Bicrossproduct Hopf quasigroups

Jennifer Klim; Shahn Majid

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 2, page 287-304
  • ISSN: 0010-2628

Abstract

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We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup k M ▹ ◂ k ( G ) from every group X with a finite subgroup G ⊂ X and IP quasigroup transversal M ⊂ X subject to certain conditions. We identify the octonions quasigroup G 𝕆 as transversal in an order 128 group X with subgroup ℤ 2 3 and hence obtain a Hopf quasigroup k G 𝕆 > ◂ k ( ℤ 2 3 ) as a particular case of our construction.

How to cite

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Klim, Jennifer, and Majid, Shahn. "Bicrossproduct Hopf quasigroups." Commentationes Mathematicae Universitatis Carolinae 51.2 (2010): 287-304. <http://eudml.org/doc/37761>.

@article{Klim2010,
abstract = {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.},
author = {Klim, Jennifer, Majid, Shahn},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {IP loop; octonions; quantum group; quasiHopf algebra; monoidal category; finite group; coset; loop; octonions; quantum group; quasiHopf algebra; monoidal category; finite group; coset},
language = {eng},
number = {2},
pages = {287-304},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Bicrossproduct Hopf quasigroups},
url = {http://eudml.org/doc/37761},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Klim, Jennifer
AU - Majid, Shahn
TI - Bicrossproduct Hopf quasigroups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 2
SP - 287
EP - 304
AB - 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.
LA - eng
KW - IP loop; octonions; quantum group; quasiHopf algebra; monoidal category; finite group; coset; loop; octonions; quantum group; quasiHopf algebra; monoidal category; finite group; coset
UR - http://eudml.org/doc/37761
ER -

References

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  1. Albuquerque H., Majid S, 10.1006/jabr.1998.7850, J. Algebra 220 (1999), 188–224. Zbl0999.17006MR1713433DOI10.1006/jabr.1998.7850
  2. Beggs E.J., 10.1016/S0022-4049(02)00119-6, J. Pure and Applied Algebra 177 (2003), 5–41. Zbl1037.18004MR1948835DOI10.1016/S0022-4049(02)00119-6
  3. Drinfeld V.G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419–1457. MR1047964
  4. Klim J., Majid S., Hopf quasigroups and the algebraic 7 -sphere, J. Algebra(to appear). MR2629701
  5. Majid S., Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, 1995. Zbl0857.17009MR1381692
  6. Perez-Izquierdo J., Shestakov I.P., 10.1016/S0021-8693(03)00389-2, J. Algebra 272 (2004), 379–393. Zbl1077.17027MR2029038DOI10.1016/S0021-8693(03)00389-2
  7. Smith J.D.H., Introduction to Quasigroups and their Representations, Taylor & Francis, 2006. Zbl1122.20035MR2268350
  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. Zbl1064.20011MR1830177

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