On indicators of Hopf algebras. (Q2351745)

Let \(n\) be a natural number. In the paper under review the author establishes several results on the \(n\)-th Frobenius-Schur indicator \(\nu_n(H)\) of a finite dimensional Hopf algebra \(H\), as introduced by \textit{Y. Kashina, S. Montgomery} and \textit{S.-H. Ng} [in Isr. J. Math. 188, 57-89 (2012; Zbl 1260.16027)]. Furthermore, the author extends the definition of \(\nu_n(H)\) for all \(n\in\mathbb Z\) and proves that for \(n\leq 0\), \(\nu_n(H)\) is still a gauge invariant of \(H\). For negative values of \(n\) it is shown that \(\nu_n(H)=\nu_{-1}(H)\nu_{-n}(H)\) and moreover \(\nu_n(H)=\nu_{-n}(H)\) if \(H\) is unimodular. Relations with the linear recursiveness of the sequence \(\nu_n(H)\), proved by Kashina, Montgomery and Ng, are investigated. One of the main results of the paper is the cyclotomic integrality of \(\nu_n(H)\), more precisely, it is shown that for all \(n\geq 1\), \(\nu_n(H)\in\mathbb Z[e^{2\pi\sqrt{-1}}/N]\), where \(N=nm\) and \(m\) is the order of the second power of the antipode of \(H\). The formula \(\nu_n(D(H))=|\nu_n(H)|^2\) is also established for all \(n\geq 1\), proving a conjecture made by Kashina, Montgomery and Ng [in loc. cit.]. It is also shown that if \(H\) is a filtered finite dimensional Hopf algebra, that is, \(H\) is a Hopf algebra in the symmetric monoidal category of filtered vector spaces, then for all \(n\geq 1\), \(\nu_n(H)=\nu_n(\text{gr\,}H)\), where \(\text{gr\,}H\) is the associated graded Hopf algebra, which is a Hopf algebra in the symmetric monoidal category of graded vector spaces. This is applied to study the indicators of the finite dimensional pointed Hopf algebras \(u(\mathcal D,\lambda,\mu)\), described by \textit{N. Andruskiewitsch} and \textit{H.-J. Schneider} [Ann. Math. (2) 171, No. 1, 375-417 (2010; Zbl 1208.16028)]. A factorization formula for the second indicator, which coincides with the trace of the antipode, is given in this context. As an application of the results of the paper, the author determines the \(n\)-th indicator of a Taft algebra and of the small quantum group \(u_q(\mathfrak{sl}_2)\), for all \(n\geq 1\).