On a family of Hopf algebras of dimension 72. (English) Zbl 1282.16036

Andruskiewitsch, Nicolás; Vay, Cristian

Bull. Belg. Math. Soc. - Simon Stevin 19, No. 3, 415-443 (2012).

The authors apply the lifting method to investigate a family of Hopf algebras of dimension \(72\) whose coradical is isomorphic to the algebra of functions on \(\mathbb S_3\). It is determined the lattice of submodules of the so-called Verma modules. As a consequence the authors classify all simple modules. It is shown that these Hopf algebras are unimodular (as well as their duals) but not quasitriangular; also, they are cocycle deformations of each other.
It is believed that the representation theory of Hopf algebras with coradical which is isomorphic to the algebra of functions on a group \(G\) might be helpful to study Nichols algebras and deformations.

Reviewer: Yang Shilin (Beijing)

MSC:

16T05

Hopf algebras and their applications

Keywords:

representations of Hopf algebras; lifting method; finite-dimensional Hopf algebras; pointed Hopf algebras; coradicals; simple modules; Nichols algebras