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Keywords:
loop; quasigroup; sphere; Hilbert space; spherical geometry
loop; quasigroup; sphere; Hilbert space; spherical geometry
Summary:
On the unit sphere $\Bbb S$ in a real Hilbert space $\bold H$, we derive a binary operation $\odot $ such that $(\Bbb S,\odot )$ is a power-associative Kikkawa left loop with two-sided identity $\bold e_{0}$, i.e., it has the left inverse, automorphic inverse, and $A_l$ properties. The operation $\odot $ is compatible with the symmetric space structure of $\Bbb S$. $(\Bbb S,\odot )$ is not a loop, and the right translations which fail to be injective are easily characterized. $(\Bbb S,\odot )$ satisfies the left power alternative and left Bol identities ``almost everywhere'' but not everywhere. Left translations are everywhere analytic; right translations are analytic except at $-\bold e_{0}$ where they have a nonremovable discontinuity. The orthogonal group $O(\bold H)$ is a semidirect product of $(\Bbb S,\odot )$ with its automorphism group. The left loop structure of $(\Bbb S,\odot )$ gives some insight into spherical geometry.
References:
[1] Bruck R.: A Survey of Binary Systems. (3rd printing, corrected), Springer-Verlag, Berlin, 1971. MR 0093552 | Zbl 0141.01401
[2] Chein O., Pflugfelder H.O., Smith J.D.H. (eds.): Quasigroups and Loops: Theory and Applications. Sigma Series in Pure Mathematics, Vol. 8, Heldermann Verlag, Berlin, 1990. MR 1125806 | Zbl 0719.20036
[3] Gabrieli E., Karzel H.: Reflection geometries over loops. Results Math. 32 (1997), 61-65. MR 1464673 | Zbl 0922.51007
[4] Gabrieli E., Karzel H.: Point-reflection geometries, geometric K-loops and unitary geometries. Results Math. 32 (1997), 66-72. MR 1464674 | Zbl 0922.51006
[5] Gabrieli E., Karzel H.: The reflection structures of generalized co-Minkowski spaces leading to K-loops. Results Math. 32 (1997), 73-79. MR 1464675 | Zbl 0923.51014
[6] Glauberman G.: On loops of odd order. J. Algebra 1 (1964), 374-396. MR 0175991 | Zbl 0123.01502
[7] Helgason S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York, 1978. MR 0514561 | Zbl 0993.53002
[8] Karzel H.: Zusammenhänge zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und $2$-Strukturen mit Rechtecksaxiom. Abh. Math. Sem. Univ. Hamburg 32 (1968), 191-206. MR 0240715 | Zbl 0162.24101
[9] Kiechle H.: Theory of $K$-loops. Habilitationsschrift, Universität Hamburg, Hamburg, Germany, 1998. Zbl 0997.20059
[10] Kikkawa M.: On some quasigroups of algebraic models of symmetric spaces. Mem. Fac. Lit. Sci. Shimane Univ. (Nat. Sci.) 6 (1973), 9-13. MR 0327962 | Zbl 0264.53028
[11] Kikkawa M.: The geometry of homogeneous Lie loops. Hiroshima Math. J. 5 (1975), 141-179. MR 0383301
[12] Kinyon M.K., Jones O.: Loops and semidirect products. to appear in Comm. Algebra. MR 1772003 | Zbl 0974.20049
[13] Klingenberg W.P.A.: Riemannian Geometry. 2nd ed., Studies in Mathematics I, de Gruyter, New York, 1995. MR 1330918 | Zbl 1073.53006
[14] Kepka T.: A construction of Bruck loops. Comment. Math. Univ. Carolinae 25 (1984), 591-595. MR 0782010 | Zbl 0563.20053
[15] Kreuzer A.: Inner mappings of Bol loops. Math. Proc. Cambridge Philos. Soc. 123 (1998), 53-57. MR 1474864
[16] Loos O.: Symmetric Spaces. vol. 1, Benjamin, New York, 1969. Zbl 0228.53034
[17] Miheev P.O., Sabinin L.V.: Quasigroups and differential geometry. Chapter XII in 2, pp.357-430. MR 1125818 | Zbl 0721.53018
[18] Nesterov A.I., Sabinin L.V.: Smooth loops, generalized coherent states and geometric phases. Int. J. Theor. Phys. 36 (1997), 1981-1989. MR 1476347 | Zbl 0883.22020
[19] Pflugfelder H.O.: Quasigroups and Loops: Introduction. Sigma Series in Pure Mathematics, Vol. 7, Heldermann Verlag, Berlin, 1990. MR 1125767 | Zbl 0715.20043
[20] Robinson D.A.: Bol loops. Ph.D. Dissertation, University of Wisconsin, Madison, Wisconsin, 1964. Zbl 0803.20053
[21] Robinson D.A.: A loop-theoretic study of right-sided quasigroups. Ann. Soc. Sci. Bruxelles Sér. I 93 (1979), 7-16. MR 0552166 | Zbl 0414.20058
[22] Sabinin L.V.: On the equivalence of categories of loops and homogeneous spaces. Soviet Math. Dokl. 13 (1972), 970-974. Zbl 0291.18006
[23] Sabinin L.V.: Methods of nonassociative algebra in differential geometry (Russian). Supplement to the Russian transl. of S. Kobayashi, K. Nomizu, ``Foundations of Differential Geometry'', vol. 1, Nauka, Moscow, 1981, pp. 293-339. MR 0628734
[24] Sabinin L.V.: Smooth Quasigroups and Loops. Kluwer Academic Press, Dordrecht, 1999. MR 1727714 | Zbl 1038.20051
[25] Sabinin L.V., Sabinina L.L., Sbitneva L.: On the notion of gyrogroup. Aequationes Math. 56 (1998), 11-17. MR 1628291 | Zbl 0923.20051
[26] Smith J.D.H.: A left loop on the $15$-sphere. J. Algebra 176 (1995), 128-138. MR 1345298 | Zbl 0841.17004
[27] Ungar A.A.: The relativistic noncommutative nonassociative group of velocities and the Thomas rotation. Results Math. 16 (1989), 168-179. MR 1020224 | Zbl 0693.20067
[28] Ungar A.A.: Weakly associative groups. Results Math. 17 (1990), 149-168. MR 1039282 | Zbl 0699.20055
[29] Ungar A.A.: Thomas precession and its associated grouplike structure. Amer. J. Phys. 59 (1991), 824-834. MR 1126776
[30] Ungar A.A.: The holomorphic automorphism group of the complex disk. Aequationes Math. 47 (1994), 240-254. MR 1268034 | Zbl 0799.20032
[31] Ungar A.A.: Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics. Found. Phys. 27 (1997), 881-951. MR 1477047
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