Hopf algebra extensions arising from semi-direct products of groups
Let \(G=N\rtimes H\) be a semidirect product of finite groups and let \(k\) be a field. The object of study in this paper is the cohomology classifying Hopf algebra extensions of the form \((kN)^*\to C\to kH\). Such extensions, arising from a ``matched pair'' of groups, were studied by \textit{G. I. Kac} [Mat. Sb., N. Ser. 76(118), 473-496 (1968; Zbl 0205.03301)], and more recently by, for example, \textit{A. Masuoka} [Trans. Am. Math. Soc. 352, No. 8, 3837-3879 (2000; Zbl 0960.16045)]. An exact sequence (first studied by Kac) connecting group cohomology with the above Hopf algebra extension is studied here in an explicit way, using e.g. work of \textit{K.-i. Tahara} [Math. Z. 129, 365-379 (1972; Zbl 0238.20068)]. Several classes of examples are calculated in detail.
scientific article; zbMATH DE number 1797480