一般线性李代数的抛物子代数上保李积的非线性可逆映射

Non-linear Invertible Maps on Parabolic Subalgebras of the General Linear Lie Algebras Preserving Lie Products

设F是特征为零的域,gl(n,F)为域F上的一般线性李代数,Tn为域F上全体n×n阶上三角矩阵李代数,称gl(n,F)中包含Tn的所有子代数为gl(n,F)的抛物子代数.决定出gl(n,F)上的任意标准抛物子代数P的形式,证明了任意抛物子代数P上的映射φ是保李积的非线性可逆映射当且仅当存在可逆矩阵T∈P,映射χ:P→F和域F的自同构f,使得φ([aij])=T[f(aij)]T-1+χ([aij])I或φ([aij])=-R(T[f(aij)]T-1)tR-1+χ([aij])I,对任意的[aij]∈P,其中R=∑ni=1(-1)iE1,n+1-i,χ满足对任意的A∈P={[x,y]x,y∈P},总有χ(A)=0.

Let F be an arbitrary field with characteristics zero,gl（n,F） the general linear Lie algebra of all n×n matrices,and let Tn be the Lie algebra of all upper n×n matrices.A subalgebra P of gl（n,F） containing Tn is called a parabolic subalgebra of gl（n,F）.Decide the form of arbitrary parabolic subalgebra P of gl（n,F）and prove that a non-linear bijective map φon P preserves Lie products if and only if there exist an invertible matrix T∈P,a function χ：P→F satisying χ（A）=0 for every matrix A∈P={｜x,y∈P},and an automorphism f of the field F,such that φ（）=T[f（aij）]T-1＋χ（）I, or φ（）=-R（T[f（aij）]T-1）tR-1＋χ（）I,for all ∈P,where R=∑ni=1（-1）iE1,n＋1-i.