Abstract
Continuous Moufang transformations are introduced and discussed. Commutation relations for infinitesimal Moufang transformations are established. The resulting Lie algebra has quite impressive structure equations, well known from the theory of alternative algebras.
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Moufang, R.: Zur Struktur von Alternativkörpern, Math. Ann. 110 (1935), 416-430.
Bruck, R. H.: A Survey of Binary Systems, Springer-Verlag, Berlin, Heidelberg, New York, 1971.
Belousov, V. D.: Foundations of the Theory of Quasigroups and Loops, Nauka, Moscow, 1967 (in Russian).
Schafer, R. D.: Representations of alternative algebras, Trans. Amer. Math. Soc. 72 (1952), 1-17.
Lukierski, J. and Minnaert, P.: Seven-spheres from octonions and geometric torsion, Phys. Lett. B 129 (1983), 392-396.
Sudbery, A.: Division algebras, (pseudo)orthogonal groups and spinors, J. Phys.: Math. Gen. 17 (1984), 939-955.
Mal'tsev, A. I.: Analytical loops, Math. Sb. 36 (1955), 569-576 (in Russian).
Pontryagin, L. S.: Topologycal Groups, Gordon and Breach, New York, 1966.
Sagle, A. A.: Malcev algebras, Trans. Amer. Math. Soc. 101 (1961), 426-458.
Akivis, M. A. and Shelekhov, A. M.: On local differentiable quasigroups and connections associated with three-webs of multi-dimensional surfaces, Sib. Mat. J. 12 (1971), 1181-1191 (in Russian).
Kuz'min, E. N.: On a connection between the Mal'tsev algebras and analytic Moufang loops, Algebra i Logika 10 (1971), 3-22 (in Russian).
Kerdman, F. S.: Analytical Moufang loops globally, Algebra i Logika 18 (1979), 523-555 (in Russian).
Kerdman, F. S.: On analytical Moufang loops globally, Dokl. AN SSSR 249 (1979), 533-536 (in Russian).
Paal, E.: Moufang transformations, Trans. Inst. of Phys. Estonian Acad. Sci. 62 (1987), 142-158 (in Russian).
Paal, E.: Birepresentations of derivative Moufang loops, Proc. Estonian Acad. Sci., Phys. Math. 40 (1991), 105-111.
Paal, E.: Derivative Moufang transformations, in V. V. Dodonov and V. I. Man'ko (eds), Group Theoretical Methods in Physics, Proc. XVIII Int. Colloq. Held at Moscow, USSR, June 4-9, 1990, Springer-Verlag, Berlin, Heidelberg, New York, 1991, pp. 573-574.
Paal, E.: An introduction to the Moufang symmetry, Prepr. F-42, Inst. of Phys., Tartu, 1987 (in Russian).
Yamaguti, K.: Note on Malcev algebras, Kumamoto J. Sci. Ser. A 5 (1962), 203-207.
Yamaguti, K.: On the theory of Malcev algebras, Kumamoto J. Sci. Ser. A 6 (1963), 9-45.
Loos, O.: ¨ Über eine Beziehung zwischen Malcev-Algebren und Lie-Tripelsysteme, Pacific J. Math. 18 (1966), 553-562.
Paal, E.: Moufang loops and generalized Lie equations, Proc. Estonian Acad. Sci. Phys. Math. 41 (1992), 125-132.
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Paal, E. Continuous Moufang Transformations. Acta Applicandae Mathematicae 50, 77–91 (1998). https://doi.org/10.1023/A:1005811017268
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DOI: https://doi.org/10.1023/A:1005811017268