Quaternion M (?) Bius Transform The Classification And Dimensional Clifford Algebra Equation

In this papers, we mainly discuss the classification of quaternionic Mobius transformations, the real matrix representations of quaternion and four-dimensional Clifford algebra. We obtain the necessary and sufficient conditions for the quaternion and four-dimensional Clifford algebra equation ax=xb+c and ax=(?)b+c is resolvable.In Chapter 1, we introduce some background of our topic.In Chapter 2, we discuss the classification of quaternionic Mobius transformations, the solution of quaternion equation ax=xb+c and ax=(?)b+c.In Chapter 3, we discuss the commutator of two nontrivial elements of U(1,2;F) which share a-unique fixed point. When F denotes the field C of complex numbers, the commutator of them is parabolic element or the identity. When F denotes the division ring of real quaternions H, we provide a theorem to distinguish between elliptic and parabolic element. These results are generalizations of counterparts in the setting of Mobius groups.In Chapter 4, we dicuss the real matrix representations of four-dimensional of Clifford algebra. Using the real matrix reprentations of four-dimensional Clifford numbers, we obtain the conditions of the solvability of four-dimensional Clifford algebra equations ax=xb+c and ax=(?)b+c.