Calabi-Yau pointed Hopf algebras of finite Cartan type

Abstract

We study the Calabi–Yau property of pointed Hopf algebra $U(\mathcal{D},\lambda )$ of finite Cartan type. It turns out that this class of pointed Hopf algebras constructed by N. Andruskiewitsch and H.-J. Schneider contains many Calabi–Yau Hopf algebras. To give concrete examples of new Calab–Yau Hopf algebras, we classify the Calabi–Yau pointed Hopf algebras $U(\mathcal{D},\lambda )$ of dimension less than 5.