Cocommutative Calabi-Yau Hopf algebras and deformations

TL;DR

This paper investigates the Calabi-Yau property of cocommutative Hopf algebras using homological integrals, characterizes when skew-group algebras of universal enveloping algebras are Calabi-Yau, and classifies low-dimensional cases.

Contribution

It provides new criteria for Calabi-Yau properties in cocommutative Hopf algebras and classifies 3-dimensional cases, advancing understanding of their structure and deformations.

Findings

Skew-group algebra of a universal enveloping algebra is Calabi-Yau iff the algebra is Calabi-Yau and G ⊆ SL(g)

Classification of Noetherian cocommutative Calabi-Yau Hopf algebras of dimension ≤ 3

Characterization of Calabi-Yau Sridharan enveloping algebras and listing of all 3-dimensional cases

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 $\g$ with a finite subgroup $G$ of automorphisms of $\g$ 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 $SL(\g)$. 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.