Cohomological Maschke's Theorem for Generalized Digroups

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 $\ddot{I}$-side, while the obstruction to full splitting is described by cocycles and identified with $Ex{t}_{Rep(D)}^1(Q,W)$. We also derive a spectral sequence with consequences for splitting and non-semisimplicity.