Geometry of skew-Hermitian matrices

Abstract

Let D be a division ring with an involution ¯ . Assume that F = { a ∈ D : a = a ¯ } is a proper subfield of D and is contained in the center of D. Let SH n ( D ) be the set of n × n skew-Hermitian matrices over D. If H 1 , H 2 ∈ SH n ( D ) and rank( H 1 − H 2) = 1, H 1 and H 2 are said to be adjacent. The fundamental theorem of the geometry of skew-Hermitian matrices over D is proved: Let n ⩾ 2 and A be a bijective map of SH n ( D ) to itself, which preserves the adjacency. Then A is of the form A ( X ) = α t P ¯ X σ P + H 0 ∀ X ∈ SH n ( D ) , where α ∈ F*, P ∈ GL n ( D), H 0 ∈ SH n ( D ) , and σ is an automorphism of D.