Primitive, -primitive and test elements of free Lie algebra and free color Lie superalgebra

ABSTRACTPhD. THESISPRIMITIVE, ∆-PRIMITIVE AND TEST ELEMENTS OF FREE LIEALGEBRA AND FREE COLOR LIE SUPERALGEBRADilek KAHYALARDEPARTMENT OF MATHEMATICSINSTITUTE OF NATURAL AND APPLIED SCIENCESUNIVERSITY OF ÇUKUROVASupervisor Prof. Dr. Naime EKİCİYear: 2006, Pages: 39Jury : Prof.Dr. Naime EKİCİProf.Dr. Yusuf ÜNLÜAsst. Prof. Dr. Gökhan ÇUVALCIOĞLUAsst. Prof. Dr. Perihan DİNÇ ARTUTAsst. Prof. Dr Zerrin ESMERLİGİLLet L be a free color Lie superalgebra generated by a finite setX = {x1 ,..., xn } . In this study it has been given necessary and sufficient conditions forany subalgebra R of L to be equal to L . From this result it has been shown that theelements specially choosen elements u1 , u2 , u3 are the test elements for L . Inaddition, it has been obtained almost primitive and test elements of L . Besides, theseries of generic elements of L has been found. Finally, by the use of doubleJacobian matrix, it has been given a necessary and sufficient condition for theelement u to be ∆ -primitive in the free Lie algebra. It has been proved that theautomorphism group of the algebra Fn acts transitively on the set of all ∆ -primitiveelements.Key Words : Free color Lie superalgebra, generic elements, almost primitiveelements, test elements.II