An overview on real division algebras
DOI:
https://doi.org/10.14244/lajm.v2i02.17Keywords:
Real algebras, topological K-theoryAbstract
In this paper, we expose initial concepts of real division algebras, providing historical noteson $\R$, $\C$, $\Ham$ and $\Oct$. This is done to emphasize the relevance of topological K-theory
through the Bott-Milnor-Kervaire Theorem, which is at the end. For completeness, we also present
two classical results about the main division algebras.
References
Atiyah M. K-theory. Boca Raton: CRC Press; 1967. 240 p.
Baez J C. The Octonions. Bulletin (New Series) of the American Mathematical Society. 2022;39(2): 145-205. DOI: https://doi.org/10.1090/S0273-0979-01-00934-X
Buchmann A. A Brief History of Quaternions and the Theory of Holomorphic Functions of Quaternionic Variables. ArXiv [Internet]. 2011. [cited 2022 Jun 13]; 1111:6088. Available from: https://arxiv.org/abs/1111.6088
Clemente GL. Ordinary and Twisted K-Theory [Thesis (Master’s degree)]. São Carlos: Universidade Federal de São Carlos; 2022 [cited 2022 Jun 10]. 436 p. Available from: https://repositorio.ufscar.br/handle/ufscar/15841
Hatcher A. Vector bundles and K-theory. [book on the internet]. 2017 [cited 2022 Jun 13]. 124 p. Available from: https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf
Hurwitz A. Über die Composition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. Göttingen (1898) pp. 309-316. Available from: https://gdz.sub.uni-goettingen.de/download/pdf/PPN252457811_1898/LOG_0034.pdf
Merino O. A Short History of Complex Numbers. [paper on the internet]. 2006 [cited 2022 Jun 13]. 5 p. Available from: https://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf
Palais R.S. The Classification of Real Division Algebras. American Mathematical Monthly, 1968, 75:366-8. DOI: https://doi.org/10.2307/2313414
Sudbery A. Quaternionic Analysis. Math. Proc. Camb. Phil. Soc. 1979; 85:199-225. DOI: https://doi.org/10.1017/S0305004100055638
Weiss I. The Real Numbers - A Survey of Constructions. ArXiv [Internet]. 2015. [cited 2022 Jun 13]; 1506:03467. Available from: https://arxiv.org/abs/1506.03467,
Zorn M. Theorie der alternativen Ringe, Abh. Math. Sem. Univ. Hamburg 8 (1930), 123-147 p. DOI: https://doi.org/10.1007/BF02940993
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Gabriel Longatto Clemente
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors may enter into separate contractual agreements for the non-exclusive distribution of the journal's published version of the work (for example, posting it in an institutional repository or publishing it in a book), with an acknowledgment of its initial publication in this journal.