TY - JOUR
T1 - AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA
AU - Klınker, Frank
PY - 2015
DA - April
JF - International Electronic Journal of Geometry
JO - Int. Electron. J. Geom.
PB - Kazım İLARSLAN
WT - DergiPark
SN - 1307-5624
SP - 94
EP - 104
VL - 8
IS - 1
LA - en
AB - 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 universalcover of the special orthogonal group SO(V, g) is given by the so called spin groupSpin(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 inwhich spin groups and related structures are best described is the Clifford algebraC`(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 accidentalisomorphisms to such in dimension three to six, see Table 1. The isomorphisms area consequence of the classification of Lie algebras and can for example be seen byrecalling 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. Dueto the fact that the complexifications of the orthogonal groups are independent ofthe signature of the pseudo-Riemannian metric the groups in each column of Table1 are real forms of the same complex group for fixed dimension.
KW - Spin group
KW - represenation
KW - covering map
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UR - https://dergipark.org.tr/en/pub/iejg/issue/47115/592802
L1 - https://dergipark.org.tr/en/download/article-file/763278
ER -