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Global Left Loop Structures on Spheres | | Michael K. Kinyon
; | | Date: |
21 Oct 1999 | | Journal: | Comment. Math. Univ. Carolinae 41 (2000), no. 2, 325-346 | | Subject: | Group Theory; Differential Geometry MSC-class: 20N05 | math.GR math.DG | | Abstract: | On the unit sphere $mathbb{S}$ in a real Hilbert space $mathbf{H}$, we derive a binary operation $odot$ such that $(mathbb{S},odot)$ is a power-associative Kikkawa left loop with two-sided identity $mathbf{e}_0$, i.e., it has the left inverse, automorphic inverse, and $A_l$ properties. The operation $odot$ is compatible with the symmetric space structure of $mathbb{S}$. $(mathbb{S},odot)$ is not a loop, and the right translations which fail to be injective are easily characterized. $(mathbb{S},odot)$ 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 $-mathbf{e}_0$ where they have a nonremovable discontinuity. The orthogonal group $O(mathbf{H})$ is a semidirect product of $(mathbb{S},odot)$ with its automorphism group (cf. http://www.arxiv.org/abs/math.GR/9907085). The left loop structure of $(mathbb{S},odot)$ gives some insight into spherical geometry. | | Source: | arXiv, math.GR/9910111 | | Services: | Forum | Review | PDF | Favorites | |
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