What is the relationship between the root lattice and weight lattice?

Question

Let $Q$ and $P$ denote the $\mathbb{Z}$-span of the simple roots and fundamental weights respectively. What is the relationship between $Q$ and $P$? Does $P$ contain $Q$?

Answer 1

The roots are "special kinds" of weights: they are the weights that correspond to the adjoint representation. The fundamental representations are the "seed" of all other representations: all other representations can be found as direct summands of tensor powers of the fundamental representations.

It follows that the root lattice is a subset of the weight lattice. This statement is equivalent to the statement that the weights of representations appearing in tensor powers of the adjoint are a subset of the weights of representations appearing in tensor powers of the fundamental ones, i.e. all representations.

But $Q$ being a subset of $P$ is not the only interesting relationship between them. For example, the weight lattice is dual to the coroot lattice, which is closely related to the root lattice. For self-dual lattices such as the $E_8$ root lattice, the weight lattice and root lattice may coincide. In the $E_8$ case, this is equivalent to the fact that the fundamental $248$-dimensional representation is the adjoint representation.