| Title:
|
Kikkawa loops and homogeneous loops (English) |
| Author:
|
Kikkawa, Michihiko |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
45 |
| Issue:
|
2 |
| Year:
|
2004 |
| Pages:
|
279-285 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
loops |
| Keyword:
|
Lie triple algebras |
| Keyword:
|
symmetric spaces |
| MSC:
|
01A70 |
| MSC:
|
05B07 |
| MSC:
|
20N05 |
| MSC:
|
22A30 |
| MSC:
|
53C35 |
| MSC:
|
Other |
| idZBL:
|
Zbl 1101.20039 |
| idMR:
|
MR2075276 |
| . |
| Date available:
|
2009-05-05T16:45:08Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119457 |
| . |
| Reference:
|
[1] Akivis M.: Local algebras of a multidimensional web (in Russian).Sibir. Mat. Z. 17 (1976), 5-11. MR 0405261 |
| Reference:
|
[2] Chein O., Pflugfelder H.O., Smith J.D.H.: Quasigroups and Loops: Theory and Applications.Heldermann Verlag, 1990. Zbl 0719.20036, MR 1125806 |
| Reference:
|
[3] Kiechle H.: Theory of K-Loops.Lecture Note in Mathematics 1778, Springer, 2002. Zbl 0997.20059, MR 1899153 |
| Reference:
|
[4] Kikkawa M.: On local loops in affine manifolds.J. Sci. Hiroshima Univ. A-I:28 (1964), 199-207. Zbl 0141.19603, MR 0187173 |
| Reference:
|
[5] Kikkawa M.: On some quasigroups of algebraic models of symmetric spaces.Mem. Fac. Lit. Sci. Shimane Univ. Nat. Sci. 6 (1973), 9-13. Zbl 0264.53028, MR 0327962 |
| Reference:
|
[6] Kikkawa M.: On some quasigroups of algebraic models of symmetric spaces II.Mem. Fac. Lit. Sci. Shimane Univ. Nat. Sci. 7 (1974), 29-35. Zbl 0279.53046, MR 0342638 |
| Reference:
|
[7] Kikkawa M.: Geometry of homogeneous Lie loops.Hiroshima Math. J. 5 (1975), 141-179. Zbl 0304.53037, MR 0383301 |
| Reference:
|
[8] Kikkawa M.: A note on subloops of a homogeneous Lie loop and subsystems of its Lie triple algebra.Hiroshima Math. J. 5 (1975), 439-446. Zbl 0312.53036, MR 0388280 |
| Reference:
|
[9] Kikkawa M.: On some quasigroups of algebraic models of symmetric spaces III.Mem. Fac. Lit. Sci. Shimane Univ. Nat. Sci. 9 (1975), 7-12. Zbl 0322.20035, MR 0412316 |
| Reference:
|
[10] Kikkawa M.: On the left translations of homogeneous loops.Mem. Fac. Lit. Sci. Shimane Univ. Nat. Sci. 10 (1976), 19-25. Zbl 0364.20076, MR 0432795 |
| Reference:
|
[11] Kikkawa M.: Projectivity of homogeneous left loops on Lie groups I.Mem. Fac. Sci. Shimane Univ. 23 (1989), 17-22. Zbl 0703.22003, MR 1044715 |
| Reference:
|
[12] Kikkawa M.: Projectivity of homogeneous left loops.Proc. International Symposium on Non-associative Algebras and Related Topics, World Scientific, 1991, pp.77-99. Zbl 0788.53042, MR 1150252 |
| Reference:
|
[13] Kikkawa M.: Remarks on Akivis left loops.Mem. Fac. Sci. Shimane Univ. 29 (1995), 1-9. Zbl 0843.22008, MR 1374956 |
| Reference:
|
[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. Zbl 1032.17058, MR 1688604 |
| Reference:
|
[15] Loos O.: Symmetric Spaces I.Benjamin, 1969. Zbl 0175.48601 |
| Reference:
|
[16] Nomizu K.: Invariant affine connections on homogeneous spaces.Amer. J. Math. 76 (1954), 33-65. Zbl 0059.15805, MR 0059050 |
| Reference:
|
[17] Sabinin L.V.: Geometry of loops (in Russian).Mat. Zametki 12 (1972), 605-616. MR 0340461 |
| Reference:
|
[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. MR 1149512 |
| . |
