On the representations of a family of pointed Hopf algebras

Abstract

For each $\ell \ge 1$ and $\lambda ,\mu \in \mathbb{k}$, we study the representations of a family of pointed Hopf algebras ${\mathcal{A}}_{\lambda ,\mu }$. These arise as Hopf cocycle deformations of the graded algebra ${\mathcal{F}\mathcal{K}}_3\#\mathbb{k}{\mathbb{G}}_{3,\ell }$, where ${\mathcal{F}\mathcal{K}}_3$ is the Fomin-Kirillov algebra and ${\mathbb{G}}_{3,\ell }$ is a given non-abelian finite group. We compute the simple modules, their projective covers and formulate a description of tensor products. We observe that our results are fundamentally different according to the shape of the Hopf cocycle involved in the deformation.

39 pages