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

Given a nontrivial torsion-free abelian group (A, +, 0), a field F of characteristic 0, and a nondegenerate bi-additive skew-symmetric map theta:A × A → F, we study the Lie algebra image over F with basis {ex: x ε A-45 degree rule{0}} and multiplication [ex, ey] = theta(x, y)ex + y. We show that image 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 image.