The Calabi–Yau property of smash products

Let H be a semisimple Hopf algebra and R a braided Hopf algebra in the category of Yetter–Drinfeld modules over H. When R is a Calabi–Yau algebra, a necessary and sufficient condition for to be a Calabi–Yau Hopf algebra is given. Conversely, when H is the group algebra of a finite group and the smash product is a Calabi–Yau algebra, we give a necessary and sufficient condition for the algebra R to be a Calabi–Yau algebra.