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I am confused by the relations between roots, coroots and weights in Lie algebras. Let $G$ be a Lie group, $T$ its maximal split torus and $\mathfrak{g}$ its Lie algebra. Here is what I have in mind (and I am always convinced by examples in $GL(n)$):

  • a root $\alpha$ is an "eigenvalue" of the adjoint action of the torus: $ad(t) x = \alpha(t) x$ for all $t \in \mathfrak{t}:=Lie(T)$ and $x \in \mathfrak{g}$. Thus $\alpha \in \mathfrak{t}^\star$.
  • a coroot $\alpha^\vee$ associated to $\alpha$ is defined, for all root $\beta$, by $\alpha^\vee(\beta) = 2(\alpha, \beta)/(\alpha, \alpha)$. It is therefore an element of $\mathfrak{t}^{\star \star} = \mathfrak{t}$.
  • a weight is a Lie algebra "character", i.e. a morphism $\mathfrak{g} \to \mathbb{R}$, i.e. an element of $\mathfrak{g}^\star$.

The relation between weights and roots remains unclear. If one is given a set of simple roots $\Delta$, it can get by the coroot construction the set of corresponding simple soroots $\Delta^\vee$, basis of $\mathfrak{t}^{\star\star}=\mathfrak{t}$.

How can we construct from these sets/basis of weights or coweights? I saw some statements that $\{\varpi_\alpha \ : \ \alpha \in \Delta\}$ is a set of simple weights (and similarly with coweights), but I don't know what is $\varpi_{\alpha}$.

I think I am confusing both. Why do we consider (co)roots and (co)weights?

Lyer Lier
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  • A weight is also an element of $\mathfrak t^{\ast}$. It is an element whose pairing with each coroot is an integer. – D_S Mar 22 '21 at 05:50
  • Roots are the VIPs among the weights. – Torsten Schoeneberg Mar 22 '21 at 15:18
  • @D_S so weights are not really the "characters" as I defined above? Are the weights defined after having a notion of coroots and using it? – Lyer Lier Mar 23 '21 at 02:29
  • Yes I think so. Similarly "coweights" are the elements of $\mathfrak t^{\ast \ast} = \mathfrak t$ whose pairing with each root is an integer. – D_S Mar 23 '21 at 04:47

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