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IP loop; octonions; quantum group; quasiHopf algebra; monoidal category; finite group; coset
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.
[1] Albuquerque H., Majid S: Quasialgebra structure of the octonions. J. Algebra 220 (1999), 188–224. DOI 10.1006/jabr.1998.7850 | MR 1713433 | Zbl 0999.17006
[2] Beggs E.J.: Making non-trivially associated tensor categories from left coset representatives. J. Pure and Applied Algebra 177 (2003), 5–41. DOI 10.1016/S0022-4049(02)00119-6 | MR 1948835 | Zbl 1037.18004
[3] Drinfeld V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1 (1990), 1419–1457. MR 1047964
[4] Klim J., Majid S.: Hopf quasigroups and the algebraic $7$-sphere. J. Algebra(to appear). MR 2629701
[5] Majid S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge, 1995. MR 1381692 | Zbl 0857.17009
[6] Perez-Izquierdo J., Shestakov I.P.: An envelope for Malcev algebras. J. Algebra 272 (2004), 379–393. DOI 10.1016/S0021-8693(03)00389-2 | MR 2029038 | Zbl 1077.17027
[7] Smith J.D.H.: Introduction to Quasigroups and their Representations. Taylor & Francis, 2006. MR 2268350 | Zbl 1122.20035
[8] Zhu Y.: Hecke algebras and representation ring of Hopf algebras. AMS/IP Stud. Adv. Math. 20, Amer. Math. Soc., Providence, RI, 2001, pp. 219–227. MR 1830177 | Zbl 1064.20011
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