The Quasi-Automorphism And Automorphism Of Heisenberg Lie Algebras

Automorphism is an important part of the structure theory of Lie algebras,and many research results have been obtained.Heisenberg Lie algebras are an important class of nilpotent Lie algebras,but the research progress of their automorphisms is slow for a long time.In 2007,Zhang Haishan et al.obtained the necessary and sufficient conditions for the automorphism of Heisenberg Lie algebras,which makes this research an important step.On this basis,in 2018,Zhang Yan et al.further discussed the structure of the automorphism group of Heisenberg Lie Algebras in the form of matrix,and obtained the decomposition structure of the automorphism group of Heisenberg Lie Algebras in the case of 5 dimensions.In this paper,we study structure of the automorphisms of(2n+1)-dimensional Heisenberg Lie Algebras,We skillfully use the elementary transformation of block matrix to characterize the decomposition structure of the automorphism group.In recent years,some scholars put forward the concept of quasi-automorphism,which is actually a generalization of automorphism.In 2019,Liao Yang et al.studied the quasi-automorphisms of general linear Lie algebras over gl(n,F)algebraically closed fields,proved that the quasi-automorphisms of general linear Lie algebras are consistent with their bidirectional commutativity,and finally obtained the accurate characterization of gl(n,F)quasi-automorphisms.As an important class of nilpotent Lie algebras,the research on quasi-automorphisms of Heisenberg Lie algebras has been slow.In this paper,we give all quasi-automorphisms of 3-dimensional Heisenberg Lie algebras.