On The Lie Triple Derivation Algebras Of Some Classes Of Lie Algebras

Lie derivations are important research objects in the theory of the structure ofLie algebras. Lie triple derivations, as the nature generalization of Lie derivations,are attracting mathematicians'interests. In this thesis, we study the decomposi-tions of Lie triple derivations and the structures of Lie triple derivation algebras fora class of finite dimensional Lie algebras and several classes of infinite dimensionalLie algebras. We give some examples in both aspects to describe the relationsbetween Lie derivation algebras and Lie triple derivation algebras. Meanwhile, weprovide some clues to investigate the relations for other Lie algebras.In Chapter 2, Based on the unique decomposition of Lie derivations given byDengyin Wang over the Lie algebra consisting of strictly upper triangular matri-ces, we study the decompositions of Lie triple derivations, which, in some extent,generalize their results. Meanwhile, we solve it in another way differing to theirs.We also provide an available method to solve the similar problems. Besides, it isshown that its Lie triple derivation algebra is a solvable Lie algebra. As a corol-lary, we obtain that Lie derivation algebra is also solvable. Meanwhile, for n≥3,we compute the codimensions of Lie derivation algebras in Lie triple derivationalgebras. Then we reveal on a higher level that Lie derivation algebra is a propersubalgebra of Lie triple derivation algebra for the Lie algebra consisting of strictlyupper triangular matrices.In Chapter 3, we study Lie triple derivation algebras of Virasoro-like algebrasand q-analog Virasoro-like algebras. We prove that their Lie triple derivations arethe inner derivations. However, the triple derivations are the sum of inner deriva-tions and some classes of outer derivations for their derived algebras. Thereforewe obtain the result that Lie triple derivations are Lie derivations. Meanwhile, arealization of Virasoro-like algebras is determined.In Chapter 4, we generalize a theorem of Farnsteiner on the derivation algebrasof finitely generated graded Lie algebras. We prove that, for finitely generated G-graded Lie algebras, Lie triple derivation algebras are also G-graded. And then itis proved that their Lie triple derivations are the sum of inner derivations and someouter derivations. Thus, we show that Lie triple derivation algebras are equal toLie derivation algebras for Schro¨dinger-Virasoro algebras and W-algebra W(2, 2).In Chapter 5, Aiat et al. proved that each Jordan derivation is an ordinaryderivation on triangular algebras in 2005. We generalize their results and show that the generalized Jordan derivations on triangular algebras are generalizedderivations by comparing the elements in both sides.