Generalized Cartan type \(W\) Lie algebras in characteristic zero (Q1369268)

In another paper [Trans. Am. Math. Soc. 350, 643-664 (1998)] the authors studied generalized Witt algebras over a field \(F\) of characteristic 0. These generalizations \(W(A,T,\varphi)\) of Witt algebras are associated with an abelian group \(A\), a vector space \(T\), and a pairing \(\varphi:T\times A\to F\) that is \(F\)-linear in the first variable and additive in the second. In this paper the authors define a subalgebra \(W_d\) of \(W(A,T,\varphi)\) associated with a certain type of injective map \(d:I\to T\) for an index set \(I\). They determine necessary and sufficient conditions for \(W_d\) to be simple, and when \(W_d\) is simple, they study its derivations, automorphisms, and a gradation, and also prove that \(H^2(W_d,F)=0\).