Clifford Algebra And M(?)bius Groups

The main aim of this dissertation is to disscuss some properties of Mobius transformations, Mobius groups and Clifford algebra from algebraic viewpoint. This dissertation is arranged as follows.In Chapter 1, we provide some backgroud information about our research and the statement of our main results.In Chapter 2, we introduce some basic concepts and properties of some backgroud materials, including the Mobius transformations, Clifford algebra and the hyperbolic space.In Chapter 3, at first, by using the real matrix representation of quaternions, we obtain the Kronecker formula of quaternionic matrices. By using this obtained formula, the necessary and sufficient condition for the existence of solutions of the equation AqXBq = D and the explicit form of its general solutions are obtained. In particular, the Cramer formula of quaternionic linear equations is derived. Secondly, by using the complex matrix representation of quaternions, the necessary and sufficient conditions for the existence of Hermite solutions of the equation AXB = D and the explicit form of its general solutions are obtained. As an application, the necessary and sufficient conditions for the existence of Hermite solutions of equation A*X*B* ± BXA = D and the explicit form of its general solutions are obtained.In Chapter 4, we consider quaternionic Mobius transformations preserving the unit ball in the quaternions H We give an explicit expression for the fixed points of g in terms of a, b, c and d and we use this to classify quaternionic Mobius transformations into six categories determined by their dynamics.In Chapter 5, We consider 4-dimensional Mobius transformations g(x) — (ax + b) (cx + d)-1 by the identification with the matrix group PS△L(2,H) of quaternoinic 2-by-2 matrices with Dieudonne determinant det△ equal to 1. By using some similar methods stablished by Ahlfors, we obtain a classification of 4-dimensional Mobius transformations. These results are pertain to the results about quaternionic Mobius transformations in Chapter 4 and the classic results obttained by using traces of matrices in the case of the complex plane.In Chapter 6, by using real quaternions and their real matrix representations, we obtain some real matrix representations of 4-dimensional Clifford numbers and some properties of these representations. We propose the concept of Moore-Penrose inverse of Clifford numbers and obtain a necessary and sufficient condition for Clifford numbers to be invertible and the explicit form of their inverse. As an application, we obtain an explicit expression of the general solution of ax=xb in terms of a and b in 4-dimensional Clifford algebra C4.