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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.

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Also posted at math.stackexchange.com/questions/420442/… –  Chris Godsil Jun 14 '13 at 22:12
Can you make your question more mathematically precise? –  Abhinav Kumar Jun 18 '13 at 19:23

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