On stable twist 2-groups of quasi-Hopf algebras

Abstract

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 $\OAut(\C{A})^{\rev}$. Given a group homomorphism $ρ: G \to \ost(A)$, we determine the obstruction in $H^3(G,Z_G(A))$ to lift $ρ$ to a $G$-action on $\C{A}$. 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 $\OAut(\C{A})$, 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 $\C{A}$ by $\ost(A)$ produce a new family of Hopf algebras $\N(n,ξ)$ of dimension $2nm$, where $ξ$ is a scalar of order $m\mid n$ 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.