Quaternionic Version of Rotation Groups

Quaternionic version of rotation group SO(3) has been constructed. We construct a quatenionic version of rotation operation that act to a quaternionic version of a space coordinate vector. The computation are done for every rotation about each coordinate axes (x,y, and z). The rotated quaternionic space coordinate vector contain some unknown constants which determine the quaternionic rotation operator. By solving for that constants, we get the expression of the quaternionics version of the rotation operator. Finally the generators of the group were obtained. The commutation of the generator were also computed.

Total conjugate of q is given by or may be written as For every quaternion q, we can make an anti-hermitian quaternion qiq † which may be related to a point (x, y, z) in 3 dimensional space through (5) Copyright c 2015, JNSMR, ISSN: 2460-4453 Hence we get Although it was abandoned by some people, but lately the research in quaternion theory was developed again.Among who devotes his/her research in quaternion theory is Stefano De Leo.He has introduced some quaternionic group with its generator (de Leo, 1999).He showed some group generators without showing its derivation.
In this paper we will derive generators and its commutation relations of quaternionic version of rotation groups and Lorentz group directly from the definition of the groups respectively.The groups that will be discussed in this paper is quaternionic version of rotation groups U (2), SU (2), O(3), SO(3) and quaternionic version of Lorentz groups SO o (3, 1), SL(2, C).

R.C, H−RIGHT LINEAR OPERATORS
Before we discuss about quaternionic groups, we need to introduce about quaternionic operators and the space containing it.There are two kind of operator action, left and right action.Due to uncommutativity of quaternionic multiplication, we need to pay attention to the distinction of the two operators.
Left action of 1, i, j, k is represented by action of left operator on q ∈ H, while its right action is repesented by action of right operator The action of L µ and R µ are defined as respectively, where h µ ≡ (1, i, j, k).Both operators satisfies the following relation and the following commutation relation We need to introduce another operator that we called as barred operator | and is defined by In addition to the barred operator, the simultaneous left and right action can be represented by Moreover, we alse define the following sets and it is clear that The addition operation in H L ⊗ H R is given by and the multiplication operation is given by For the sake of simplicity, we introduce the notation to represents elements in H L ⊗H R that are right linear with respect to field X.
are right R-linear operators and satisfy The set of all O R is clearly equal to because it satisfy The set of all O as in (23) will be denoted by H L ⊗C R .Due to the associativity of quaternion multiplication, operator O H defined by is a right linear operator with respect to H in H since it follows that Copyright c 2015, JNSMR, ISSN: 2460-4453 In the last definition, we have used abuse of notation because H is not a field.Hence it follows that The direct product of two right R-linear is defined by The conjugation operation for left and right operators are defined by while conjugation operation for O R is defined by Hence, the conjugate of eq.( 29) is (32) In this section, we will construct the quaternionic version of rotation group SO(3).We start by defining a set U (1, H L ) containing elements of H L satisfying As indicated in the definition, the set naturally get a multiplication operation inherited from those in H L .It is easy to check that U (1, H L ) is a group with the given multiplication operation.For every v ∈ U (1, H L ), its inverse v −1 = v † is satisfy condition (33) also.According to the definition, it is clear that U (1, H L ) ⊂ H L .Now we define actions of u to quaternion q and its anti-hermitian qiq † by q → uq qiq † → uqiq † u † . (34) We want to show that a transformation of V = (x, y, z) T ∈ R 3 by an element of SO( 3) is equivalen to a transformation of q = ξ + jζ by an element of U (1, H L ).Now we assume that q = uq, or more explicitly we can write Here, A and B are elements of C such that u = A+jB is an element of U (1, H L ).From eq.( 35), we get By using eq.( 5), we obtain Now we have to find three pairs of values of A and B such that each A + jB are related to transformations R 1 , R 2 , and R 3 in SO(3), where R 1 , R 2 , and R 3 are defined by In order to obtain transformation u x which is related to R 1 , the first condition that must be satisfied by A and B is then we obtain, for every θ ∈ R, that is an element of U (1, H L ) that transforms q equivalently with transformations of R 1 (θ) to V ∈ R 3 .