Geometry of skew-Hermitian matrices Original Research Article

Let D be a division ring with an involution image. Assume that image is a proper subfield of D and is contained in the center of D. Let SHn(D) be the set of n × n skew-Hermitian matrices over D. If H1,H2set membership, variantSHn(D) and rank(H1 − H2) = 1, H1 and H2 are said to be adjacent. The fundamental theorem of the geometry of skew-Hermitian matrices over D is proved: Let n greater-or-equal, slanted 2 and A be a bijective map of SHn(D) to itself, which preserves the adjacency. Then A is of the form imageA(X)=αtP¯XσP+H0for allXset membership, variantSHn(D), where α set membership, variant F*, P set membership, variant GLn(D), H0set membership, variantSHn(D), and σ is an automorphism of D.