The authors introduce the notion of homological integral on Hopf algebras, the underlying algebras of which are Artin-Schelter Gorenstein, and use it to study regular Hopf algebras.
Recall that an integral in a Hopf algebra \(H\) can be considered as a non-zero element of \(\operatorname{Hom}_H(k,H)\). \(H\), as an algebra, is called Artin-Schelter Gorenstein if it has finite (left and right) injective dimension \(d\) and satisfies the following conditions (for left and right \(H\)-modules): \(\text{Ext}_H^d(k,H)\cong k\), \(\text{Ext}^i_H(k,H)=0\), for all \(i\neq d\). Let \(H\) be such a Hopf algebra. The space \(\text{Ext}^d_H(k,H)\) is called the left or right homological integral of \(H\) and denoted by \(\int^l\) or \(\int^r\), depending on whether \(k\) and \(H\) are considered as left or right \(H\)-modules.
In the first part of the work the authors show that a Noetherian, finitely generated, PI Hopf algebra \(H\) has finite global dimension iff the following two conditions are satisfied. Condition 1: The map \(\text{Ext}^d_H(\int^r,\varepsilon)\colon\text{Ext}^d_H(\int^r,H)\to\text{Ext}^d_H(\int^r,k)\) is an isomorphism, where \(\int^r\) denotes the right homological integral of \(H\). Condition 2: For every simple left \(H\)-module \(T\not\cong\int^r\), \(\text{Ext}^d_H(T,k)=0\).
The second part of the work is devoted to the study of Hopf algebras of Gelfand-Kirillov dimension 1 and of finite global dimension. It is shown in particular that if \(H\) is prime and \(\int^r=\int^l\) then it is commutative, if moreover \(k\) is algebraically closed then \(H\) is either \(k[x]\) or \(k[x^{\pm 1}]\).