Hopf automorphisms and twisted extensions. (Q2803555)

This paper deals with the smash coproduct \(K\) of a Hopf algebra \(A\) over a field \(k\) and \(k^G\) the Hopf algebra of functions of \(G\) to \(k\), where \(G\) is a finite group that acts on \(A\). Let \(\tau\) be an automorphism of \(A\), the twisted exponent with respect to \(\tau\) is the smallest positive integer \(n\) which is a multiple of the order of \(\tau\) and \(x^{[n,\tau]}=\sum_xx_1(\tau\cdot x_2)\cdots(\tau\cdot x_n)=\varepsilon(x)1\) where the \(x_i\)'s are the components of \(\Delta_A\) applied \(n-1\) times to \(x\).NEWLINENEWLINE The first result shows that the exponent of \(K\) is the least common multiple of exponent of \(G\) and the exponents of \(A\) with respect to the automorphism induced by \(g\in G\). Given a \(K\)-module \(M\), it is shown that \(\nu_m^K(M)=\sum_{x\in G,\;x^m=1}\nu_{m,x^{-1}}^A(M_x)\), where \(\nu_m(K)\) denotes the \(m\)th Frobenius-Schur indicator of \(M\), \(\nu_{m,x^{-1}}^A\) the \(m\)th twisted Frobenius-Schur indicator and \(M_x=p_x\cdot M\) for \(\{p_x\mid x\in G\}\) a basis of \(k^G\). Next, the notion of twisted indicator is extended to non-semisimple Hopf algebras and a connection between the twisted indicator of the Hopf algebra and the indicator of the smash coproduct is stablished. Finally a non-semisimple example is studied.