The Hurwitz Problem On Composition Of Quadratic Formss

We introduce a series of Zn2-graded quasialgebras Pn(m) which generalizes Clifford algebras, higher octonions, and higher Cayley algebras. The con-structed series of algebras and their minor perturbations are applied to con-tribute explicit solutions to the Hurwitz problem on compositions of quadratic forms. In particular, we provide explicit expressions of the well-known Hurwitz-Radon square identities in a uniform way, recover the Yuzvinsky-Lam-Smith formulas, confirm the third family of admissible triples proposed by Yuzvin-sky in 1984, improve the two infinite families of solutions obtained recently by Lenzhen, Morier-Genoud and Ovsienko, and construct several new infinite families of solutions.An outline of the paper is as follows:Chapter 1 is to introduce the history and development of the Hurwitz problem as well as our recent research progress.Chapter 2 is devoted to the basic definitions of group graded quasialge-bras and multiplicative pairs. Some interesting Z2n-graded quasialgebras are recalled and the series of algebras Pn(m) are introduced.The uniform and explicit expression of the Hurwitz-Radon square identi-ties is then given in Chapter 3.In Chapter 4, simpler constructions of Yuzvinsky-Lam-Smith formulas are provided and Yuzvinsky’s third family of admissible triples is verified.In Chapter 5, some improvements of the Lenzhen-Morier-Genoud-Ovsienko formulas are presented. The duality of two basic methods of generating ad-missible triples is also mentioned in passing.Three new infinite series of admissible triples are shown in Chapter 6.