Elementary And ф-free Lie Triple Systems

This thesis mainly discusses two questions. In the first question, we concern Heisen-berg n-Lie algebras, prove that two Heisenberg n-Lie algebras may not be isomorphic if they have the same dimension. We also determine the structures of automorphism groups of (3m+1)-dimensional Heisenberg 3-Lie algebras and(mn+1)-dimensional Heisenberg n-Lie algebras, and give the concrete expression of the automorphisms. In the second question, we give the concepts of elementary andΦ-free Lie triple systems, and investigate the structures of them and the relation between them.Section 1 recalls some definitions and notations used in section 2. Section 2 deter-mines the structures of automorphism groups of (3m+1)-dimensional Heisenberg 3-Lie algebras and (mn+1)-dimensional Heisenberg n-Lie algebras and gives the concrete expression of the automorphisms. Section 3 gives the notions and some basic facts of Lie triple systems. Section 4 discusses the structures of elementary Lie triple systems. Section 5 describes some properties ofΦ-free Lie triple systems. Section 6 describes the relations between elementary andΦ-free Lie triple systems.