For a finite-dimensional Hopf algebra $\mathsf {A}$ , the McKay matrix $\mathsf {M}_{\mathsf {V}}$ of an $\mathsf {A}$ -module $\mathsf {V}$ encodes the relations for tensoring the simple $\mathsf {A}$ -modules with $\mathsf {V}$ . We prove results about the eigenvalues and the right and left (generalized) eigenvectors of $\mathsf {M}_{\mathsf {V}}$ by relating them to characters. We show how the projective McKay matrix $\mathsf {Q}_{\mathsf {V}}$ obtained by tensoring the projective indecomposable modules of $\mathsf {A}$ with $\mathsf {V}$ is related to the McKay matrix of the dual module of $\mathsf {V}$ . We illustrate these results for the Drinfeld double $\mathsf {D}_n$ of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of $\mathsf {M}_{\mathsf {V}}$ and $\mathsf {Q}_{\mathsf {V}}$ in terms of several kinds of Chebyshev polynomials. For the matrix $\mathsf {N}_{\mathsf {V}}$ that encodes the fusion rules for tensoring $\mathsf {V}$ with a basis of projective indecomposable $\mathsf {D}_n$ -modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.

Let $p$, $q$ be prime numbers with ${{p}^{2}}\,<\,q,\,n\,\in \,\mathbb{N}$, and $H$ a semisimple Hopf algebra of dimension $p{{q}^{n}}$ over an algebraically closed field of characteristic 0. This paper proves that $H$ must possess one of the following two structures: (1) $H$ is semisolvable; (2) $H$ is a Radford biproduct $R\#kG$, where $kG$ is the group algebra of group $G$ of order $p$ and $R$ is a semisimple Yetter–Drinfeld Hopf algebra in $_{kG}^{kG}y\mathcal{D}$ of dimension ${{q}^{n}}$.