Some infinite-dimensional simple Lie algebras in characteristic 0 related to those of Block

AbstractGiven a nontrivial torsion-free abelian group (A, +, 0), a field F of characteristic 0, and a nondegenerate bi-additive skew-symmetric map ϑ:A × A → F, we study the Lie algebra L(A, ϑ) over F with basis {ex: x ϵ A⧹{0}} and multiplication [ex, ey] = ϑ(x, y)ex + y. We show that L(A, ϑ) is simple, determine its derivations, and show that the locally finite derivations D have the form D(ex) = μ(x)ex, μ ϵ Hom(A, F). We describe all isomorphisms between two such algebras. Finally, we compute H2 (L, F)