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Year 2019, Volume: 12 Issue: 1, 1 - 8, 27.03.2019

Murat Bekar

https://doi.org/10.36890/iejg.545737

Abstract

References

  • [1] Abłamowicz, R. and Sobczyk, G., Lectures on Clifford (Geometric) Algebras and Applications. Birkhäuser, Boston, 2004.
  • [2] Abłamowicz, R., Computations with Clifford and Grassmann Algebras, Adv. Appl. Clifford Algebras, 19 (2009), 499-545.
  • [3] Abłamowicz, R. and Fauser, B., On the Transposition Anti-Involution in Real Clifford Algebras I: The Transposition Map, Linear Multilinear A, 59 (2011), 1313-1358.
  • [4] Abłamowicz, R. and Fauser, B., On the Transposition Anti-Involution in Real Clifford Algebras II: Stabilizer Groups of Primitive Idempotents, Linear Multilinear A, 59 (2011), 1359-1381.
  • [5] Abłamowicz, R. and Fauser, B., On the Transposition Anti-Involution in Real Clifford Algebras III: The Automorphism Group of the Transposition Scalar Product on Spinor Spaces, Linear Multilinear A, 60 (2012), 621-644.
  • [6] Aslan, S. and Yayli, Y., Canal Surfaces with Quaternions, Adv. Appl. Clifford Algebras, 26 (2016), 31-38.
  • [7] Bekar, M. and Yayli, Y., Semi-Euclidean Quasi-Elliptic Planar Motion, Int. J. Geom. Methods Mod. Phys., 13 (2016), 1650089 (11 pages), DOI: 10.1142/S0219887816500894.
  • [8] Bekar, M. and Yayli, Y., Lie Aigebra of Unit Tangent Bundle, Adv. Appl. Clifford Algebras, 27 (2017), 965–975.
  • [9] Ell, T. A. and Sangwine, S. J., Quaternion Involutions and Anti-Involutions, Comput. Math. Appl., 53 (2007), 137-143.
  • [10] Es, H., First and Second Acceleration Poles in Lorentzian Homothetic Motions, Commun. Fac. Sci. Univ. Ank. Ser. A 1 Math. Stat., 67 (2018), 19-28.
  • [11] Hahn, A. J., Quadratic Algebras, Clifford Algebras and Arithmetic Witt Groups, Springer-Verlag, New York, 1994.
  • [12] Hamilton, W. R., On a New Species of Imaginary Quantities Connected with the Theory of Quaternions, P. Roy. Irish Academy, 2 (1844), 424-434.
  • [13] Jafari, M., Split Semi-Quaternions Algebra in Semi-Euclidean 4-Space, Cumhuriyet Science Journal, 36 36 (2015), 70–77.
  • [14] Kuipers, J. B., Quaternions and Rotation Sequences. Princeton University Press, New Jersey, 1999.
  • [15] Kula, L. and Yayli, Y., Split Quaternions and Rotations in Semi-Euclidean Space E^4_2 , J. Korean Math. Soc., 6 (2007), 1313-1327.
  • [16] Lopez, R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, Int. Electron. J. Geom., 7 (2014), 44-107.
  • [17] Ward, J. P., Quaternions and Cayley Algebras and Applications, Kluwer Academic Publishers, Dordrecht, 1996.

Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space

Year 2019, Volume: 12 Issue: 1, 1 - 8, 27.03.2019

Murat Bekar

https://doi.org/10.36890/iejg.545737

Abstract

In this paper, a one-to-one correspondence between the set of unit split semi-quaternions and
unit tangent bundle of semi-Euclidean plane is given. It is shown that the set of unit split semiquaternions
based on the group operation of multiplication is a Lie group. The Lie algebra of
this group, consisting of the vector space matrix of the angular velocity vectors, is also considered.
Planar rotations in Euclidean plane are expressed using split semi-quaternions. Some examples are
given to illustrate the findings.

