Faithful Representations of Lie Groups II
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The present paper is a continuation of Part I. In the introduction of Part I we have explained our main problems and sketched their results. In the present part we shall proceed to give complete proofs of them.
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Weyl's theorem says that the fundamental group of a compact semisimple group is finite, equivalently its first Betti number is zero, and its universal covering group is compact.
“The last statement is given more precision by an important theorem of H. Weyl, saying that the fundamental group of a compact semi-simple group is finite (or that the first Betti number is zero), in other words, that the universal covering group is still compact.”
From Topology of Lie groups (1952) · cites this paper as [43]
In the proof following Theorem 3 (page unavailable), Gotô invokes Weyl’s theorem to say that the compact form of a complex semisimple Lie algebra generates a compact group, then discusses a simply connected group generated by that form. The copy does not state the claimed finiteness of the fundamental group or the equivalence with a zero first Betti number, and it does not explicitly say that the universal covering group of every compact semisimple group is compact. (LLM, conf=0.75)
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Research works printed but not linked: 3
- On the classification of real simple Lie groups, Rec. Math., Vol. 5 (1939). https://doi.org/10.1017/S002776300002287X Published online by Cambridge University Press M. Goto
- Faithful representations of Lie groups I, Mathematica Japonicae, Vol. 1 (1949).
- The classical groups, Princeton (1939). https://doi.org/10.1017/S002776300002287X Published online by Cambridge University Press
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Samelson, H. (1952) · Bulletin of the American Mathematical Society · doi:10.1090/s0002-9904-1952-09544-6 · Cited by 219 · see citing works
- Authors
- Gotô, M.
- Affiliations
- Nagoya University (JP)
- Corresponding author
- Gotô, Morikuni
- Venue
- Nagoya Mathematical Journal
- Publisher
- Cambridge University Press (CUP)
- Published
- 1950-06
- Volume
- 1
- Pages
- 91-107
- ISSN
- 0027-7630 (ISSN-L)
- 2152-6842
- OpenAlex
- W1993477707
- Cited by
- 41
- References
- 0
- Citations by year
- 2024: 1 · 2023: 1 · 2022: 2 · 2021: 3 · 2016: 1 · 2015: 7 · 2013: 1
- FWCI
- 1.00
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Nagoya Mathematical Journal (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Algebra and Geometry
- Physical Sciences › Mathematics › Algebra and Number Theory › Advanced Topics in Algebra
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Finite Group Theory Research
- Keywords
- faithful representation, Lie groups, representation theory of Lie groups
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