I am studying representations of complex semisimple Lie algebras, their root system, weight spaces etc.
I am a very beginner so this question is about relating key definitions to one another.
Question 1: How are the concepts of weights and roots related? Do you have any intuitive way to interpret these? I see that roots "live" in the Lie algebra, while weights "live" in its representation space, but I fail to connect this in a bigger picture.
My understanding:
Generally, roots live in an Euclidean vector space, they are like generalized eigenvalues, they span the vector space. The root system generates hyperplanes in the space and if we make reflections along these hyperplanes, these reflections generate the Weyl group of this root system.
If the Euclidean vector space is a semisimple Lie algebra $\mathfrak{g}$ and $\mathfrak{h}$ its cartan subalgebra, the roots of $\mathfrak{g}$ are nonzero elements $\alpha$ such that we can take a nonzero vector $X$ from $\mathfrak{g}$ and any vector $H$ from $\mathfrak{h}$ makes $[X, H] = \alpha(H) X$ true.
Question 2: What is the intuition behind this equality? Can I visualize $X$ and $H$ as some vectors and $\alpha$ also as some vector? Then how can we put $H$ "into" $\alpha$, what does $\alpha(H)$ mean?
As for weights, they live in the representation space of the $\mathfrak{g}$. Weights are linear functionals such that corresponding weight spaces are nonzero. Again, I am not really sure, how to interpret the weight space definition $V_\lambda = \{v \in V \mid \forall H \in \mathfrak{h}: H * v = \lambda(H) * v\}$ and the expression $H * v = \lambda(H) * v$.
Thank you very much for any intuition to this.
Best Answer
Actually, the roots live in the dual space of $\mathfrak h$. In other words, roots are linear maps from $\mathfrak h$ into the field $\Bbbk$ over which you are working. They are the elements $\alpha\in\mathfrak h^*\setminus\{0\}$ such that, for some $X\in\mathfrak g$, $(\forall H\in\mathfrak h):[H,X]=\alpha(H)X$. So, basically the roots are the non-zero eigenvalues of the maps$$\begin{array}{ccc}\mathfrak g&\longrightarrow&\mathfrak g\\X&\mapsto&[H,X]\end{array}$$($H\in\mathfrak g$). And the weights have a similar definition, but this time we are working with an arbitrary representation of $\mathfrak g$ (and we don't have the restriction that they cannot be equal to $0$).