Sedenion extension loops and frames of hypercomplex 2[superscript n]-ons

2004-01-01
Kivunge, Benard
Jonathan Smith
Organizational Units
Organizational Unit
Mathematics
Abstract

This research work is mainly on hypercomplex numbers with major emphasis on the 16 dimensional sedenions. In Chapter 1, a literature review on current and past work on the area of hypercomplex numbers is given. Basic definitions to be used in the rest of the work are given. We introduce the sedenions and the different doubling formulas for obtaining sedenions from the octonions and general 2n-ons from 2n -1-ons. We also discuss some advantages and disadvantages of these doubling formulas.;In Chapter 2 the sedenions are considered from the point of view of loop theory. Although the sedenions do not directly yield loops in the way that Moufang loops are obtained from octonions, the left loop of non-zero sedenions does contain new two-sided subloops as sedenion extensions. These loops are constructed abstractly as extensions of subloops L of the octonions. The chapter examines the satisfaction by these extensions of various standard loop-theoretical identities. Equivalent conditions for the extensions to be groups are discussed. Consequences arising from these conditions are also given. For example, it turns out that they are all power-associative, even though the full left loop of all non-zero sedenions is not itself power-associative.;Chapter 3 gives a powerful result, the multiplication of the frames of the 2n-ons fits in the projective geometry PG(n - 1, 2). In doing so, it is convenient to consider this projective geometry from several different viewpoints: as a design, in terms of Nim addition, and as the geometry of linear subspaces of a vector space. It is then shown that the number of nontrivial subloops of order 2k inside the loop formed by the frame of the 2n-ons, the set of basic elements together with their negatives, is given by nk 2=2n-1 2n-1-1 2n-2-1&cdots;2 n-k+1-12 1-122-1 23-1&cdots; 2k-1.;In Chapter 4, Hadamard matrices are discussed. It is shown that the sign matrix of the frame multiplication in the 2n-ons under the Smith-Conway or Cayley-Dickson process is a skew Hadamard matrix. These matrices are shown to be equivalent to Kronecker products when n ≤ 3. Chapter 5 gives some open problems for future research.