Let D be a division ring with an involution ?, $ \mathcal{H}_2 $ (D) be the set of 2 × 2 Hermitian matrices over D. Let ad(A,B) = rank(A ? B) be the arithmetic distance between A, B ∈ $ \mathcal{H}_2 $ (D). In t...Let D be a division ring with an involution ?, $ \mathcal{H}_2 $ (D) be the set of 2 × 2 Hermitian matrices over D. Let ad(A,B) = rank(A ? B) be the arithmetic distance between A, B ∈ $ \mathcal{H}_2 $ (D). In this paper, the fundamental theorem of the geometry of 2 × 2 Hermitian matrices over D (char(D) ≠ = 2) is proved: if φ: $ \mathcal{H}_2 $ (D) → $ \mathcal{H}_2 $ (D) is the adjacency preserving bijective map, then φ is of the form φ(X) = $ ^t \bar P $ X σ P +φ(0), where P ∈ GL 2(D), σ is a quasi-automorphism of D. The quasi-automorphism of D is studied, and further results are obtained.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 10671026)
文摘Let D be a division ring with an involution ?, $ \mathcal{H}_2 $ (D) be the set of 2 × 2 Hermitian matrices over D. Let ad(A,B) = rank(A ? B) be the arithmetic distance between A, B ∈ $ \mathcal{H}_2 $ (D). In this paper, the fundamental theorem of the geometry of 2 × 2 Hermitian matrices over D (char(D) ≠ = 2) is proved: if φ: $ \mathcal{H}_2 $ (D) → $ \mathcal{H}_2 $ (D) is the adjacency preserving bijective map, then φ is of the form φ(X) = $ ^t \bar P $ X σ P +φ(0), where P ∈ GL 2(D), σ is a quasi-automorphism of D. The quasi-automorphism of D is studied, and further results are obtained.
