The Complexity of Crossed Products

Let $H$ be a finite-dimensional Hopf algebra, $A$ be a finite-dimensional algebra measured by $H$ and $A\mathbin{\#_\sigma}H$ be a crossed product. In this paper, we first show that if $H$ is semisimple as well as its dual $H^*$, then the complexity of $A\mathbin{\#_\sigma} H$ is equal to that of $A$. Furthermore, we prove that the complexity of a finite-dimensional Hopf algebra $H$ is equal to the complexity of the trivial module $_Hk$. As an application, we prove that the complexity of Sweedler's 4-dimensional Hopf algebra $H_4$ is equal to $1$.