Let <mi mathvariant="fraktur">g be a complex linear Lie algebra. Assume that the center <m

ntaraxq 2022-07-02 Answered
Let g be a complex linear Lie algebra. Assume that the center z of g is trivial
Let r be the radical of g. If r is abelian, then g is semisimple?
What if g is the Lie algebra of an algebraic complex linear group?
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Answers (2)

Mekjulleymg
Answered 2022-07-03 Author has 14 answers
Your question whether g is semisimple is equivalent to whether necessarily r = 0. The answer is no. A counterexample is given by
g = { ( a b d c a e 0 0 0 ) : a , b , c , d , e C }
where the radical
r = { ( 0 0 d 0 0 e 0 0 0 ) : d , e C }
is two-dimensional.
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Cierra Castillo
Answered 2022-07-04 Author has 6 answers

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