Geometry of 2×2 Hermitian matrices over any division ring |
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基金项目: | supported by National Natural Science Foundation of China (Grant No. 10671026) |
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摘 要: | 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.
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Geometry of 2 × 2 Hermitian matrices over any division ring |
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Authors: | LiPing Huang |
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Institution: | HUANG LiPing School of Mathematics , Computing Science,Changsha University of Science , Technology,Changsha 410004,China |
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Abstract: | 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. |
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Keywords: | division ring with involution Hermitian matrices geometry of matrices quasiautomorphism |
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