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Journal of the London Mathematical Society 2000 62(1):71-84; doi:10.1112/S0024610700008954
© 2000 by London Mathematical Society
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© The London Mathematical Society

A Class of Infinite Dimensional Simple Lie Algebras

Kaiming Zhao

Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences Beijing 100080, China, zhao{at}iss06.iss.ac.cn

Received 11 February 1999. Revision received 9 July 1999.

Let A be an abelian group, F be a field of characteristic 0, and {alpha}, ß be linearly independent additive maps from A to F, and let {delta}isinker({alpha})\{0}. Then there is a Lie algebra L = L(A, {alpha}, ß, {delta}) = {oplus}xisinA Fex under the product

[ex, ey]]={alpha}(x–y)ex+y+({alpha}{wedge}ß) (x, y) ex+y–{delta}.

If, further, ß({delta}) = 1, and ß(A) = Z, there is a subalgebra L+:=L(A+, {alpha}, ß, {delta}) = {oplus}xisinA+ Fex, where A+ = {xisinA|ß(x)≥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.


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