Triple Derivations On Infinite-dimensional Lie(Super-) Algebras

In this paper,we study the triple derivations on Lie color algebras,the Block Lie algebras and the Lie triple derivations on trivial extension algebras.Our main work is divided into the following three parts.In the first part,we study the triple derivations on Lie color algebras.Let ￡ be a Lie color algebra over a commutative ring R.If 1/2∈R,L is perfect and has zero center,then we have that every triple derivation on ￡ is a derivation and every triple derivation on the derived algebras Der(L)is an inner derivation,which are some extension of the results of J.H.Zhou[57]and J.Zhou et al.[56].Moreover,we also consider the triple homomorphisms on Lie color algebras and obtain that under some assumptions,every triple homomorphism between two Lie color algebras is either a homomorphism or an anti-homomorphism or a direct sum of a homomorphism and an anti-homomorphism.In the second part,we study the triple derivations on the Block Lie algebras B(q).We prove that every triple derivation on B(q)is a derivation.Namely,TDer(B(q))=Der(B(q)).In the third part,we investigate the Lie triple derivations on trivial extension algebras H x X.Due to the fact that the Lie triple derivation on H x X may not be a derivation([49],Counterexample 3.5),we provide some sufficient conditions under which the Lie triple derivation L on H x X is proper.Namely,L can be written as a sum of a derivation and a center valued map.We apply our results to triangular algebras and show some examples(see 5.3.1,Example 1)illuminating the necessity of our assumptions.