Products of Involutions in Steinberg Group over Skew Fields

摘要

Consider the stable Steinberg group St(K) over a skew field K. An element x is called an involution if x~2 = 1. In this paper, an involution is allowed to be the identity. The authors prove that an element A of GL_n(K) up to conjugation can be represented as BC, where B is lower triangular and C is simultaneously upper triangular. Furthermore, B and C can be chosen so that the elements in the main diagonal of B are β_1, β_2, • • • ,β_n, and of C are γ_1,γ_2,••• , γ_n c_n, where c_n ∈ [K~*,K~*] and Π_(j=1)~n β_jγ_j = det A. It is also proved that every element δ in St(K) is a product of 10 involutions.