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Loops, their cores and symmetric spaces. (English) Zbl 0909.20052
Some relations between global symmetric spaces, global smooth Bol and Moufang loops and the corresponding 3-webs are investigated. Properties of groups generated by the left and right translations of differentiable connected Moufang loops are described and some propositions for symmetric space structure connected with a differentiable Bol loop are proved. One corollary of the theory is that every differentiable connected Moufang loop is analytic. Some well-known local results of the Bol loop theory are obtained in an original way. In particular, generalization of the classical Hausdorff-Campbell formulae for the class of the left alternative local loops is found. Every connected differentiable Bol loop satisfying the identity $$x(y^2x)=y(x^2y)$$ must be an abelian group.

##### MSC:
 20N05 Loops, quasigroups 53C35 Differential geometry of symmetric spaces 22A30 Other topological algebraic systems and their representations 53A60 Differential geometry of webs
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##### References:
 [1] M. A. Akivis and A. M. Shelekhov,Geometry and Algebra of Multidimensional Three-Webs, Kluwer Academic Publishers, Dordrecht, 1992. · Zbl 0771.53001 [2] A. Barlotti and K. Strambach,The geometry of binary systems, Advances in Mathematics49 (1983), 1–105. · Zbl 0518.20064 · doi:10.1016/0001-8708(83)90013-0 [3] V. D. Belousov,Foundations of the Theory of Quasigroups and Loops (Russian), Nauka, Moscow, 1976. [4] R. H. Bruck,A Survey of Binary Systems, Ergebnisse der Mathematik 20, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. · Zbl 0081.01704 [5] M. Funk and P. T. Nagy,On collineations groups generated by Bol reflections, Journal of Geometry48 (1993), 63–78. · Zbl 0793.51001 · doi:10.1007/BF01226801 [6] G. Glaubermann,On loops of odd order, Journal of Algebra1 (1964), 374–396. · Zbl 0123.01502 · doi:10.1016/0021-8693(64)90017-1 [7] H. R. Halder,Dimension der Bahnen lokal kompakter Gruppen, Archiv der Mathematik22 (1971), 302–303. · Zbl 0218.54033 · doi:10.1007/BF01222579 [8] E. Hewitt and K. Ross,Abstract Harmonic Analysis I, Springer-Verlag, Berlin-Heidelberg-New York, 1963. · Zbl 0115.10603 [9] G. Hochschild,The Structure of Lie Groups, Holden Day, San Francisco, 1965. · Zbl 0131.02702 [10] K. H. Hofmann and K. Strambach,Topological and analytical loops, inQuasigroups and Loops: Theory and Applications (O. Chien, H.O. Pflugfelder and J.D.H. Smith, eds.), Sigma Series in Pure Math.8, Heldermann-Verlag, Berlin, 1990, pp. 205–262. · Zbl 0747.22004 [11] S. N. Hudson,Topological loops with invariant uniformities, Transactions of the American Mathematical Society109 (1963), 181–190. · Zbl 0115.02501 · doi:10.1090/S0002-9947-1963-0155302-5 [12] S. N. Hudson,Transformation groups in the theory of topological loops, Proceedings of the American Mathematical Society15 (1964), 872–877;Errata, ibid17 (1966), 770. · Zbl 0133.16203 · doi:10.1090/S0002-9939-1964-0167962-X [13] M. Kikkawa,Geometry of homogeneous Lie loops, Hiroshima Mathematical Journal5 (1975), 141–179. · Zbl 0304.53037 [14] E. N. Kuz’min,Malcev algebras and their representations (Russian), Algebra i Logika7 (1968), no. 4, 48–69. [15] E. N. Kuz’min,Malcev algebras of dimension five over a field of zero characteristic (Russian), Algebra i Logika9 (1970), no. 5, 691–700. [16] E. N. Kuz’min,The connection between Malcev algebras and analytic Moufang loops (Russian), Algebra i Logika10 (1971), no. 1, 3–22. [17] E. N. Kuz’min,Levi’s theorem for Malcev algebras (Russian), Algebra i Logika16 (1977), no. 4, 424–431. [18] O. Loos,Symmetric Spaces, Vol. 1, Benjamin, New York, 1969. · Zbl 0175.48601 [19] A. I. Malcev,Analytical loops (Russian), Matematicheskii Sbornik36 (1955), 569–676. [20] P. O. Miheev and L. V. Sabinin,The Theory of Smooth Bol Loops, Friendship of Nations University, Moscow, 1985. [21] P. O. Miheev and L. V. Sabinin,Quasigroups and Differential Geometry, Chapter XII inQuasigroups and Loops: Theory and Applications (O. Chein, H.O. Pflugfelder and J.D.H. Smith, eds.), Sigma Series in Pure Math.8, Heldermann-Verlag, Berlin, 1990, pp. 357–430. [22] D. Montgomery and L. Zippin,Topological Transformation Groups, Wiley Interscience Publishers, New York, 1955. · Zbl 0068.01904 [23] G. D. Mostow,The extensibility of local Lie groups of transformations and groups on surfaces, Annals of Mathematics52 (1950), 606–636. · Zbl 0040.15204 · doi:10.2307/1969437 [24] G. D. Mostow,Some new decompositions for semi-simple groups, Memoirs of the American Mathematical Society14 (1955), 31–54. · Zbl 0064.25901 [25] P. T. Nagy,3-nets with maximal family of two-dimensional subnets, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg61 (1991), 203–211. · Zbl 0767.53012 · doi:10.1007/BF02950765 [26] P. T. Nagy and K. Strambach,Loops as invariant sections in groups and their geometry, Canadian Journal of Mathematics46 (1994), 1027–1056. · Zbl 0814.20055 · doi:10.4153/CJM-1994-059-8 [27] P. T. Nagy and K. Strambach,Sharply transitive sections in Lie groups: A Lie theory of smooth loops, in preparation. [28] R. D. Schafer,An Introduction to Non-associative Algebras, Academic Press, New York, 1966. · Zbl 0145.25601 [29] H. Scheerer,Restklassenräume kompakter zusammenhängender Mannigfaltigkeiten mit Schnitt, Mathematische Annalen206 (1973), 144–155. · Zbl 0256.22007 · doi:10.1007/BF01430981
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