When root system is defined on some vector space [tex]V[/tex], the definition uses numbers(adsbygoogle = window.adsbygoogle || []).push({});

[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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Understanding root systems

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**