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A CLASS OF INFINITE DIMENSIONAL SIMPLE LIE ALGEBRAS

Published online by Cambridge University Press:  30 October 2000

KAIMING ZHAO
Affiliation:
Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China; [email protected]
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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 ]xAFex under the product

formula here

If, further, β(δ) = 1, and β(A) = Z, there is a subalgebra L+:=L(A+, α, β, δ) = [oplus ]xA+Fex, where A+ = {xA|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.

Type
Research Article
Copyright
The London Mathematical Society 2000

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