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Gyrogroups, the grouplike loops in the service of hyperbolic geometry and Einstein’s special theory of relativity. (English) Zbl 1147.20055

The author has previously published a series of papers and books, in which he has developed the theory of gyrogroups. In his book “Beyond the Einstein addition law and its gyroscopic Thomas precession. The theory of gyrogroups and gyrovector spaces.” [Fundamental Theories of Physics 117. Dordrecht: Kluwer Academic Publishers (2001; Zbl 0972.83002)] one finds a lucid presentatation of a mayor part of this material. Gyrogroups are a special class of loops, which is suited for applications in hyperbolic geometry and relativity theory. In the paper under review he gives a survey on this work containing the geometric approach. The closing sections of the paper give a perspective how one could treat the problem of dark matter and questions from quantum informatics with these methods.
Reviewer’s remarks: In the article under review the author indicates that the gyrocommutative gyrogroups are the Bruck loops, i.e. the (left) Bol loops satifying the (left) Bruck identity. This matter was discussed by L. V. Sabinin, L. L. Sabinina, L. V. Sbitneva [Aequationes Math. 56, No. 1-2, 11-17 (1998; Zbl 0923.20051)]. These loops have entered the world of algebra in works of R. H. Bruck [A survey of binary systems (Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. Heft 20.) Berlin-Göttingen-Heidelberg: Springer-Verlag (1958; Zbl 0081.01704), Chapter VII], G. Glauberman [J. Algebra 1, 374-396 (1964; Zbl 0123.01502), J. Algebra 8, 393-414 (1968; Zbl 0155.03901)] and D. A. Robinson [Ann. Soc. Sci. Bruxelles, Sér. I 93, 7-16 (1979; Zbl 0414.20058)]. One finds detailed information about Bruck loops in the book of H. Kiechle [Theory of K-loops. Lect. Notes Math. 1778. Berlin: Springer (2002; Zbl 0997.20059)]. It is known, that (abstract) symmetric spaces are Bruck loops. Motivated by Ungar’s publications, the fact that the relativistic addition of velocities defines a Bruck loop has been explained also by other authors, e.g. L. V. Sabinin and P. O. Mikheev [Usp. Mat. Nauk 48, No. 5(293), 183-184 (1993; Zbl 0814.53057)], L. V. Sabinin and A. Nesterov [Hadronic J. 20, No. 3, 219-237 (1997; Zbl 0936.20056)].

MSC:

20N05 Loops, quasigroups
83A05 Special relativity
53A60 Differential geometry of webs
51P05 Classical or axiomatic geometry and physics
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