On Quadratic and Nonquadratic Forms :

Hurwitz transformations are defined as specific automorphisms of a CayleyDickson algebra. These transformations generate quadratic and nonquadratic forms. We investigate here the Hurwitz transformations corresponding to Cayley-Dickson algebras of dimensions 2m = 2, 4 and 8. The Hurwitz transformations which lead to quadratic forms are discussed from geometrical and Lie-algebraic points of view. Applications to number theory and dynamical systems are briefly examined. ∗ Work presented both to the Symposium “Symmetries in Science IX” held at the Cloister Mehrerau (Bregenz, Austria, 6-10 August 1996) and to the Symposium “III Catalan Days on Applied Mathematics” held at the Institut d’Estudis Ilerdencs (Lleida, Spain, 27-29 November 1996). To be published in Symmetries in Science IX, ed. Bruno Gruber (Plenum Press, New York, 1997). ON QUADRATIC AND NONQUADRATIC FORMS: APPLICATION TO NONBIJECTIVE R → R TRANSFORMATIONS Maurice Kibler Institut de Physique Nucléaire de Lyon IN2P3-CNRS et Université Claude Bernard 43 Boulevard du 11 Novembre 1918 F-69622 Villeurbanne Cedex France