Hurwitz pairs equipped with complex structures

Abstract

In 1923 there appeared a paper of A. Hurwitz, where he solved the problem of finding all the pairs of positive integers (n,p) and all systems of c ^k_jα ε ℝ, j, k=1, …,n; α=1,…,p, p≤n, such that the set of bilinear forms η_j=x_α c ^k_jα y_k satisfied the condition Σ_j η ²_j = Σ_α x ²_α Σ_k y ²_k . The present authors reformulate the Hurwitz problem and its solution in the language of Clifford algebras, introducing the notion of the Hurwitz pair of two real unitary vector spaces V and S. Then they study the dependence of these pairs on induced symplectic decomposition and arrive in a natural way at a complex geometry induced by a given symplectic decomposition in connection with distinguishing a fixed direction in S, what introduces a kind of anisotropy.