Characteristic classes for cohomology of split Hopf algebra extensions

We introduce characteristic classes for the spectral sequence associated to a split short exact sequence of Hopf algebras. These classes can be seen as obstructions for the vanishing of differentials in the spectral sequence. We give a decomposition theorem and interpret our results in the settings of group and Lie algebra extensions. As applications, we derive several results concerning the collapse of the (Lyndon–)Hochschild–Serre spectral sequence and the order of characteristic classes.