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Keywords:
IP loop; octonions; quantum group; quasiHopf algebra; monoidal category; finite group; coset
IP loop; octonions; quantum group; quasiHopf algebra; monoidal category; finite group; coset
Summary:
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.
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