Irreducible representations of crossed products
In 1975, I. Kaplansky conjectured that if \(H\) is a finite-dimensional semisimple Hopf algebra over an algebraically closed field, then the dimension of each irreducible \(H\)-module divides the dimension of \(H\). The main result of this paper is a verification of this conjecture for fields of characteristic 0 and Hopf algebras of prime power dimension. The proof is based on a rather interesting general approach. Let \(k\) be an algebraically closed field and let \(H\) be a finite-dimensional Hopf algebra over \(k\) with the property that the dimension of each irreducible \(H\)-module divides the dimension of \(H\). If \(G\) is a finite group whose order is not divisible by the characteristic of \(k\), then it is shown here that any crossed product of the form \(H\#kG\) or \(H\#kG^*\) has this same property with respect to irreducible modules. In particular, this divisibility property is satisfied by Hopf algebras constructed as a sequence of crossed products involving group algebras and their duals. Finally, it is shown that if \(\text{char }k=0\), then any Hopf algebra of prime power dimension is of the above form, thereby yielding the main result.
scientific article; zbMATH DE number 1308202
scientific article