Division algebras, extended supersymmetries and applications

Abstract I present here some new results which make explicit the role of the division algebras R , C , H , O in the construction and classification of, respectively, N = 1, 2, 4, 8 global supersymmetric quantum mechanical and classical dynamical systems. In particular an N = 8 Malcev superaffine algebra is introduced and its relation to the non-associative N = 8 SCA is discussed. A list of present and possible future applications is given.