Supersymmetric Hypermatrix Lie Algebra and Hypermatrix Groups Generated by the Dihedral Set $D_3$

This work is an investigation into the structure and properties of supersymmetric hypermatrix Lie algebra generated by elements of the dihedral group $D_3$. It is based on previous work on the subject of supersymmetric Lie algebra (Schreiber, 2012). In preview work I used several new algebraic tools; namely cubic hypermatrices (including special arrangements of such hypermatrices) and I obtained an algebraic structure associated with the basis of the Lie algebra $sl_2$, and I showed that the basis elements $sl_2$ are generators of infinite periodic hypermatrix Lie algebraic structures with semisimple sub-algebras. The generated algebra has been shown to be an extended Lie hypermatrix algebra that has a classical Lie algebra decomposition composed of hypermatrices with periodic properties. The generators of higher dimensional Lie algebra were shown to be special supersymmetric, anti-symmetric and certain skew-symmetric hypermatrices. The present work takes a different look at the structure of periodic hypermatrix Lie algebra by using elements generating the classical dihedral group $D_3$. Using cubic dihedral symmetric hypermatrices (type: even-even, odd-odd, even-odd odd-even permutation) to generate Lie hypermatrix algebra I show that the extended dihedral algebra is a Lie hypermatrix algebras with special hypermatrix group properties, semisimple, symmetric, skew-symmetric, anti-symmetric, and anti-clockwise symmetric properties.