut(CA), together with an explicit description of the corresponding tensor autoequivalences. We apply this framework to pointed Hopf algebras A that arise as central extensions of Taft algebras by cyclic groups, and we explicitly determine all of their tensor autoequivalences. The equivariantizations of CA by ost(A) produce a new family of Hopf algebras N(n,ξ) of dimension 2nm, where ξ is a scalar of order m| n such that neither N(n,ξ) nor its dual has the Chevalley property. In dimension 24, we prove that N(6,−1) and its dual are precisely the Hopf algebras H for which neither H nor H^* has the Chevalley property. This completes the classification of nonsemisimple Hopf algebras of dimension 24.
ER  -