Geometry of 2×2 Hermitian matrices over any division ring

Let D be a division ring with an involution-,H2(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 ∈ H2(D) . In this paper,the fundamental theorem of the geometry of 2 × 2 Hermitian matrices over D(char(D) = 2) is proved:if  :H2(D) → H2(D) is the adjacency preserving bijective map,then  is of the form (X) = tP XσP +(0) ,where P ∈ GL2(D) ,σ is a quasi-automorphism of D. The quasi-automorphism of D is studied,and further results are obtained.

Geometry of 2 × 2 Hermitian matrices over any division ring

Let D be a division ring with an involution ^?, $ \mathcal{H}_2 $ \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 $ \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 $ \mathcal{H}_2 (D) → $ \mathcal{H}_2 $ \mathcal{H}_2 (D) is the adjacency preserving bijective map, then φ is of the form φ(X) = $ ^t \bar P $ ^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.