Keywords

Lie algebra pseudo-rotation (boost) split semi-quaternion unit tangent bundle

References

  • [1] Abłamowicz, R. and Sobczyk, G., Lectures on Clifford (Geometric) Algebras and Applications. Birkhäuser, Boston, 2004.
  • [2] Abłamowicz, R., Computations with Clifford and Grassmann Algebras, Adv. Appl. Clifford Algebras, 19 (2009), 499-545.
  • [3] Abłamowicz, R. and Fauser, B., On the Transposition Anti-Involution in Real Clifford Algebras I: The Transposition Map, Linear Multilinear A, 59 (2011), 1313-1358.
  • [4] Abłamowicz, R. and Fauser, B., On the Transposition Anti-Involution in Real Clifford Algebras II: Stabilizer Groups of Primitive Idempotents, Linear Multilinear A, 59 (2011), 1359-1381.
  • [5] Abłamowicz, R. and Fauser, B., On the Transposition Anti-Involution in Real Clifford Algebras III: The Automorphism Group of the Transposition Scalar Product on Spinor Spaces, Linear Multilinear A, 60 (2012), 621-644.
  • [6] Aslan, S. and Yayli, Y., Canal Surfaces with Quaternions, Adv. Appl. Clifford Algebras, 26 (2016), 31-38.
  • [7] Bekar, M. and Yayli, Y., Semi-Euclidean Quasi-Elliptic Planar Motion, Int. J. Geom. Methods Mod. Phys., 13 (2016), 1650089 (11 pages), DOI: 10.1142/S0219887816500894.
  • [8] Bekar, M. and Yayli, Y., Lie Aigebra of Unit Tangent Bundle, Adv. Appl. Clifford Algebras, 27 (2017), 965–975.
  • [9] Ell, T. A. and Sangwine, S. J., Quaternion Involutions and Anti-Involutions, Comput. Math. Appl., 53 (2007), 137-143.
  • [10] Es, H., First and Second Acceleration Poles in Lorentzian Homothetic Motions, Commun. Fac. Sci. Univ. Ank. Ser. A 1 Math. Stat., 67 (2018), 19-28.
  • [11] Hahn, A. J., Quadratic Algebras, Clifford Algebras and Arithmetic Witt Groups, Springer-Verlag, New York, 1994.
  • [12] Hamilton, W. R., On a New Species of Imaginary Quantities Connected with the Theory of Quaternions, P. Roy. Irish Academy, 2 (1844), 424-434.
  • [13] Jafari, M., Split Semi-Quaternions Algebra in Semi-Euclidean 4-Space, Cumhuriyet Science Journal, 36 36 (2015), 70–77.
  • [14] Kuipers, J. B., Quaternions and Rotation Sequences. Princeton University Press, New Jersey, 1999.
  • [15] Kula, L. and Yayli, Y., Split Quaternions and Rotations in Semi-Euclidean Space E^4_2 , J. Korean Math. Soc., 6 (2007), 1313-1327.
  • [16] Lopez, R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, Int. Electron. J. Geom., 7 (2014), 44-107.
  • [17] Ward, J. P., Quaternions and Cayley Algebras and Applications, Kluwer Academic Publishers, Dordrecht, 1996.
There are 17 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Murat Bekar

Publication Date March 27, 2019
Published in Issue Year 2019 Volume: 12 Issue: 1

Cite

APA Bekar, M. (2019). Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space. International Electronic Journal of Geometry, 12(1), 1-8. https://doi.org/10.36890/iejg.545737
AMA Bekar M. Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space. Int. Electron. J. Geom. March 2019;12(1):1-8. doi:10.36890/iejg.545737
Chicago Bekar, Murat. “Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space”. International Electronic Journal of Geometry 12, no. 1 (March 2019): 1-8. https://doi.org/10.36890/iejg.545737.
EndNote Bekar M (March 1, 2019) Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space. International Electronic Journal of Geometry 12 1 1–8.
IEEE M. Bekar, “Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space”, Int. Electron. J. Geom., vol. 12, no. 1, pp. 1–8, 2019, doi: 10.36890/iejg.545737.
ISNAD Bekar, Murat. “Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space”. International Electronic Journal of Geometry 12/1 (March 2019), 1-8. https://doi.org/10.36890/iejg.545737.
JAMA Bekar M. Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space. Int. Electron. J. Geom. 2019;12:1–8.
MLA Bekar, Murat. “Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space”. International Electronic Journal of Geometry, vol. 12, no. 1, 2019, pp. 1-8, doi:10.36890/iejg.545737.
Vancouver Bekar M. Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space. Int. Electron. J. Geom. 2019;12(1):1-8.
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