Dual of a rational lattice

Let $𝐿$ be a Rational lattice with basis ${𝛼 𝑖} 𝑛 𝑖 = 1$ The dual of $𝐿$ is the set #m/def/geo

𝐿 \circ = {𝛼 \in 𝐿 ℚ : ⟨ 𝛼, 𝐿 ⟩ \subseteq ℤ}

which is a rational lattice iff $𝐿$ is nondegenerate, in which case the dual basis is defined by

⟨ 𝛼 \circ 𝑖, 𝛼 𝑗 ⟩ = 𝛿 𝑖 𝑗

$𝐿$ is called self-dual iff $𝐿 \circ = 𝐿$ .

Properties

Let $𝐿$ be a nondegenerate lattice with Gram matrix $𝐺$ . The Gram matrix of $𝐿 \circ$ is $𝐺 - 1$ .

#state/develop | #lang/en | #SemBr