# relations between root lattice and weight lattice

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

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$Q$ being a subset of $P$ is not the only interesting relationship between them. For example, the weight lattice is dual to the coroot lattice - where the latter 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 one, too.