Cohomological Maschke's Theorem for Generalized Digroups

TL;DR

This paper extends Maschke's theorem to generalized digroups by constructing an associative algebra and analyzing representation categories, cocycles, and spectral sequences.

Contribution

It introduces a new framework for Maschke-type phenomena in generalized digroups, including an associative enveloping algebra and splitting criteria.

Findings

Representation category equivalent to modules over the constructed algebra

Splitting occurs under a Maschke-type condition on the group component

Spectral sequence provides insights into splitting and non-semisimplicity

Abstract

We study Maschke-type phenomena in the representation theory of generalized digroups. For a generalized digroup $D$, we construct an associative enveloping algebra $A_D$ and prove that $Rep(D)$ is equivalent to the category of left $A_D$-modules. Under a Maschke-type condition on the group component, we show that short exact sequences split on the $\rho$-side, while the obstruction to full splitting is described by cocycles and identified with $Ext^1_{Rep(D)}(Q,W)$. We also derive a spectral sequence with consequences for splitting and non-semisimplicity.