A Further Generalization of Algebras of Cartan Type W

In this paper, a class of infinite-dimensional Lie algebras, called algebras of Cartan type *, over a field F of characteristic 0 is studied. These algebras are derivations of the field F(x1, x2,…,xn). We prove that algebras of Cartan type * are simple. We also prove that any derivation on an algebra of this type with one indeterminate is an inner derivation.