Adv. Appl. Cliﬀord Algebras (2018) 28:26
2018 Springer International Publishing AG,
part of Springer Nature
published online February 24, 2018
Applied Cliﬀord Algebras
The Hyperbolic Spinor Representation
of Transformations in R
by Means of Split
Mustafa Tarak¸cio˘glu, T¨ulay Eri¸sir
Communicated by Wolfgang Spr¨ossig
Abstract. In this study, ﬁrstly, we give a diﬀerent approach to the rela-
tionship between the split quaternions and rotations in Minkowski space
. In addition, we obtain an automorphism of the split quaternion alge-
corresponding to a rotation in R
. Then, we give the relationship
between the hyperbolic spinors and rotations in R
. Finally, we associate
to a split quaternion with a hyperbolic spinor by means of a transfor-
mation. In this way, we show that the rotation of a rigid body in the
Minkowski 3-space R
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 repre-
sentation) of transformation in the 3-dimensional Minkowski space R
expressed by means of split quaternions.
Mathematics Subject Classiﬁcation. 11R52, 15A66, 51B20.
Keywords. Lie groups, Hyperbolic spinors, Split quaternions.
Lie group that is also a diﬀerentiable manifold is a group which has property
that the group operations are compatible with the smooth structure. Sophus
Lie, found out the theory of continuous transformation groups, denomi-
nated the Lie groups. Nominately, in 1893, ﬁrstly, the term “groupes de
Lie” appeared in the thesis of Lie’s student . Lie groups represent the
best-developed theory of continuous symmetry of mathematical objects and
structures. Moreover, we can say that the set of unit split quaternions is an
interesting example of Lie groups.