jostpuur
Jul9-08, 08:42 AM
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?
\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?