AN EXPLICIT CONSTRUCTION OF REAL PRE-HURWITZ ALGEBRAS

A real algebra with a generators {e_α}_α=1,…,p-1 satisfying the relations: e_αe_β + e_βe_α = -2δ_αβI(α,β = 1, 2, …, p - 1) is called a pre-Hurwitz algebra, if it has a matrix representation such that ^t(e_α) = - e_α{α = 1,2,…,p - 1). Here we notice that a pre-Hurwitz algebra need not satisfy the irreducibility condition. In this paper, an explicit construction of generators of all pre-Hurwitz algebras is given and their irreducibility is discussed. The main results are given in Theorem I, II and III.