Even lattice

A rational lattice $𝐿$ is said to be even iff $⟨ 𝛼, 𝛼 ⟩ \in 2 ℤ$ for all $𝛼 \in 𝐿$ . #m/def/geo It immediately follows from polarization that $𝐿$ is integral.

Properties

Associated Lie algebra of a positive definite even lattice

Associated elementary 2-group

An even lattice $𝐿$ of rank $𝑛$ has an associated elementary abelian 2-group $ˇ 𝐿 = 𝐿 / 2 𝐿$ of dimension $𝑛$ , where we write $ˇ 𝛼 = 𝛼 + 2 𝐿$ . We have a canonical alternating $ℤ$ -bilinear map

𝑐 0 : 𝐿 \times 𝐿 \to ℤ 2 (𝛼, 𝛽) \mapsto ⟨ 𝛼, 𝛽 ⟩ + 2 ℤ

which induces the alternating $ℤ 2$ -bilinear form

𝑐 1 : ˇ 𝐿 \times ˇ 𝐿 \to ℤ 2 (ˇ 𝛼, ˇ 𝛽) \mapsto ⟨ 𝛼, 𝛽 ⟩ + ℤ 2

Similarly we have a canonical map

𝑞 0 : 𝐿 \to ℤ 2 𝛼 \mapsto 12 ⟨ 𝛼, 𝛼 ⟩ + 2 ℤ

which induces the quadratic form

𝑞 1 : ˇ 𝐿 \to ℤ 2 ˇ 𝛼 \mapsto 12 ⟨ 𝛼, 𝛼 ⟩ + 2 ℤ

so that $𝑐 1$ is the polar form of $𝑞 1$ . Now $𝑞 1$ or equivalently $𝑐 1$ is nondegenerate iff the Gram matrix has odd determinant, in particular if $𝐿$ is a unimodular lattice.

#state/tidy | #lang/en | #SemBr