Rational lattice

A rational lattice $𝐿$ of rank $𝑛$ is the $ℤ$ -span of a basis of an $𝑛$ -dimensional quadratic space $𝐿 ℚ$ over $ℚ$ . #m/def/geo Equivalently, a rational lattice $𝐿$ is a rank $𝑛$ $ℤ$ -module with a symmetric $ℤ$ -bilinear map

⟨ \cdot, \cdot ⟩ : 𝐿 \times 𝐿 \to ℚ

where for any field $𝐾$ with $c h a r 𝐾 = 0$ we identify $𝐿 𝐾 = 𝐾 \otimes ℤ 𝐿$ ,¹ which is made a quadratic space under the extension of $⟨ \cdot, \cdot ⟩$ . The following notation is also useful for subsets of a given quadrance

𝐿 𝑚 = {𝛼 \in 𝐿 : ⟨ 𝛼, 𝛼 ⟩ = 𝑚}

Further terminology

Given a basis ${𝛼 𝑖} 𝑛 𝑖 = 1$ of $𝐿$ , the Gram matrix is given by $𝐺 𝑖 𝑗 = ⟨ 𝛼 𝑖, 𝛼 𝑗 ⟩$ .
$𝐿$ is nondegenerate iff $𝐿 ℚ$ is nondegenerate iff $d e t 𝐺 \neq 0$ .
$𝐿$ is integral iff $⟨ 𝛼, 𝛽 ⟩ \in ℤ$ for all $𝛼 \in 𝐿$ iff $𝐺$ is integral.
$𝐿$ is even iff $⟨ 𝛼, 𝛼 ⟩ \in 2 ℤ$ for all $𝛼 \in 𝐿$ , which implies integral by polarization.
$𝐿$ is positive definite iff $⟨ 𝛼, 𝛼 ⟩ > 0$ for all nonzero $𝛼 \in 𝐿$ .
$𝐿$ is unimodular iff $| d e t 𝐺 | = 1$ .
Dual of a rational lattice
Self-dual rational lattice
Theta function of a positive definite lattice

Properties

A discrete Z-submodule of a quadratic space is a rational lattice

Rational lattice

Further terminology

Properties

See also

Footnotes