Supersymmetric Hypermatrix Lie Algebra and Hypermatrix Groups Generated by the Dihedral Set D3

This work is an investigation into the structure and properties of supersymmetric hypermatrix Lie algebra generated by elements of the dihedral group D3. 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 sl2, and I showed that the basis elements sl2 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 D3. 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.