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Geometry of 2 × 2 Hermitian matrices over any division ring

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Abstract

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.

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Authors and Affiliations

  1. School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, 410004, China

    LiPing Huang

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  1. LiPing Huang
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Correspondence to LiPing Huang.

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This work was supported by National Natural Science Foundation of China (Grant No. 10671026)

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Huang, L. Geometry of 2 × 2 Hermitian matrices over any division ring. Sci. China Ser. A-Math. 52, 2404–2418 (2009). https://doi.org/10.1007/s11425-009-0070-2

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  • DOI: https://doi.org/10.1007/s11425-009-0070-2

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