Further Study Of Geometry Of Hermitian Matrices Over Division Rings

The study of the geometry of matrices was initiated by Hua L.-K.in the middle forties of the last century.In recent decades,it has a great development.In2009,Huang L.-P.proved the fundamental theorem of geometry of2×2Hermitian matrices over any division ring.In2011,Huang L.-P.proved the fundamental theorem of geometry of n×n Hermitian ma.trices over a division ring,i.e.,to characterize adjacency preserving maps from an n×n Hermitian matrices space over a division ring with an involution to itself.Based on these work,the purpose of this paper is to prove the fundamental theorem of geometry of Hermitian matrices over any two division rings.This paper has four chapters.In Chapter1,we give a brief introduction of the background of research and main result. In Chapter2,we discuss the properties of maximal sets of rank1and rank2,and give some lemmas about them.In Chapter3, we prove the fundamental theorem of geometry of2×2Hermitian matrices over any two division rings. Let Hn(D.-)be the set of al],n×n Hermitian matrices over a division ring D with an involution.In Chapter4,we prove the following fundamental theorem of geometry of Hermitian matrices over two division rings:Let D’and D be division rings with involutions*and-such that|F’|≥3,where F’={x∈D’:x=x*},F={x∈D:x=x}.Let Z’ be the center of D’ and Z the center of D.Assume that m,n are integers≥2.Let φ:Hm(D’,*)�'Hn(D,-)be an adjacency preserving bijective map.Then m=n.If n≥3,or n=2with D’≠(a,b/Z’) where Z’=F’,then φ is of the form φ(X)=t(hP)XτP+φ(0),(?) X∈Hn(D’,*), where O≠h∈F,P∈GLn(D),τ is an isomorphism from D’ to D which satisfies xτ=h(x*)τh-1,x∈D’.If n=2with D’=(a,b/Z’) where Z’=F’,then φ is of the form either as above or φ(X)=t(h,P)XρP+φ(O),(?)∈H2(D’,*),where h∈F*,P∈GL2(D):ρ is an isomorphism from D’ to D which satisfies xρ=(x*)ρ,x∈D’.