Notes on \(W\)-type Lie algebras (Q1881609)

For \(F\) a field of characteristic zero the authors denote by \(W ( {n\atop m})\) the Lie algebra with elements \(\{ f\partial_i \mid f \in F [x_1^{\pm 1},\dots,x_n^{\pm 1},x_{n+1},\dots,x_m]\}\) with \([f\partial_i, g\partial_j] = f\partial_i (g)\partial_j - g\partial_j (f)\partial_i\). They define a Lie algebra \(W[1] = W_0 \oplus W_c\) where \(W_0 = W ( {0\atop m} )\) is called the 0-homogeneous component of \(W ( { n \atop m} )\). For \(G\) an abelian group and \(W(G)\) a \(G\)-graded Lie algebra with 0-homogeneous component \(W(G)_0 = W[1]\) they establish sufficient conditions for \(W(G)\) to be simple.