Structures Of Split Lie Type Algebras

There are three parts in this thesis.Firstly, we study ?-Hom-Jordan Lie algebras. To begin with, we define a split regular?-Hom-Jordan Lie algebra and its connections of roots. Furthermore, by the techniques of connections of roots, we get a sufficient and necessary condition of a simple split regular ?-Hom-Jordan Lie algebra being of maximal length with a symmetric root system. We also get a sufficient condition of a split regular ?-Hom-Jordan Lie algebra, which can be decomposed into the direct sum of its simple ideals.Secondly, we study Hom-Leibniz algebras and Hom-Lie color algebras. To begin with,we define a split regular Hom-Leibniz algebra and the connections of roots for this algebra.By developing techniques of connections of roots, we obtain a sufficient condition of a split regular Hom-Leibniz algebra with a symmetric root system, which can be decomposed into the direct sum of its ideals. Then we define a (?) J-connection. By using of the(?)J-connection,we obtain a sufficient and necessary condition of a simple split regular Hom-Leibniz algebra being of maximal length with a symmetric root system. Moreover, we define a split regular Hom-Lie color algebra and the connections of roots for this algebra. By the techniques of connections of roots, we obtain a sufficient and necessary condition of a simple split regular Hom-Lie color algebra being of maximal length with a symmetric root system. We also get a sufficient condition of a split regular Hom-Lie color algebra, which can be decomposed into the direct sum of its simple ideals.Thirdly, we study Leibniz triple systems. To begin with, using the universal Leibniz envelope of a Leibniz triple system, we define a split Leibniz triple system and the connec-tions of roots for this triple system. By the techniques of connections of roots, we obtain a sufficient condition of a split Leibniz triple system with a symmetric root system, which can be decomposed into the direct sum of its ideals. Then we define a(?)J-connection. By using of the (?) J-connection, we obtain a sufficient and necessary condition of a simple split Leibniz triple system being of maximal length with a symmetric root system. Moreover, as the natural extension of a split Leibniz triple system, we define a graded Leibniz triple sys-tem and its graded connections. By means of constructing techniques of graded connections,we get the support of a simple graded Leibniz triple system, which has all of its elements connected. We also get a sufficient condition of a graded Leibniz triple system with a trivial annihilator, which can be decomposed into the direct sum of its ideals.