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 $H_0=L_0$ and $\dim H=2\dim L$ 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 $(\sigma ,\delta ,u,v)$ 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.