Investigating Sedenion Extension Loops and Frames of General Hypercomplex Numbers
Abstract
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
constructedby doubling octonions is the 24 dimensional sedenions. The oldest
method of constructing these algebras is the CayleyDickson 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 twosided 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 nonabelian 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 antiautomorphism
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 .