G2 Matrix Manifold: A Software Construct

TL;DR

This paper develops symbolic, numeric, and graphic computational tools in Mathematica to construct and analyze G2 structures, octonions, and related algebraic properties, including BCH formulas and maximal tori.

Contribution

It introduces novel algorithms for octonion exponentiation, logarithms, and BCH computations, and explores the geometric and algebraic properties of G2 structures.

Findings

Verified validity of vector product in specific dimensions

Developed symbolic exponential computations for G2 basis

Uncovered examples and counterexamples of maximal tori

Abstract

An ensemble of symbolic, numeric and graphic computations developed to construct the Octonionic and compact G2 structures in Mathematica 8.0. Cayley-Dickenson Construction symbolically applied from Reals to Octonions. Baker- Campbell-Hausdorff formula (BCH) in bracket form verified for Octonions. Algorithms for both exponentiation and logarithm of Octonions developed. Exclusive validity of vector Product verified for 0, 1, 3 and 7 dimensions. Symbolic exponential computations carried out for two distinct g2 basis(s) and arbitrary precision BCH for G2 was coded. Example and counter-example Maximal Torus for G2 was uncovered. Densely coiled shapes of actions of G2 rendered. Kolmogorov Complexity for BCH investigated and upper bounds computed: Complexity of non-commutative non- associative algebraic expression is at most the Complexity of corresponding commutative associative algebra plus K(BCH).