Previous |  Up |  Next

Article

MSC: 53B20, 53B30, 53C21
Keywords:
Group of invertible elements; algebra of quaternions; principal locally trivial bundle; 2-dimensional subalgebras; structural group; unit; Hopf fibration
Summary:
We have, that all two-dimensional subspaces of the algebra of quaternions, containing a unit, are 2-dimensional subalgebras isomorphic to the algebra $\mathbb{C}$ of complex numbers. It was proved in the papers of N. E. Belova. In the present article we consider a 2-dimensional subalgebra $(i)$ of complex numbers with basis ${1, i}$ and we construct the principal locally trivial bundle which is isomorphic to the Hopf fibration.
References:
[1] Alekseevsky, D. V., Marchiafava, S., Pontecorvo, M.: Compatible complex structures on almost quaternionic manifolds. Trans. Amer. Math. Soc. 351, 3 (1999), 997–1014. DOI 10.1090/S0002-9947-99-02201-1 | MR 1475674 | Zbl 0933.53017
[2] Bělohlávková, J., Mikeš, J., Pokorná, O.: On 4-planar mappings of special almost antiquaternionic spaces. Rend. Circ. Mat. Palermo, II. Ser. 66 (2001), 97–103. MR 1826682 | Zbl 1013.53022
[3] Bělohlávková, J., Mikeš, J., Pokorná, O.: 4-planar mappings of almost quaternionic and almost antiquaternionic spaces. Gen. Math. 5 (1997), 101–108. MR 1723598 | Zbl 0969.53006
[4] Belova, N. E.: Bundles of Algebras of Dimension 4. Kazan. Dep. in VINITI, Kazan University 3037-B99, Kazan, 1999.
[5] Belova, N. E.: Bundles defined by associative algebras. Diss. on scientific degree candidate Sci., Science competition, Kazan University, Kazan, 2001.
[6] Berger, M.: Geometry I. Springer, New York–Berlin–Heidelberg, 1987. Zbl 0606.51001
[7] Bushmanova, G. V., Norden, A. P.: Elements of conformal geometry. Kazan University, Kazan, 1972. MR 0370386
[8] Dubrovin, B. A., Novikov, S. P., Fomenko, A. T.: Modern Geometry. Methods and Applications. Nauka, Moscow, 1979. MR 0566582
[9] Hrdina, J., Slovák, J.: Generalized planar curves and quaternionic geometry. Ann. Global Anal. Geom. 29, 4 (2006), 343–354. DOI 10.1007/s10455-006-9023-y | MR 2251428 | Zbl 1097.53008
[10] Hinterleitner, I.: 4-planar mappings of quaternionic Kähler manifolds. In: Geometric methods in physics, 31 workshop, Białowieża, Poland, June 24–30, 2012. Selected papers based on the presentations at the workshop, Birkhäuser/Springer, Basel, (2013), 187–193. MR 3364004 | Zbl 1308.53073
[11] Kurbatova, I. N.: 4-quasi-planar mappings of almost quaternion manifolds. Sov. Math. 30 (1986), 100–104, transl. from Izv. Vyssh. Uchebn. Zaved., Mat., 1 (1986), 75–78. Zbl 0602.53029
[12] Kuzmina, I. A., Mikeš, J., Vanžurová, A.: The projectivization of conformal models of fibrations determined by the algebra of quaternions. Acta Univ. Palacki. Olomuc., Fac. Rer. Nat., Math. 50, 1 (2011), 91–98. MR 2920701 | Zbl 1252.53023
[13] Kuzmina, I., Mikeš, J.: On pseudoconformal models of fibrations determined by the algebra of antiquaternions and projectivization of them. Ann. Math. Inform. 42 (2013), 57–64. MR 3148230 | Zbl 1299.53042
[14] Mikeš, J., Bělohlávková, J., Pokorná, O.: On special 4-planar mappings of almost Hermitian quaternionic spaces. In: Proc. 2nd Meeting on Quaternionic Structures in Math. and Phys., electronic only, Roma, Italy, September 6–10, 1999, Dip. di Matematica “Guido Castelnuovo”, Univ. di Roma “La Sapienza”, Rome, (2001), 265–271. Zbl 1032.53008
[15] Mikeš, J., Jukl, M., Juklová, L.: Some results on traceless decomposition of tensors. J. Math. Sci., New York 174, 5 (2011), 627–640; translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz., 124, 1 (2010). DOI 10.1007/s10958-011-0321-y | MR 2882720 | Zbl 1332.53020
[16] Jukl, M., Juklová, L., Mikeš, J.: Applications of local algebras of differentiable manifolds. J. Math. Sci., New York 207, 3 (2015), 485–511; translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz., 126 (2013), 219–261. DOI 10.1007/s10958-015-2381-x | Zbl 1317.13057
[17] Norden, A. P.: Spaces of Affine Connection. Nauka, Moscow, 1976. MR 0467565
[18] Postnikov, M. M.: Lectures on the Geometry. Semester IV. Differential Geometry. Nauka, Moscow, 1988. MR 0985587
[19] Rozenfeld, B. A.: Higher-dimensional Spaces. Nauka, Moscow, 1966.
[20] Rozenfeld, B. A.: Geometry of Lie Groups. Kluwer, Dordrecht–Boston–London, 1997.
[21] Shapukov, B. N.: Connections on a differential fibred bundle. Tr. Geom. Sem. Kazan. Univ. 12 (1980), 97–109. MR 0622541
[22] Vishnevsky, V. V., Shirokov, A. P., Shurygin, V. V.: Spaces over Algebras (Prostranstva nad algebrami). Kazan University Press, Kazan’, 1985, (in Russian).
Partner of
EuDML logo