Weights in the dual Cartan

Definition (where weights live)

Let $\mathfrak g$ be a complex semisimple Lie algebra and $\mathfrak h\subset\mathfrak g$ a Cartan subalgebra . A weight of a representation is, by definition, an element of the dual space $\mathfrak h^\ast$, i.e. a linear functional on $\mathfrak h$. The root system $\Delta$ of $\mathfrak g$ is also a subset of $\mathfrak h^\ast$ (see root systems ), so weights and roots live in the same ambient vector space and can be compared geometrically.

Integral and dominant weights (standard semisimple setup)

Fix a set of simple roots and corresponding coroots $H_\alpha\in\mathfrak h$ (defined so that $\beta(H_\alpha)$ are the Cartan integers). The integral weight lattice is

A weight is dominant if $\lambda(H_\alpha)\ge 0$ for all simple roots. Highest-weight classification says finite-dimensional irreducibles are parametrized by dominant integral weights (compare highest-weight classification ).

Link with compact groups and maximal tori

If $G$ is a compact connected Lie group with maximal torus $T$ (see the maximal torus theorem ), then characters $T\to U(1)$ differentiate to linear functionals on $\mathfrak t=\mathrm{Lie}(T)$, producing an integral lattice in $\mathfrak t^\ast$. After complexification, this lattice matches the integral weights in $\mathfrak h^\ast$ for the complexified Lie algebra. The Weyl group acts on these lattices and preserves integrality.