Derivations, isomorphisms, and second cohomology of generalized Witt algebras

Generalized Witt algebras, over a field $F$ of characteristic $0$, were defined by Kawamoto about 12 years ago. Using different notations from Kawamoto's, we give an essentially equivalent definition of generalized Witt algebras $W=W(A,T,\varphi)$ over $F$, where the ingredients are an abelian group $A$, a vector space $T$ over $F$, and a map $\varphi:T\times A\to K$ which is linear in the first variable and additive in the second one. In this paper, the derivations of any generalized Witt algebra $W=$
$W(A,T,\varphi)$, with the right kernel of $\varphi$ being $0$, are explicitly described; the isomorphisms between any two simple generalized Witt algebras are completely determined; and the second cohomology group $H^2(W,F)$ for any simple generalized Witt algebra is computed. The derivations, the automorphisms and the second cohomology groups of some special generalized Witt algebras have been studied by several other authors as indicated in the references.