Cocommutative Hopf algebra actions and the Connes spectrum
The Connes spectrum, as introduced [in Contemp. Math. 130, 311-334 (1992; Zbl 0782.16021)] by the authors and the reviewer, is too technically to formally define here. Roughly speaking, if \(H\) is a finite-dimensional, semisimple Hopf algebra over a splitting field \(K\), then \(\text{CS}(A,H)\) is a subset of the irreducible representations of \(H\) determined by the action of \(H\) on the \(H\)-module algebra \(A\). Indeed, the Connes spectrum \(\text{CS}(A,H)\) is intimately related to the smash product \(A \# H\) and can be used to study the simplicity and primeness of the latter ring. This paper proves a lovely, rather surprising result, namely if \(H\) is cocommutative and if \(A\) is \(H\)-prime, then \(\text{CS}(A,H)\) is closed under multiplication. In other words, if \(\pi,\chi \in\text{CS}(A,H)\), then all irreducible constituents of the tensor representation \(\pi \otimes \chi\) are also contained in the Connes spectrum. As a consequence, in the case of group algebras, it follows that \(\text{CS}(A,K[G])\) consists precisely of all the irreducible \(K[G]\)- modules which factor through \(K[G/N]\) for some fixed normal subgroup \(N\) of \(G\). The paper then goes on to study the relationship between this Connes kernel \(N=\text{CK}(A,K[G])\) and the action of \(G\) on \(A\).
finite-dimensional, semisimple Hopf algebra
scientific article; zbMATH DE number 589953