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On the Radical of a Lie Algebra

1950; American Mathematical Society; Volume: 1; Issue: 1 Linguagem: Inglês

10.2307/2032424

1088-6826

Harish-chandra Harish-Chandra,

Advanced Topics in Algebra

harish-chandraLet 8 be a Lie algebra over a field K of characteristic zero.For any X€E8 we denote, as usual, the linear mapping Y-*[X, Y] of 8 into itself by ad X.Let T be the radical of 8. Consider the set 91 consisting of all iV£r such that ad N is nilpotent.It was shown in a recent paper1 that 9t is the unique maximal nilpotent ideal2 of 8. Further if D is a derivation of T then DTCN.For any X, Y, Z& put B(X, Y) =s£(ad X ad Y) and T(X, Y, Z) = s/>(ad [X, Y] adZ).Then B(X, Y) is a symmetric bilinear form on 8 while T(X, Y, Z) is a skewsymmetric trilinear form.It is easily verified that they are both invariant under all derivations of 8, that is, B(DX, Y) + B(X, DY) = 0, T(DX, Y, Z) -f T(X, DY, Z) 4-T(X, Y, DZ) = 0 for any derivation D and X, Y, Z£8.An ideal SDc in 8 is called characteristic if -DSD?C 3JJ for every derivation D of 8. Our first theorem may now be stated as follows: Theorem 1.An element X of 8 belongs to the radical T if and only if T(X, Y, Z) =0for all Y, Z£8.3As an immediate corollary we get the following: