On radically graded finite-dimensional quasi-Hopf algebras

In this paper we continue the structure theory of finite dimensional quasi-Hopf algebras started in our previous papers. First, we completely describe the class of radically graded finite dimensional quasi-Hopf algebras over $\mathbb C$, whose radical has prime codimension. As a corollary we obtain that if $p>2$ is a prime then any finite tensor category over $\mathbb C$ with exactly $p$ simple objects which are all invertible must have Frobenius–Perron dimension $p^N$, $N=1$, 2, 3, 4, 5 or 7. Second, we construct new examples of finite dimensional quasi-Hopf algebras which are not twist equivalent to a Hopf algebra. For instance, to every finite dimensional simple Lie algebra $\mathfrak g$ and a positive integer $n$, we attach a quasi-Hopf algebra of dimension $n^{\dim\mathfrak g}$.