Root lattice - Knowlpedia

Definition

Let $\Phi\subset E$ be a crystallographic root system. Its root lattice is the free abelian subgroup

Q=Q(\Phi):=\mathbb Z\Phi =\left\{\sum_{\alpha\in\Phi}n_\alpha\alpha:n_\alpha\in\mathbb Z\right\} \subset E.

If $\Delta=\{\alpha_1,\ldots,\alpha_r\}$ is any system of simple roots, then

Q=\mathbb Z\alpha_1\oplus\cdots\oplus\mathbb Z\alpha_r.

Relation to weights

Every root pairs integrally with every coroot, so the root lattice is contained in the weight lattice:

Q\subseteq P.

The quotient $P/Q$ is finite. For a connected simply connected compact semisimple group, it is naturally dual to the center; equivalently, its character data measure which highest weights are trivial on central subgroups.

If $V_\lambda$ is an irreducible highest-weight representation, every weight $\mu$ of $V_\lambda$ satisfies

\lambda-\mu\in Q.

More precisely, after choosing positive roots, $\lambda-\mu$ is a nonnegative integer combination of simple roots. Thus all weights of an irreducible module occupy one coset of $Q$ inside $P$ .

Examples

For type $A_1$ , with root $\alpha$ and fundamental weight $\omega=\alpha/2$ ,

Q=\mathbb Z\alpha=2\mathbb Z\omega, \qquad P=\mathbb Z\omega, \qquad P/Q\cong\mathbb Z/2\mathbb Z.

For type $A_{n-1}$ , realize the roots as $e_i-e_j$ in the hyperplane $\sum_i x_i=0\subset\mathbb R^n$ . Then

Q=\{(a_1,\ldots,a_n)\in\mathbb Z^n:\textstyle\sum_i a_i=0\}.

Lie algebra and group data

The root lattice depends only on the root system, hence on the complex semisimple Lie algebra. It does not distinguish the different global Lie groups with that Lie algebra. Those global forms are detected by an intermediate character lattice between $Q$ and $P$ : the adjoint form uses $Q$ , while the simply connected form uses $P$ .

References

James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, §§9–13. Publisher record.
Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, Springer, 2002, Chapter VI, §1. Publisher record.