The Subalgebra Structure of the Cayley-Dickson Algebra of Dimension 32 (trigintaduonion)

TL;DR

This paper investigates the subalgebra structure of the 32-dimensional trigintaduonion algebra by analyzing its generated loop, revealing a complex hierarchy of normal subloops and subalgebras of various dimensions.

Contribution

It provides a detailed analysis of the subalgebra structure of the trigintaduonion algebra using loop theory, which is a novel approach in this context.

Findings

Identifies 373 non-trivial subloops of various orders

Shows all subloops are normal

Establishes the subalgebra dimensions from subloop analysis

Abstract

The Cayley-Dickson algebras R (real numbers), C (complex numbers), H (quaternions), O (octonions), S (sedenions), and T (trigintaduonions) have attracted the attention of several mathematicians and physicists because of their important applications in both pure mathematics and theoretical physics. This paper deals with the determination of the subalgebra structure of the algebra T by analyzing the loop T_L of order 64 generated by its 32 basis elements. The analysis shows that T_L is a non-associative finite invertible loop (NAFIL) with 373 non-trivial subloops of orders 32, 16, 8, 4, and 2 all of which are normal. These subloops generate subalgebras of T of dimensions 16, 8, 4, 2, and 1 which form the elements of its structure.