Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $\Lambda$ be an unimodular even lattice of signature $(m,n)$. By a classifying theorem by Milnor, $\Lambda$ must be of the form $U^k\oplus E_8(\pm 1)^l$, where $U$ is the hyperbolic plane.

Now consider $\Lambda_{\mathbb{R}}:=\Lambda\otimes \mathbb{R}$. I know that $\Lambda_\mathbb{R}$ is isometric to $\mathbb{R}^{m+n}$ with the quadratic diagonal form of signature $(m,n)$.

My question is: suppose we want to visualise $\Lambda$ as a subset of $\mathbb{R}^{m+n}$ through the given isometry, is it legitimate to see it as $\mathbb{Z}^{m+n}$?

Of course my question concerns only the point of view of sets and "how the points are placed", because of course the quadratic form restricted to $\Lambda$ must be even.

share|improve this question
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.