A CLASS OF INFINITE DIMENSIONAL SIMPLE LIE ALGEBRAS

Abstract

Let A be an abelian group, F be a field of characteristic 0, and α, β be linearly independent additive maps from A to F, and let δ∈ker(α)\{0}. Then there is a Lie algebra L = L(A, α, β, δ) = [oplus ]_x∈AFe_x under the product

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If, further, β(δ) = 1, and β(A) = Z, there is a subalgebra L⁺:=L(A⁺, α, β, δ) = [oplus ]_x∈A⁺Fe_x, where A⁺ = {x∈A|β(x)[ges ]0}. The necessary and sufficient conditions are given for L' = [L, L] and L⁺ to be simple, and all semi-simple elements in L' and L⁺ are determined. It is shown that L' and L⁺ cannot be isomorphic to any other known Lie algebras and L' is not isomorphic to any L⁺ , and all isomorphisms between two L' and all isomorphisms between two L⁺ are explicitly described.