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Understanding root systems

  1. Jul 9, 2008 #1
    When root system is defined on some vector space [tex]V[/tex], the definition uses numbers

    [tex]
    \langle x,y\rangle = 2\frac{(x,y)}{\|y\|^2},\quad x,y\in V
    [/tex]

    where [tex](\cdot,\cdot)[/tex] is the inner product. This number is handy, because a reflection of vector [tex]y[/tex] with respect to a hyper plane orthogonal to [tex]x[/tex] is given by

    [tex]
    y\mapsto y - \langle y,x\rangle x.
    [/tex]

    This all makes sense, but when the root system is defined in context of Lie algebras, I'm getting a bit lost with notations. When [tex]\mathfrak{g}[/tex] is some semisimple Lie group, and [tex]\mathfrak{h}[/tex] is a Cartan subalgebra, the roots are linear forms [tex]\gamma\in\mathfrak{h}^*[/tex]. Since the Killing form is nondegenerate, for each root [tex]\gamma[/tex] there exists a vector [tex]h_{\lambda}\in\mathfrak{h}[/tex] so that [tex]\gamma(x)=(h_{\gamma},x)[/tex], where [tex](\cdot,\cdot)[/tex] is the Killing form. This is then used to define a bilinear form onto [tex]\mathfrak{h}^*[/tex] by setting [tex](\gamma_1,\gamma_2)=(h_{\gamma_1},h_{\gamma_2})[/tex]. However, this does not define an inner product, so the numbers [tex]\langle \gamma_1, \gamma_2\rangle [/tex] don't seem to have an interpretation with reflections now.

    Have I now understood something incorrectly, or is it the case that root systems are always defined with respect to some bilinear form, which is not necessarily an inner product?
     
  2. jcsd
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