Geometry of n × n(n ≥ 3) Hermitian Matrices Over Any Division Ring with an Involution and Its Applications

Abstract

Let D be a division ring with an involution ⁻ and char(D) ≠ 2, F = {x ∈ D: x = }. Let ℋ _n (D) be the set of all n × n hermitian matrices over D. Two hermitian matrices H ₁ and H ₂ are said to be “adjacent” if rank(H ₁ − H ₂) = 1. The fundamental theorem of geometry of hermitian matrices over D is proved: If n ≥ 3 and 𝒜 is a bijective map from ℋ _n (D) to itself such that 𝒜 preserves the adjacency, then ∀ X ∈ ℋ _n (D), where a ∈ F*, P ∈ GL _n (D), and ρ is an automorphism of D which satisfies for all x ∈ D. The application of the fundamental theorem to algebra and geometry is discussed. For example, every Jordan isomorphism or additive rank-1-preserving surjective map on ℋ _n (D) is characterized.

Key Words:

• Adjacency
• Division ring with an involution
• Geometry of matrices
• Hermitian matrices
• Jordan isomorphism

2000 Mathematics Subject Classification:

• 14L35
• 14L24
• 51D20
• 15A33

ACKNOWLEDGMENTS

The author thanks professor Zhe-Xian Wan for his kindest suggestions.

Project 10671026 supported by National Natural Science Foundation of China.

Notes

Communicated by E. I. Zelmanov.