Gyrogroups were discovered by Ungar in modelling the Einstein velocity addition rule in relativity. Have they been shown to be useful elsewhere in mathematics (or mathematical physics)?
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$\begingroup$ The link with definition of gyrogroup is quite poor: axiom 1 says "there exists at least one element 0 such that...", and axiom 2 refers to 0. Either 0 is part of the data — i.e., is a constant, aka 0-ary law— (and it should be said), or it's not, and it should be corrected as well. $\endgroup$– YCorMay 10, 2021 at 14:51
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2 Answers
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Gyrogroups have been used in the theory of signal processing:
M. Ferreira, Spherical continuous wavelet transforms arising from sections of the Lorentz group. [publishers version]
answered Dec 1, 2020 at 10:04
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$\begingroup$ I understand the distaste for Elsevier, but I think that the "paywall version" is not literally paywalled—it seems that I can access it freely, from off-campus. $\endgroup$– LSpiceMay 10, 2021 at 14:21
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1$\begingroup$ you are right, I will remove that notice, thanks. $\endgroup$ May 10, 2021 at 14:42
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Gyrocommutative gyrogroups were independently introduced by Bruck, and are now called Bruck loops or K-loops in the quasigroup literature. He presumably invented them for some reason, but I couldn't find out a quick answer why.
Bruck loops do arise naturally in the study of sharply 3-transitive groups. You can associate a generalized ring, known as a neardomain, to every sharply 3-transitive group. The addition operation is then a Bruck loop. Kiechle has some slides, Is every Neardomain a Nearfield?, that mention the connection. In the finite case, the additive loop of a neardomain is forced to be an abelian group operation, but recently Rips, Segev, and Tent have shown that there are infinitely many infinite examples. See Tent - Infinite sharply multiply transitive groups for more.
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$\begingroup$ I guess Bruck introduced the eponymous loops in Contributions to the theory of loops? $\endgroup$– LSpiceMay 10, 2021 at 14:25
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$\begingroup$ @LSpice I didn't see it there, though the paper is very long (110 pages) so I might have missed it. The closest I came to seeing it is in II.2, but that's about Moufang loops. $\endgroup$– arsmathMay 10, 2021 at 19:14
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$\begingroup$ Not based on my knowledge of Bruck loops; I did a MathSciNet search for "Bruck loop" and followed what I thought (missed the 2nd page) was the oldest result, which says to see Robinson - Bol loops. I thought that the paper I linked was the only Bruck reference there, but once again I seem to have missed more. Anyway, it was a long-winded way of doing my part before asking you for the ref. 😁 $\endgroup$– LSpiceMay 10, 2021 at 20:11
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1$\begingroup$ @LSpice I looked at the Robinson paper, and he gives an example of a Bruck loop (example 2.2), and refers to his Ph.D. thesis for more examples (right after Theorem 2.8). He doesn't name them, though. Maybe someone later named them in honor of Bruck, without a super-tight connection? $\endgroup$– arsmathMay 11, 2021 at 7:56
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$\begingroup$ Bruck discussed the loops later named after him in passing in his book Survey of Binary Systems. Bol loops, as named by Robinson, are a larger class that includes both Bruck loops and Moufang loops. Glauberman later studied Bruck loops of odd order, but called them "B-loops". The name "Bruck loop" itself dates back at least to 1970. $\endgroup$ Feb 4, 2023 at 4:08