Ring Division Algebras, Self-Duality and Supersymmetry

Abstract

We argue that once octonions are formulated as soft Lie algebras, they may be safely used and the non-associativity can be overcame. The necessary points are: (a) Fixing the direction of action by introducing the \delta operator. (b) Closing the \delta algebra by using structure functions f_{ijk} (\phi). (c) Representation of the \delta algebra can be developed. The E or E(\phi) can be found and their structure functions can be computed easily. There may be different applications of soft seven sphere in physics. We have given two cases where the ring division algebras occupies a special position. Self-duality and Simple supersymmetric Yang-Mills theories are two promising places where soft seven sphere prove to be useful and essential.