单李代数的抛物子代数上保括积的非线性可逆映射

Non-linear Invertible Maps on Parabolic Subalgebras Preserving Lie Products

设g为复数域C上的单李代数,Pπ是g的标准抛物子代数.证明了李代数Pπ上的映射φ是保括积的非线性双射当且仅当φ可以表示为李代数Pπ上内自同构、图自同构、对角自同构、复数域上自同构诱导的映射的乘积.由此推导出李代数Pπ上的自同构可表示为李代数Pπ上内自同构、图自同构、对角自同构的乘积.

Let g be a complex simple Lie algebra,Pπ is standard parabolic subalgebra of g.Prove that a non-linear bijective map φ on Pπ preserves Lie products if and only if it is a product of an inner automorphism,a graph automorphism,a diagonal automorphism and a bijective map extended by an automorphism of the complex field.As its corrolary,any automorphism of Pπ is a product of an inner automorphism,a graph automorphism,a diagonal automorphism.