Investigating Sedenion Extension Loops and Frames of General Hypercomplex Numbers
Njuguna, Lydia Nyambura
MetadataShow full item record
A sequence of algebras over the field of real numbers can be constructed, each with twice the dimension of the previous one. The algebra constructed by doubling complex numbers is the 22 - dimensional quaternions. Next we have the 23 - dimensional octonions, constructed by doubling the quaternions.The algebra constructed-by doubling octonions is the 24 -dimensional sedenions. The oldest method of constructing these algebras is the Cayley-Dickson formula. Since they extendthe complex numbers, they are called hypercomplex numbers in general for dimensions greater than 24 . Our main emphasis is on the general Z"-ons. Split extensions of 2n -ons are studied from the point of view of Loop Theory. Multiplication of basis elements of complex, quaternion, octonion and sedenion split extensions using the Jonathan Sooth doubling formula is done. It is shown thatNim addition gives a way of determining the subscripts for the products of the basis elements for the split extensions. This result is extended to split extensions of general2n - ons. Subloops of sedenions are also investigated, and it is shown that they do not necessarily reduce to either subloops of octonions or sedenion split extensions. This is done by showing the existence of two-sided subloops of sedenions that are neither subloops of octonions or sedenion split extensions. It is well known that when L, the multiplicative sub loop of octonions is abelian, the sedenion split extension L.x SO formed is a group. The structure of Lx SO when L is non-abelian is studied: IJ? 'particular, it is shown that Lx SO fails to satisfy group properties. When the satisfaction of standard Loop theoretical properties is investigated, L x SO is seen to satisfy the Jordan identity and flexible properties. It however fails to satisfy the left alternative, right alternative and anti-automorphism properties. Finally, multiplication of the basis elements of complex numbers, quatemions, octonions and sedenions is done using the Jonathan Smith doubling formula. In each case, it is shown that the formula gives a way of constructing Hadamard matrices, as did the Sooth- Conway formula. This result is generalized for genera12n - ons .