Positive-definite even unimodular lattice

Let $n\ge 1$ and let $V$ be an $n$-dimensional real vector space with a positive-definite inner product $⟨\cdot ,\cdot ⟩$. An orthonormal basis of $V$ gives a linear inner-product-preserving isomorphism $U:V\to {\mathbb{R}}^n$. A full lattice in $V$ is a subgroup $\mathrm{\Lambda }\subseteq V$ such that $U(\mathrm{\Lambda })$ is a full Euclidean lattice: equivalently, $U(\mathrm{\Lambda })=\mathbb{Z}{b}_1\oplus \cdots \oplus \mathbb{Z}{b}_n$ for a real basis $b_1,\dots ,b_n$ of ${\mathbb{R}}^n$. This condition is independent of the orthonormal basis, because a change between two such bases is an invertible orthogonal linear map and therefore carries real bases and their integer spans to real bases and their integer spans.

The dual lattice is ${\mathrm{\Lambda }}^{\ast }:=\{y\in V:⟨y,x⟩\in \mathbb{Z}\text{ for every }x\in \mathrm{\Lambda }\}$. The lattice is integral if $⟨x,y⟩\in \mathbb{Z}$ for every $x,y\in \mathrm{\Lambda }$, even if $⟨x,x⟩\in 2\mathbb{Z}$ for every $x\in \mathrm{\Lambda }$, and unimodular if $\mathrm{\Lambda }={\mathrm{\Lambda }}^{\ast }$. A positive-definite even unimodular lattice of rank $n$ is a full lattice in an $n$-dimensional positive-definite real inner-product space that is integral, even, and unimodular.

Full Euclidean lattice and covolume

Real and complex inner product spaces, with the inner product linear in the first argument

Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases

Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans

Evenness implies integrality: for $x,y\in \mathrm{\Lambda }$, the polarization identity gives $2⟨x,y⟩=⟨x+y,x+y⟩-⟨x,x⟩-⟨y,y⟩\in 2\mathbb{Z}$, hence $⟨x,y⟩\in \mathbb{Z}$. The definition retains both conditions so each is visible in later hypotheses.