I'm reading a paper$^\dagger$ and having a bit of trouble understanding some of their notation. The notation is introduced on the bottom of the first page.
Let $N \cong \mathbf{Z}^n$ be a lattice, and let $P \subseteq N_{\mathbf{Q}} := N \otimes _{\mathbf{Z}} {\mathbf{Q}}$ be an $n$-dimensional lattice polytope
Does anybody know what this notation means, specifically what it means to have the integers or rationals as a subscript?
$^\dagger$ Alexander M. Kasprzyk, Benjamin Nill, Reflexive polytopes of higher index and the number $12$.
Are you familiar with tensor products in general? Because the subscript for lattices is defined right there: $N_{\mathbb Q}$ is the $\mathbb Q$-vector space obtained by tensoring $N$ with the rationals, a process often referred to as extending scalars. Basically, think of $\mathbb Z^n$ as sitting in $\mathbb Q^n$, and now consider a set of generators for $N$; under this inclusion you can think of them as vectors and consider their $\mathbb Q$-span. In particular, if $N$ has full rank then $N_{\mathbb Q}$ is the whole $\mathbb Q^n$.
The symbol $\otimes_{\mathbb Z}$ just means that we are taking the tensor product over $\mathbb Z$, which is a necessary part of the data of a tensor product; roughly it means that we are "building the new $\mathbb Q$-structure over the skeleton of the existing $\mathbb Z$-structure on $N$, as opposed to any more specialized or fundamental structure (of which there are not really examples in this case, but if you consider e.g. two complex vector spaces and take their tensor product over $\mathbb Q$, that would be less specialized than tensoring over $\mathbb C$ but more specialized than tensoring over $\mathbb Z$).