Cocommutative Calabi–Yau Hopf algebras and deformations

Abstract

The Calabi–Yau property of cocommutative Hopf algebras is discussed by using the homological integral, a recently introduced tool for studying infinite dimensional AS-Gorenstein Hopf algebras. It is shown that the skew-group algebra of a universal enveloping algebra of a finite dimensional Lie algebra $\mathfrak{g}$ with a finite subgroup G of automorphisms of $\mathfrak{g}$ is Calabi–Yau if and only if the universal enveloping algebra itself is Calabi–Yau and G is a subgroup of the special linear group $\mathit{SL}(\mathfrak{g})$. The Noetherian cocommutative Calabi–Yau Hopf algebras of dimension not larger than 3 are described. The Calabi–Yau property of Sridharan enveloping algebras of finite dimensional Lie algebras is also discussed. We obtain some equivalent conditions for a Sridharan enveloping algebra to be Calabi–Yau, and then partly answer a question proposed by Berger. We list all the nonisomorphic 3-dimensional Calabi–Yau Sridharan enveloping algebras.