Characteristic classes for cohomology of split Hopf algebra extensions

TL;DR

This paper introduces characteristic classes for spectral sequences of split Hopf algebra extensions, revealing their role as obstructions and providing decomposition results, with applications to group and Lie algebra extensions.

Contribution

It develops a new framework of characteristic classes for spectral sequences in Hopf algebra extensions, linking them to obstructions and spectral sequence collapse phenomena.

Findings

Characteristic classes act as obstructions to differential vanishing.

Decomposition theorem for spectral sequences in Hopf algebra extensions.

Results apply to group and Lie algebra extension spectral sequences.

Abstract

We introduce characteristic classes for the spectral sequence associated to a split short exact sequence of Hopf algebras. We show that these characteristic classes can be seen as obstructions for the vanishing of differentials in the spectral sequence and prove a decomposition theorem. We also interpret our results in the settings of group and Lie algebra extensions and prove some interesting corollaries concerning the collapse of the (Lyndon-)Hochschild-Serre spectral sequence and the order of characteristic classes.