Understanding root systems

  • Thread starter jostpuur
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Main Question or Discussion Point

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?
 

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