| The generalized Witt algebra is an important kind of infinite-dimensional Liealgebras. Recently Osborn, Dokovic, Kaiming Zhao, Xiaoping Xu and Yucai Su havecome up with some results of simple Cartan type W Lie algebras in characteristic zero.Passmanmann has come up with some valuable results of the simplicity of generalizedWitt algebras.On what have already been achieved, in the present paper, we construct a class ofgeneralized Witt algebras and study its some simple subalgebras W_d which isanalogues of the classical infinite dimensional simple Lie algebras of Cartan type W.In the former half of the paper, we describe the centre of W and the simplicitytheorem. Roughly speaking, it says that if charF=0, then (?) is simple if and onlyif A≠0 andφis nondegenerate. If A is a nonzero torsion-free abelian group offinite rank, Der (FA) is a simple Lie algebra and a generalized Witt algebra. Thanksto two results of R.Fransteiner on derivations of graded Lie algebras, we give explicitdescription of all derivations of W. In the later half, we introduce the subalgebraW_d of W and determine the necessary and sufficient conditions for W_d to be simple.We also show that a derivation of W_d is the sum of a locally inner derivation and aderivation of degree 0. |
