2025/02/05 by Neto, Arthur Rezende Alves, Alves, Marcelo Muniz
#16S35 #16S40 #16T05 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2502.03642
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 Hpar. It is known from a result by Dokuchaev, Exel and Piccione that when H is the algebra of a finite group G, then Hpar 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 Hpar can be written as a direct sum of unital ideals indexed by the components of the same groupoid associated to the group G.