On partial representations of pointed Hopf algebras
Arthur Rezende Alves Neto, Marcelo Muniz Alves
TL;DR
The paper develops a structural theory for partial representations of pointed Hopf algebras, showing that the partial Hopf algebra $H_{par}$ admits a direct-sum decomposition into unital ideals indexed by equivalence classes of $\mathcal{P}_1(G)$, where $G=G(H)$ is the grouplike subgroup. Central to the approach are central idempotents $\Gamma_X$ in $H_{par}$ and the base subalgebra $A_{par}$, with $H_{par} \cong \underline{A_{par} \# H}$ and $A_{par}$ decomposing as $A=\bigoplus_{X} A\,\Gamma^A_X$. The authors generalize the well-known group-algebra case to pointed Hopf algebras, derive an explicit multiplicity formula $A \cong \bigoplus_{L\le G} q(G,L) A P_L$, and illustrate the theory with rank-one nilpotent and non-nilpotent examples. These results provide a concrete, computable framework for partial representations, connect to Hopf algebroid theory, and yield new avenues for analyzing partial actions via coradical and grouplike structure.
Abstract
Partial representations of Hopf algebras were motivated by the theory of partial representations of groups. Alves, Batista e Vercruysse introduced partial representations of a Hopf algebra and showed that, as in the case of partial groups actions, a partial $H$-action on an algebra $A$ leads to a partial representation on the algebra of linear endomorphisms of $A$, and a left module $M$ over the partial smash product of $A$ by $H$ carries also a partial representation of $H$ on its algebra of linear endomorphisms. Moreover, partial representations of $H$ correspond to left modules over a Hopf algebroid $H_{par}$. It is known from a result by Dokuchaev, Exel and Piccione that when $H$ is the algebra of a finite group $G$, then $H_{par}$ is isomorphic to the algebra of a finite groupoid determined by $G$. In this work we show that if $H$ is a pointed Hopf algebra with finite group $G$ of grouplikes then $H_{par}$ can be written as a direct sum of unital ideals indexed by the components of the same groupoid associated to the group $G$.