Some remarks about Lie groups transitive on spheres and tori
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The present note pertains principally to two papers of D. Montgomery and H. Samelson [l, 2J, 1 in which the authors study compact Lie groups transitive on tori [l] and spheres [2], I will here prove in another way, generalize, and sharpen a part of their results.§1 contains the remarks to [l], §2 to [2]; they are independent of one another and the methods used in both are quite different.I recall first the definition and some simple properties of homogeneous spaces.A manifold W is a homogeneous space under the Lie group 2 G if to each element a of G there corresponds a differentiate transformation T a 'x-^T a (x)of W into itself such that: (1) T a {x) depends continuously on the pair a£G, x£:W.(2) To the product (ab) corresponds the mapping x->T Ca b) (x)
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Except for three cases, only the well-known groups can act transitively on a sphere: for even dimensions only the orthogonal group plus G2, for dimension 4n+1 the orthogonal, unimodular unitary, and unitary groups, and for dimension 4n-1 also the symplectic and full quaternion-unitary groups.
“Except for three cases only those groups can act transitively on a sphere which are well known to do so, namely for even dimensions only the orthogonal group (plus G 2 , which has 5 6 as Klein space), for dimension 4w+l the orthogonal, the unimodular unitary, and the unitary group, for dimension 4n-1 also the symplectic group and the full quaternion-unitary group (plus 5 16 = B4/B3 and 5 7 = S 3 /”
From Topology of Lie groups (1952) · cites this paper as [7]
Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.
Experimental — accuracy not yet measuredAny group that acts transitively and effectively on the torus T^n coincides with T^n.
“A similar result is that any group acting transitively and effectively on the torus T n is actually T n itself [7; 90 ].”
From Topology of Lie groups (1952) · cites this paper as [7]
Section 1 states that a Lie group acting transitively and effectively on the n-dimensional torus is isomorphic to T^n, and Borel proves a related result for compact connected Lie groups. The citing sentence says “any group” without the Lie-group qualification, so it is broader than the result in our copy. No page number is available in the indexed passages. (LLM, conf=0.86)
Experimental — accuracy not yet measuredIf a simply connected Klein space has characteristic 1, it is a single point; if characteristic 2, it is a sphere; if characteristic is a prime greater than 2, it is one of a small list of spaces.
“If a (simply connected) Klein space has characeristic 1, it is a single point [127]; if it has characteristic 2, it is a sphere; if it has characteristic a prime>2, it is one of a small list of spaces [7; 127].”
From Topology of Lie groups (1952) · cites this paper as [7]
Theorems IV and V in §2 support the sphere and finite-list claims for simply connected spaces on which a compact connected Lie group acts transitively. Theorem IV specifies even-dimensional spheres. Our copy does not state that characteristic 1 implies a single point, and the citing sentence omits the restriction on the acting group. (LLM, conf=0.86)
Experimental — accuracy not yet measuredFor spheres of even dimension, the isotropy group of a transitive transformation group must be a maximal subgroup of maximal rank.
“For spheres of even dimension another approach has been given by A. Borel [7]: the sphere has characteristic two, and two is a prime number; this implies easily that the isotropy group must be a maximal subgroup of maximal rank, and now one can make use of the complete list of such groups in §25.”
From Topology of Lie groups (1952) · cites this paper as [7]
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Cites
Transformation Groups of Spheres
Montgomery, D., & Samelson, H. (1943) · The Annals of Mathematics · doi:10.2307/1968975 · Cited by 293
(1931) · French digital mathematics library (Numdam) · Cited by 96
Spiegelungsgruppen und Aufzählung halbeinfacher Liescher Ringe
Witt, E. (1941) · Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg · doi:10.1007/bf02940749 · Cited by 90
Ein Satz �ber die Wirkungsr�ume geschlossener Liescher Gruppen
Hopf, H., & Samelson, H. (1940) · Commentarii Mathematici Helvetici · doi:10.1007/bf01378063 · Cited by 76
La théorie des groupes finis et continus et l'Analysis situs
Cartan, É. (1952) · International Linear Collider · Cited by 49
Stiefel, E. (1941) · Commentarii Mathematici Helvetici · doi:10.1007/bf02565625 · Cited by 40
Matrices with elements in a principal ideal ring
MacDuffee, C. C. (1933) · Bulletin of the American Mathematical Society · doi:10.1090/s0002-9904-1933-05681-1 · Cited by 19
Groups transitive on the 𝑛-dimensional torus
Montgomery, D., & Samelson, H. (1943) · Bulletin of the American Mathematical Society · doi:10.1090/s0002-9904-1943-07952-9 · Cited by 15
DIVERTICULA OF THE DUODENOJEJUNAL FLEXURE
Raven, R. (1939) · The Lancet · doi:10.1016/s0140-6736(00)60079-x · Cited by 2
Flagged first, then by citations
Research works printed but not linked: 2
- 4. E. Cartan, Sur les invariants intégraux de certains espaces homogènes clos . . . , Polskie Towarzystwo matematyczne, Krakow, Rocznik vol. 8 (1929) pp. 182-225
- 5. G. de Rham, Sur l’analysis situs des variétés à n dimensions. J. Math. Pures Appl. vol. 10 (1931) pp. 115-200.
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Sorted by citations
Samelson, H. (1952) · Bulletin of the American Mathematical Society · doi:10.1090/s0002-9904-1952-09544-6 · Cited by 219
“Borel [7]: the sphere has characteristic two, and two is a prime number; this implies easily that the isotropy group must be a maximal subgroup of maximal rank, and now one can make use of the complete list of such groups in §25.”
Background
- Authors
- Borel, A.
- Corresponding author
- Borel, Armand
- Venue
- Bulletin of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 1949
- Volume
- 55
- Issue
- 6
- Pages
- 580-587
- ISSN
- 0002-9904 (ISSN-L)
- 1088-9485
- 0273-0979
- OpenAlex
- W2054095663
- Semantic Scholar
- ada964993d6e7a6e899d6a6233d86732e48e3576
- Cited by
- 143
- References
- 9
- Citations by year
- 2026: 1 · 2025: 3 · 2024: 2 · 2023: 3 · 2022: 3 · 2021: 5 · 2020: 4 · 2019: 2 · 2018: 2 · 2017: 11 · 2016: 3 · 2015: 4 · 2014: 8 · 2013: 2 · 2012: 7
- FWCI
- 6.71
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- Bulletin of the American Mathematical Society (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Physical Sciences › Mathematics › Algebra and Number Theory › Advanced Algebra and Geometry
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Keywords
- compact Lie groups, transitive actions, homogeneous spaces, sphere, torus
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