Automorphisms and test elements of Lie algebras

ABSTRACTMSc THESISAUTOMORFISMS AND TEST ELEMENTS OF LIE ALGEBRASCennet ESKALDEPARTMENT OF MATHEMATICSINSTITUTE OF NATURAL AND APPLIED SCIENCESUNIVERSITY OF ÇUKUROVASupervisor: Assist.Prof.Dr. Ahmet TEMİZYÜREKYear: 2006, Pages:58Jury: Prof.Dr. Naime EKİCİAssist.Prof.Dr. Ahmet TEMİZYÜREKAssist.Prof.Dr. Perihan ARTUT (DİNÇ)Let k be a field and L be the free Lie algebra generated by the setX = {x1 , K , x n } over k . The subset {u1 , K , u k } (k ≤ n ) of L is a test set if anyendomorphism ϕ ∈ EndL , ϕ (u i ) = u i is necessarily an automorphism. Firstly,when L is a free nilpotent Lie algebra and ϕ ∈ EndL . We have investigated thatnecessary and sufficient conditions ϕ to be an automorphism of L . We have shownthat if M n is a free metabelian Lie algebra of rank n then each IA endomorphism ofM n which acts as the identity on some test set is an inner automorphism of M n .Moreover we calculate, for n ≥ 3 , the test rank of M n equals n − 1 .Key Words: Test test, free Lie algebra, endomorphismII