Products of Involutions in Steinberg Group over Skew Fields*

Abstract

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 ${\prod\limits_{j = 1}^n {\overline{{\beta _{j} \gamma _{j} }} } } = \det A.$ It is also proved that every element δ in St(K) is a product of 10 involutions.

Keywords

Steinberg group / Involution / Skew field / 15A23 / 20H25

References

[1] Gustafson Linear Algebra Appl., 1976, 13: 157

[2] Dennis J. Algebra, 1988, 118: 150

[3] Ambrosiewicz Demonstratio Math., 1991, 24: 311

[4] You Linear Algebra Appl., 1993, 186: 235

[5] Knuppel Geom. Dedicata, 1991, 38: 301

[6] Milnor, J., Introduction to Algebraic K-Theory, Princeton University Press, Princeton, 1971

[7] Silvester, J. R., Introduction to Algebraic K-Theory, Chapman and Hall, London-New York, 1981

[8] Sourour Linear Multilinear Algebra, 1986, 19: 141

[9] Hua, L. G. and Wan, Z. X., Classical Groups (in Chinese), Science and Technoloy Press, Shanghai, 1963

[10] Cohn, P. M., Skew Field Constructions, Cambridge University Press, London, 1977