Year 2015, Volume 8 , Issue 1, Pages 94 - 104 2015-04-30

AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA

Frank KLINKER [1]


In this note we present explicit and elementary formulas for the correspondence between the group of special Lorentz transformation SO+(3, 1), on the one hand, and its spin group SL(2, C), on the other hand. Although we will not mention Clifford algebra terminology explicitly, it is hidden in our calculations by using complex 2 × 2-matrices. Nevertheless, our calculations are strongly motivated by the Clifford algebra gl(4, C) of fourdimensional space-time.It is well known that for a pseudo-euclidean vector space (V, g) the universal cover of the special orthogonal group SO(V, g) is given by the so called spin group Spin(V, g). For the case V = Rp+q and g = diag(1q, −1p) we write SO(p, q) and Spin(p, q). The covering map is 2:1 for dim V > 2. The theoretic setting in which spin groups and related structures are best described is the Clifford algebra C`(V, g), see [2, 3, 8] for example. Although spin groups in general refrain from being described by classical matrix groups for dimensional reason, there are accidental isomorphisms to such in dimension three to six, see Table 1. The isomorphisms are a consequence of the classification of Lie algebras and can for example be seen by recalling the connection to Dynkin diagrams. We use the notation from [4] and recommend this book for details on the definition of the classical matrix groups. Due to the fact that the complexifications of the orthogonal groups are independent of the signature of the pseudo-Riemannian metric the groups in each column of Table 1 are real forms of the same complex group for fixed dimension.

Spin group, represenation, covering map
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  • [3] Reese Harvey, F., Spinors and Calibrations (Perspectives in Mathematics). Academic Press, 1990)
  • [4] Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces. American Mathe- matical Society, 2001.
  • [5] Hestenes, D., Space-Time Algebra (Documents on Modern Physics). Gordon and Breach Science Publishers, Inc. 1966.
  • [6] Joos, H., Zur Darstellungstheorie der inhomogenen Lorentzgruppe als Grundlage quanten- mechanischer Kinematik. Fortschritte der Physik 10 (1962), 65-146.
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  • [8] Michelson, M. -L. and Blaine Lawson, H., Spin Geometry. Princeton University Press, 1989.
Primary Language en
Subjects Mathematics
Journal Section Research Article
Authors

Author: Frank KLINKER

Dates

Publication Date : April 30, 2015

Bibtex @research article { iejg592802, journal = {International Electronic Journal of Geometry}, issn = {}, eissn = {1307-5624}, address = {}, publisher = {Kazım İLARSLAN}, year = {2015}, volume = {8}, pages = {94 - 104}, doi = {}, title = {AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA}, key = {cite}, author = {Klınker, Frank} }
APA Klınker, F . (2015). AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA . International Electronic Journal of Geometry , 8 (1) , 94-104 . Retrieved from https://dergipark.org.tr/en/pub/iejg/issue/47115/592802
MLA Klınker, F . "AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA" . International Electronic Journal of Geometry 8 (2015 ): 94-104 <https://dergipark.org.tr/en/pub/iejg/issue/47115/592802>
Chicago Klınker, F . "AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA". International Electronic Journal of Geometry 8 (2015 ): 94-104
RIS TY - JOUR T1 - AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA AU - Frank Klınker Y1 - 2015 PY - 2015 N1 - DO - T2 - International Electronic Journal of Geometry JF - Journal JO - JOR SP - 94 EP - 104 VL - 8 IS - 1 SN - -1307-5624 M3 - UR - Y2 - 2021 ER -
EndNote %0 International Electronic Journal of Geometry AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA %A Frank Klınker %T AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA %D 2015 %J International Electronic Journal of Geometry %P -1307-5624 %V 8 %N 1 %R %U
ISNAD Klınker, Frank . "AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA". International Electronic Journal of Geometry 8 / 1 (April 2015): 94-104 .
AMA Klınker F . AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA. Int. Electron. J. Geom.. 2015; 8(1): 94-104.
Vancouver Klınker F . AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA. International Electronic Journal of Geometry. 2015; 8(1): 94-104.
IEEE F. Klınker , "AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA", International Electronic Journal of Geometry, vol. 8, no. 1, pp. 94-104, Apr. 2015