Geometry of \(2 \times 2\) Hermitian matrices over any division ring (Q1047887)

Suppose \(D\) is a division ring with an involution \(^{ - }\), \(\mathcal{H}_2 (D)\) is the set of \(2 \times 2\) Hermitian matrices over \(D\). Let \(\text{ad}(A,B) = \text{rank}(A - B)\) be the arithmetic distance between \(A, B \in \mathcal{H}_2 (D)\). The author proves the fundamental theorem of the geometry of \(2 \times 2\) Hermitian matrices over \(D\) \((\text{char}(D) \neq 2)\): If \(\varphi : \mathcal{H}_2 (D) \rightarrow \mathcal{H}_2 (D)\) is the adjacency preserving bijective map, then \(\varphi \) is of the form \(\varphi (X) = {}^t \bar P X^{\sigma} P +\varphi (0)\), where \(P \in GL_{2}(D), \sigma \) is a quasi-automorphism of \(D\). The quasi-automorphism of \(D\) is also investigated.