A generalized power map for Hopf algebras (Q2762040)

Let \(H\) be a finite-dimensional Hopf algebra over a field \(k\) and let \(n\) be a positive integer. In [\textit{Y. Kashina}, Commun. Algebra 27, No. 3, 1261-1273 (1999; Zbl 0923.16032)], the author considered the map \([n]\colon H\to H\), \([n]=m^{n-1}\Delta_{n-1}\), where \(m^{n-1}\colon H^{\otimes n}\to H\) is the iterated multiplication, and \(\Delta_{n-1}\colon H\to H^{\otimes n}\) is the iterated comultiplication. In particular, she asked when there exists \(n\) such that \([n]\) is the trivial map in the sense of Hopf algebras. If \(H=kG\) is a group algebra, then the smallest such \(n\) is the exponent of \(G\).NEWLINENEWLINENEWLINEIn the present paper, the existence of such \([n]\) is established for more examples of semisimple Hopf algebras, including the Drinfeld double of a group algebra. It turns out that the question posed by Kashina in these papers was solved by Etingof and Gelaki, for semisimple Hopf algebras, via the beautiful theory of the exponent of a finite-dimensional Hopf algebra [\textit{P. Etingof} and \textit{S. Gelaki}, Math. Res. Lett. 6, No. 2, 131-140 (1999; Zbl 0954.16028)]. Indeed, the exponent of a finite-dimensional Hopf algebra is defined as the smallest \(n\) such that the map \([[n]]:=m^{n-1}(\text{id}\otimes{\mathcal S}^2\otimes\cdots\otimes{\mathcal S}^{2(n-1)})\Delta_{n-1}\) is the trivial map. Clearly, \([[n]]=[n]\) for \(H\) semisimple, but \([[n]]\) has better interpretations in terms of the Drinfeld double of \(H\). For \(H\) non-semisimple, it is expected that the exponent of \(H\) is infinite, as it is for pointed non-semisimple Hopf algebras; see also [\textit{P. Etingof} and \textit{S. Gelaki}, Math. Res. Lett. 9, No. 2-3, 277-287 (2002; Zbl 1012.16038)].NEWLINENEWLINEFor the entire collection see [Zbl 0958.00024].