On The Structure Of Lie Triple Supersystems

In this thesis, I give some properties of Lie triple supersystems and discuss complete Lie triple supersystems. Moreover, I obtain some conditions for solvable Lie triple supersystems by studying c-supplemented Lie triple supersystems.The following are the main results of this thesis:Theorem 1 Let T be a Lie triple supersystem. For m, n∈N, the following statements hold:(1) [T,Cn(T),Tn]=[T,Tn,Cn(T)]=[Cn(T),Tn,T]={0}.(2) (Tn+Cn(T))m(?)(Tn)m+Cn(T)m.Theorem 2 Let T be a Lie triple supersystem, then the following statements hold:(1) T is complete if and only if LS(T)=LD(T) and C(LS(T))={0}.(2) If T is complete, then T is simple complete if and only if it is indecom-posable.(3) If LS(T) is a complete Lie superalgebra, then T is a complete Lie triple supersystem.Theorem 3 Let T be a Lie triple supersystem. Then the following are equiva-lent:(1) T is c-supplemented.(2) T/φ(T) is completely factorisable and every subsystem ofφ(T) is an ideal of T.Theorem 4 Let T be a Lie triple supersystem, then the following statements hold:(1) If T is c-supplemented, then T is an E-algebra.(2) If all maximal nilpotent subsystems of T are c-ideals of T, then T is solvable.(3) If T T is solvable, then all maximal subsystems of T are c-supplemented in T.