Rational lattice

Dual of a rational lattice

Let 𝐿 be a Rational lattice with basis {𝛼𝑖}𝑛𝑖=1 The dual of 𝐿 is the set geo

𝐿∘={𝛼∈𝐿ℚ:⟨𝛼,𝐿⟩⊆ℤ}

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

⟨𝛼∘𝑖,𝛼𝑗⟩=𝛿𝑖𝑗

𝐿 is called self-dual iff 𝐿∘ =𝐿.

Properties

1. Let 𝐿 be a nondegenerate lattice with Gram matrix 𝐺. The Gram matrix of 𝐿∘ is 𝐺−1.

---

develop | en | SemBr