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

Abstract

This research work is mainly on hypercomplex numbers with major emphasis on the 16 dimensional sedenions. In Chapter 1, we give the literature review on current and past work on the area of hypercomplex numbers. Basic definitions are given and the sedenions are introduced. The advantages and disadvantages of the different doubling formulas are discussed. In Chapter 2, we introduce sedenion extensions and give equivalent conditions for the extensions to be groups. Consequences arising from these conditions are also given. Chapter 3 gives a powerful result, the multiplication of the frames of the 2-ons fits in the projective geometry _PG(n — 1,2). A formula for counting the number of multiplicative subloops of order 2k inside the frames is given. In Chapter 4, Hadamard matrices are discussed. It is shown that the sign matrix of the frame multiplication in the 2-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.