G2 Matrix Manifold: A Software Construct

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).