Positive definite lattice

A rational lattice $𝐿$ is said to be positive definite iff $⟨ 𝛼, 𝛼 ⟩ > 0$ for any nonzero $𝛼 \in 𝐿$ . #m/def/geo Equivalently, the quadratic space $𝐿 ℝ$ is $ℝ 𝑛, 0$ (“Euclidean space”).¹

Properties

There exist finitely many lattice points of a given norm, i.e. $| 𝐿 𝑚 | < \infty$ for any $𝑚 \in ℚ$ .
Assume $𝐿$ is integral and $𝛼, 𝛽 \in 𝐿 2$ . Then $⟨ 𝛼, 𝛽 ⟩ \in {0, \pm 1, \pm 2}$ and

⟨ 𝛼, 𝛽 ⟩ = - 2 ⟺ 𝛼 + 𝛽 = 0 ⟨ 𝛼, 𝛽 ⟩ = - 1 ⟺ 𝛼 + 𝛽 \in 𝐿 2

Proof of 1–2

Since

𝐿 𝑚 = B 𝑚 (⃗ 𝟎) \cap 𝐿

where $B 𝑚 (⃗ 𝟎)$ is compact and $𝐿$ is discrete, it follows $𝐿 𝑚$ is finite, proving ^P2.

^P2 follows from the Cauchy-Schwarz inequality.

Associated Lie algebra of a positive definite even lattice

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

1988. Vertex operator algebras and the Monster, §6.1, pp. 122–124 ↩

Positive definite lattice

Properties

Footnotes