Suppose you are given a linear combination of points of $\mathbb{Z}^n$ which corresponds to a nontrivial representation $V$ of some simple Lie algebra $g$. Is it possible to reconstruct $g$ and $V$ from this data, without knowing the canonical form on the weight lattice? (It looks like there might be some problems with dual Lie algebras --- can you at least find the pair $g, g^{\vee}$?)

Suppose you are given a linear combination of points of $\mathbb{Z}^n$ which corresponds to the weight datum of a nontrivial representation $V$ of some simple Lie algebra $g$. Is it possible to reconstruct $g$ and $V$ from this data, without knowing the canonical form on the weight lattice? (It looks like there might be some problems with dual Lie algebras --- can you at least find the pair $g, g^{\vee}$?)

Suppose you are given a linear combination of points of $\mathbb{Z}^n$ which corresponds to a nontrivial representation $V$ of some Lie algebra $g$. Is it possible to reconstruct $g$ and $V$ from this data, without knowing the canonical form on the weight lattice? (It looks like there might be some problems with dual Lie algebras --- can you at least find the pair $g, g^{\vee}$?)

Suppose you are given a linear combination of points of $\mathbb{Z}^n$ which corresponds to a nontrivial representation $V$ of some simple Lie algebra $g$. Is it possible to reconstruct $g$ and $V$ from this data, without knowing the canonical form on the weight lattice? (It looks like there might be some problems with dual Lie algebras --- can you at least find the pair $g, g^{\vee}$?)