Preserving Problems On General Linear Lie Algebras And Their Parabolic Subalgebras

Inspired by the idea, advocated by P.Semrlin in2007, to consider preserving problems about abstract algebra systems, we aim to enrich and develop preserving problems on general linear Lie algebras and their parabolic subalgebras in this thesis.The paper is organized as follows, we mainly explain relevant studying back-grounds, development and notations at the beginning.We plan to determine the invertible linear maps on the general linear Lie algebra preserving solvability in Chapter one. We construct certain invertible linear maps preserving solvability in both directions on gl(n, C), such as inner automorphisms, graph automorphisms, scalar multiplication maps, invertible linear maps induced by a linear function on gl(n, C) and so on. Meanwhile, we use these special maps to characterize arbitrary invertible linear maps preserving solvability in both directions.In Chapter Two we obtain the non-linear invertible maps preserving Lie product on parabolic subalgebras of the general linear Lie algebras. Similar to Chapter Two, we construct certain invertible maps preserving Lie product on parabolic, which are used to describe arbitrary non-linear invertible linear maps preserving Lie product on parabolic.In chapter three we focus on characterizing the non-linear invertible maps on the general linear Lie algebras preserving standard parabolic subalgebras. At first we give some basic properties of the non-linear invertible maps preserving standard parabolic subalgebras, then construct certain invertible maps preserving standard parabolic subalgebras, such as invertible linear maps induced by graph automor-phisms, maps preserving lattices, invertible linear maps induced by permissible sets and so on. Finally, we describe arbitrary non-linear invertible linear maps on the general linear Lie algebras preserving standard parabolic subalgebras.