Ternary "Quaternions" and Ternary TU(3) algebra

Abstract

To construct ternary "quaternions" following Hamilton we must introduce two "imaginary "units, $q_1$ and $q_2$ with propeties $q_1^n=1$ and $q_2^m=1$. The general is enough difficult, and we consider the $m=n=3$. This case gives us the example of non-Abelian groupas was in Hamiltonian quaternion. The Hamiltonian quaternions help us to discover the $SU(2)=S^3$ group and also the group $L(2,C)$. In ternary case we found the generalization of U(3) which we called $TU(3)$ group and also we found the the SL(3,C) group. On the matrix language we are going from binary Pauly matrices to three dimensional nine matrices which are called by nonions. This way was initiated by algebraic classification of $CY_m$-spaces for all m=3,4,...where in reflexive Newton polyhedra we found the Berger graphs which gave in the corresponding Cartan matrices the longest simple roots $B_{ii}=3,4,..$ comparing with the case of binary algebras in which the Cartan diagonal element is equal 2, {\it i.e. } $A_{ii}=2$.