Weight lattice - Knowlpedia

Definition

Let $\Phi\subset E$ be a crystallographic root system, with coroots $\alpha^\vee=2\alpha/(\alpha,\alpha)$ . Its weight lattice is

P=P(\Phi):= \{\lambda\in E:\langle\lambda,\alpha^\vee\rangle\in\mathbb Z \text{ for every }\alpha\in\Phi\}.

Equivalently, if $\alpha_1,\ldots,\alpha_r$ are the simple roots and $\omega_1,\ldots,\omega_r$ are the fundamental weights defined by

\langle\omega_i,\alpha_j^\vee\rangle=\delta_{ij},

then

P=\mathbb Z\omega_1\oplus\cdots\oplus\mathbb Z\omega_r.

Dominant weights and representations

A weight $\lambda\in P$ is dominant when

\langle\lambda,\alpha_i^\vee\rangle\geq0 \quad\text{for every simple root }\alpha_i.

The highest-weight theorem identifies dominant elements of $P$ with finite-dimensional irreducible representations of the complex semisimple Lie algebra, and with irreducible representations of its simply connected group.

The root lattice satisfies $Q\subseteq P$ , and $P/Q$ is finite. This quotient governs the possible central isogeny forms; choosing which intermediate lattice occurs is additional group-level data not specified by the Lie algebra alone.

Which weights belong to a Lie group?

The abstract weight lattice $P$ is the largest lattice allowed by the root datum. For a particular connected semisimple group $G$ with maximal torus $T$ , the actual character lattice satisfies

Q\subseteq X^*(T)\subseteq P.

An integral Lie-algebra highest weight exponentiates to a representation of $G$ precisely when it lies in $X^*(T)$ . Hence all of $P$ occurs for the simply connected form, while only $Q$ occurs for the adjoint form.

For example, type $A_1$ has $P=\mathbb Z\omega$ and $Q=2\mathbb Z\omega$ . Every nonnegative multiple of $\omega$ defines an $SU(2)$ -representation, whereas only the even multiples descend to $SO(3)$ .

Terminology caution

“A weight of a representation” is an element that actually occurs as a simultaneous eigenvalue. The weight lattice is the ambient lattice of all integral candidates; most of its elements need not occur in a fixed representation.

References

James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, §§13 and 21. Publisher record.
Brian C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015, Chapters 8 and 12. Publisher record.