For any simple complex Lie algebra $\frak{g}$, with a given choice of Cartan subalgebra $\frak{h}$, we have an associated root system $R \subseteq \frak{h}^*$. The properties of $R$ can be formalized as an **abstract root system** allowing for a classification of simple complex Lie algebras.

For any irreducible representation $V$ of $\frak{g}$ we have an associated finite set of weight vectors, which again live in $\frak{h}^*$. These do not form a root system since they are not closed under multiplication by $-1$. (I also guess they are not closed under the action of the Weyl group of the relections in weights, but I am not sure.)

Is there some abstract system, generalizing root systems that abstracts the properties of weights? If so, how does it reduce to a root sysmtem for the special case of the adjoint representation?

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