How is defined a tensor product of $\mathbb{Z}-$lattices $U$ and $V:\ $ $U\otimes V$ ? For instance $E_8\otimes_{\mathbb{Q}} E_8$, where $Е_8$ is a root lattice of exceptional Lie algebra $е_8$.
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1$\begingroup$ Your question is off-topic here, as this website is for research level math questions. I suggest trying at math.stackexchange.com . $\endgroup$– AngeloCommented Jun 3, 2013 at 17:15
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1$\begingroup$ The tensor product should be over ${\bf Z}$, not ${\bf Q}$. $\endgroup$– Noam D. ElkiesCommented Jun 3, 2013 at 20:22
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1 Answer
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You can consider the lattices embedded in Euclidean spaces $\mathbb{R^m}$ and $\mathbb{R^n}$ say. Then $U \otimes V$ is in $\mathbb{R}^m \otimes \mathbb{R}^n$ and the bilinear pairing (i.e. dot product) on this vector space is given by saying $\langle x \otimes y, x' \otimes y' \rangle = \langle x,y \rangle \cdot \langle y, y' \rangle$ for split tensors, and extending linearly to get it on the tensor product. If you don't want to consider embeddings into Euclidean spaces, just note that the lattice structure is equivalent to a quadratic or a bilinear form on the free abelian group. Then this formula defines the inner product for the tensor product. See, for instance, Section 1.10 of Martinet's book Perfect lattices in Euclidean spaces.
