Lie Subalgebras in a Certain Operator Lie Algebra with Involution  

作　　者：Shan Li SUN Xue Feng MA 

机构地区：[1]Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education,School of Mathematics and Systems Science, Be~jing University of Aeronautics and Astronautics, Beijing 10019], P. R. China

出　　处：《Acta Mathematica Sinica,English Series》2011年第8期1521-1534,共14页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China （Grant No. 10571011）

摘　　要：We show in a certain Lie-algebra, the connections between the Lie subalgebra G＋ ：= G-t-G＊ 4- [G, G＊], generated by a Lie subalgebra G, and the properties of G. This allows us to investigate some useful information about the structure of such two Lie subalgehras. Some results on the relations between the two Lie subalgebras are obtained. As an application, we get the following conclusion： Let ,4 C /3（2d） be a space of self-adjoint operators and L： ：= A ~ i^4 the corresponding complex Lie＊-algebra. G＋ = G 4- G＊ 4- [G, G＊] and G are two LM-decomposable Lie subalgebras of L： with the decomposition 6＋ = 7^（6＋） 4- S, G -- T~~ 4- 86, and T^6 C T^（6＋）. Then 6＋ is ideally finite iff T~ ：= 7~6 4- ~ 4- [T^6,7~] is a quasisolvable Lie subalgebra, S^- ：= 86 4- S~ 4- [$6, $~] is an ideally finite semisimple Lie subalgebra, and [7~6,86] = [R.~, 86] = {0}.We show in a certain Lie-algebra, the connections between the Lie subalgebra G＋ ：= G-t-G＊ 4- [G, G＊], generated by a Lie subalgebra G, and the properties of G. This allows us to investigate some useful information about the structure of such two Lie subalgehras. Some results on the relations between the two Lie subalgebras are obtained. As an application, we get the following conclusion： Let ,4 C /3（2d） be a space of self-adjoint operators and L： ：= A ~ i^4 the corresponding complex Lie＊-algebra. G＋ = G 4- G＊ 4- [G, G＊] and G are two LM-decomposable Lie subalgebras of L： with the decomposition 6＋ = 7^（6＋） 4- S, G -- T~~ 4- 86, and T^6 C T^（6＋）. Then 6＋ is ideally finite iff T~ ：= 7~6 4- ~ 4- [T^6,7~] is a quasisolvable Lie subalgebra, S^- ：= 86 4- S~ 4- [$6, $~] is an ideally finite semisimple Lie subalgebra, and [7~6,86] = [R.~, 86] = {0}.

关 键 词：Skew operator generalized scalar operator E-solvable Lie algebra ideally finite 

分 类 号：O152[理学—数学] O152.5[理学—基础数学]