# Understanding root systems

1. Jul 9, 2008

### jostpuur

When root system is defined on some vector space $$V$$, the definition uses numbers

$$\langle x,y\rangle = 2\frac{(x,y)}{\|y\|^2},\quad x,y\in V$$

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

$$y\mapsto y - \langle y,x\rangle x.$$

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 $$\mathfrak{g}$$ is some semisimple Lie group, and $$\mathfrak{h}$$ is a Cartan subalgebra, the roots are linear forms $$\gamma\in\mathfrak{h}^*$$. Since the Killing form is nondegenerate, for each root $$\gamma$$ there exists a vector $$h_{\lambda}\in\mathfrak{h}$$ so that $$\gamma(x)=(h_{\gamma},x)$$, where $$(\cdot,\cdot)$$ is the Killing form. This is then used to define a bilinear form onto $$\mathfrak{h}^*$$ by setting $$(\gamma_1,\gamma_2)=(h_{\gamma_1},h_{\gamma_2})$$. However, this does not define an inner product, so the numbers $$\langle \gamma_1, \gamma_2\rangle$$ 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?