Kikkawa loops and homogeneous loops

Michihiko Kikkawa

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 2, page 279-285
  • ISSN: 0010-2628

Abstract

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In H. Kiechle's publication ``Theory of K-loops'' [3], the name Kikkawa loops is given to symmetric loops introduced by the author in 1973. This concept started from an analogical imagination of sum of vectors in Euclidean space brought up on a sphere. In 1975, this concept was extended by him to the more general concept of homogeneous loops, and it led us to a non-associative generalization of the theory of Lie groups. In this article, the backstage of finding these concepts will be disclosed from the viewpoint how a new mathematical concept appears and grows up in imagination of a mathematician.

How to cite

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Kikkawa, Michihiko. "Kikkawa loops and homogeneous loops." Commentationes Mathematicae Universitatis Carolinae 45.2 (2004): 279-285. <http://eudml.org/doc/249347>.

@article{Kikkawa2004,
abstract = {In H. Kiechle's publication ``Theory of K-loops'' [3], the name Kikkawa loops is given to symmetric loops introduced by the author in 1973. This concept started from an analogical imagination of sum of vectors in Euclidean space brought up on a sphere. In 1975, this concept was extended by him to the more general concept of homogeneous loops, and it led us to a non-associative generalization of the theory of Lie groups. In this article, the backstage of finding these concepts will be disclosed from the viewpoint how a new mathematical concept appears and grows up in imagination of a mathematician.},
author = {Kikkawa, Michihiko},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {loops; Lie triple algebras; symmetric spaces; symmetric loops; K-loops; Kikkawa-loops; geodesic local loops; symmetric spaces; Lie triple algebras},
language = {eng},
number = {2},
pages = {279-285},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Kikkawa loops and homogeneous loops},
url = {http://eudml.org/doc/249347},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Kikkawa, Michihiko
TI - Kikkawa loops and homogeneous loops
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 2
SP - 279
EP - 285
AB - In H. Kiechle's publication ``Theory of K-loops'' [3], the name Kikkawa loops is given to symmetric loops introduced by the author in 1973. This concept started from an analogical imagination of sum of vectors in Euclidean space brought up on a sphere. In 1975, this concept was extended by him to the more general concept of homogeneous loops, and it led us to a non-associative generalization of the theory of Lie groups. In this article, the backstage of finding these concepts will be disclosed from the viewpoint how a new mathematical concept appears and grows up in imagination of a mathematician.
LA - eng
KW - loops; Lie triple algebras; symmetric spaces; symmetric loops; K-loops; Kikkawa-loops; geodesic local loops; symmetric spaces; Lie triple algebras
UR - http://eudml.org/doc/249347
ER -

References

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  2. Chein O., Pflugfelder H.O., Smith J.D.H., Quasigroups and Loops: Theory and Applications, Heldermann Verlag, 1990. Zbl0719.20036MR1125806
  3. Kiechle H., Theory of K-Loops, Lecture Note in Mathematics 1778, Springer, 2002. Zbl0997.20059MR1899153
  4. Kikkawa M., On local loops in affine manifolds, J. Sci. Hiroshima Univ. A-I:28 (1964), 199-207. (1964) Zbl0141.19603MR0187173
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  10. Kikkawa M., On the left translations of homogeneous loops, Mem. Fac. Lit. Sci. Shimane Univ. Nat. Sci. 10 (1976), 19-25. (1976) Zbl0364.20076MR0432795
  11. Kikkawa M., Projectivity of homogeneous left loops on Lie groups I, Mem. Fac. Sci. Shimane Univ. 23 (1989), 17-22. (1989) Zbl0703.22003MR1044715
  12. Kikkawa M., Projectivity of homogeneous left loops, Proc. International Symposium on Non-associative Algebras and Related Topics, World Scientific, 1991, pp.77-99. Zbl0788.53042MR1150252
  13. Kikkawa M., Remarks on Akivis left loops, Mem. Fac. Sci. Shimane Univ. 29 (1995), 1-9. (1995) Zbl0843.22008MR1374956
  14. Kikkawa M., Geometry of homogeneous left Lie loops and tangent Lie triple algebras, Mem. Fac. Sci. Engr. Shimane Univ. 32-B (1999), 69-74. (1999) Zbl1032.17058MR1688604
  15. Loos O., Symmetric Spaces I, Benjamin, 1969. Zbl0175.48601
  16. Nomizu K., Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65. (1954) Zbl0059.15805MR0059050
  17. Sabinin L.V., Geometry of loops (in Russian), Mat. Zametki 12 (1972), 605-616. (1972) MR0340461
  18. Sanami M., Kikkawa M., A class of double Lie algebras on simple Lie algebras and projectivity of simple Li groups, Mem. Fac. Sci. Shimane Univ. 25 (1991), 39-44. (1991) MR1149512

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