Abstract
In this study, firstly, we give a different approach to the relationship between the split quaternions and rotations in Minkowski space \(\mathbb {R}_1^3\). In addition, we obtain an automorphism of the split quaternion algebra \(H'\) corresponding to a rotation in \(\mathbb {R}_1^3\). Then, we give the relationship between the hyperbolic spinors and rotations in \(\mathbb {R}_1^3\). Finally, we associate to a split quaternion with a hyperbolic spinor by means of a transformation. In this way, we show that the rotation of a rigid body in the Minkowski 3-space \(\mathbb {R}_1^3\) expressed the split quaternions can be written by means of the hyperbolic spinors with two hyperbolic components. So, we obtain a new and short representation (hyperbolic spinor representation) of transformation in the 3-dimensional Minkowski space \(\mathbb {R}_1^3\) expressed by means of split quaternions.
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References
Antonuccio, F.: Hyperbolic Numbers and the Dirac Spinor (1998). arXiv:hep-th/9812036v1
Balci, Y., Erisir, T., Gungor, M.A.: Hyperbolic spinor Darboux equations of spacelike curves in Minkowski 3-space. J. Chungcheong Math. Soc. 28(4), 525–535 (2015)
Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geomerty. Academic Press, London (1975)
Carmeli, M.: Group Theory and General Relativity, Representations of the Lorentz Group and Their Applications to the Gravitational Field. Imperial College Press, McGraw-Hill, New York (1977)
Cartan, É.: The Theory of Spinors. MIT Press, Cambridge (1966)
Cockle, J.: On systems of algebra involving more than one imaginary. Philos. Mag. 35, 434–435 (1849)
Del Castillo, G.F.T., Barrales, G.S.: Spinor formulation of the differential geometry of curves. Rev. Colomb. Mat. 38, 27–34 (2004)
Erisir, T., Gungor, M.A., Tosun, M.: Geometry of the hyperbolic spinors corresponding to alternative frame. Adv. Appl. Clifford Algebras 25(4), 799–810 (2015)
Flaut, C., Shpakivskyi, V.: An efficient method for solving equations in generalized quaternion and octanion algebras. Adv. Appl. Clifford Algebras 25(2), 337–350 (2015)
Flaut, C., Ştefănescu, M.: Some equations over generalized quaternion and octanion division algebras. Bull. Math. Soc. Sci. Math. Roumanie Tome 52(100), 427–439 (2009)
Friedrich, T.: Dirac Operators in Riemannian Geometry. American Mathematical Society, Providence (2000)
Gurlebeck, K., Sprossig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Toronto (1997)
Ketenci, Z., Erisir, T., Gungor, M.A.: A construction of hyperbolic spinors according to Frenet frame in Minkowski space. Dyn. Syst. Geom. Theor. 13(2), 179–193 (2015)
Kisi, I., Tosun, M.: Spinor Darboux equations of curves in Euclidean 3-space. Math. Morav. 19(1), 87–93 (2015)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Wiley Interscience, New York (1969)
Kula, L.: Split Quaternions and Geometrical Applications, Ph.D. Thesis. Ankara University, Ankara (2003)
Kula, L., Yayli, Y.: Split quaternions and rotations in semi Euclidean space. J. Korean Math. Soc. 44, 1313–1327 (2007)
Lawson, H.B., Michelsohn, M.L.: Spin Geometry. Princeton University Press, New Jersey (1989)
O’Donnell, P.: Introduction to 2-Spinors in General Relativity, World Scientific Publishing, London (2003)
O’Neill, B.: Semi-Riemannian Geometry, with Applications to Relativity. Academic Press, New York (1983)
Pottmann, H., Wallner, J.: Computational Line Geometry. Springer, Berlin (2000)
Tresse, A.: Sur les invariants differentiels des groupes continus de transformations. Acta Math. 18, 1–88 (1894)
Unal, D., Kisi, I., Tosun, M.: Spinor Bishop equation of curves in Euclidean 3-space. Adv. Appl. Clifford Algebras 23(3), 757–765 (2013)
Vivarelli, M.D.: Development of spinors descriptions of rotational mechanics from Euler’s rigid body displacement theorem. Celest. Mech. 32, 193–207 (1984)
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Communicated by Wolfgang Sprössig
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Tarakçioğlu, M., Erişir, T., Güngör, M.A. et al. The Hyperbolic Spinor Representation of Transformations in \(\mathbb {R}_1^3\) by Means of Split Quaternions. Adv. Appl. Clifford Algebras 28, 26 (2018). https://doi.org/10.1007/s00006-018-0844-0
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DOI: https://doi.org/10.1007/s00006-018-0844-0