Would like references to a counterexample in nilpotent Lie Algebras (Wikipedia error?)

Discussion in 'Math Research' started by Jeremy Henty, Jun 18, 2008.

  1. Jeremy Henty

    Jeremy Henty Guest

    I hope this isn't too trivial for sci.math.research !

    The Wikipedia pages for Lie Algebra[1] and Cartan's criterion[2] both
    claim that a Lie Algebra is nilpotent if and only if its Killing form
    vanishes. This appears to be false: it is easy to exhibit a
    non-nilpotent Lie Algebra with a zero Killing form. (Example at the
    end of this post.)

    I have been searching for references to show the Wikipedia editors but
    found nothing. Presentations of Cartan's criterion usually refer to
    solvable and semisimple cases only. Some mention in passing that a
    nilpotent Lie Algebra has zero Killing form, but I've seen no
    discussion of the converse. So, does anyone have any references? Is
    this one of those obvious things that everyone knows and noone ever
    writes down?

    [1] http://en.wikipedia.org/wiki/Lie_algebra
    [2] http://en.wikipedia.org/wiki/Cartan's_criterion

    Regards,

    Jeremy Henty


    Counterexample.

    Let L be a 3-dimensional complex vector space with basis a, b, c .
    There's a unique alternating bilinear form [,] on L such that [ab] = b
    , [ac] = i.c , [bc] = 0 . The only non-trivial case of the Jacobi
    identity is [a[bc]] + [b[ac]] + [c[ab]] = 0 , which follows since each
    of the three terms vanishes individually. Hence L is a Lie algebra.

    Let * be the Killing form. Now, [L,L] = <b,c> and <b,c> is abelian,
    so for any x, y, [b[xy]] = [c[xy]] = 0 , so (ad b)(ad x) = (ad c)(ad
    x) = 0 , so b*x = c*x = 0 . Also, a*a = tr((ad a)^2) =
    tr(diag(0,1,i)^2) = tr(diag(0,1,-1)) = 0 , so * vanishes.

    Finally, L is not nilpotent. Eg. it has trival centre (since ker(ad
    a) = <a> , ker(ad b) = ker(ad c) = <b,c> , so the kernel of the
    adjoint representation is trivial). Hence the lower central series is
    identically zero. Alternatively, observe that the [L,L] = <b,c> and
    [L,<b,c>] = <b,c> , so the upper central series stabilises at <b,c> .
     
    Jeremy Henty, Jun 18, 2008
    #1
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  2. You are (or at least can be) a Wikipedia editor. As you are
    right you can correct the articles yourself. I would suggest
    putting the counterexample (as you've written it here) in the
    Lie algebras talk page.

    Victor Meldrew
    "I don't believe it!"
     
    victor_meldrew_666, Jun 19, 2008
    #2
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