Geodesic
Bicrossproduct Hopf quasigroups
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 2, pp. 287-304.

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We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup $kM {\triangleright\blacktriangleleft} k(G)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup $G_{\mathbb O}$ as transversal in an order 128 group $X$ with subgroup $\mathbb Z_2^3$ and hence obtain a Hopf quasigroup $kG_{\mathbb O}{{>\blacktriangleleft}} k(\mathbb Z_2^3)$ as a particular case of our construction.
Classification : 16S36, 16W50, 81R50
Keywords: IP loop; octonions; quantum group; quasiHopf algebra; monoidal category; finite group; coset 
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     title = {Bicrossproduct {Hopf} quasigroups},
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Klim, Jennifer; Majid, Shahn. Bicrossproduct Hopf quasigroups. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 2, pp. 287-304. http://geodesic.mathdoc.fr/item/CMUC_2010__51_2_a13/