Lattice subgroup

A lattice $𝐿$ of a locally compact Hausdorff abelian topological group $𝐺$ is a discrete subgroup $𝐿 \leq 𝐺$ such that the quotient $𝐺 / 𝐿$ is compact. #m/def/group The above definition generalizes and is motivated by the case where $𝐺 = ℚ 𝑛$ , where we define a Rational lattice.

Classical lattice

A classical lattice $𝐿$ is a lattice in the topological vector space $𝕂 𝑛$ where $𝕂 = ℝ$ or $𝕂 = ℚ$ , and is called complete iff $s p a n 𝕂 𝐿 = 𝕂 𝑛$ .

Let $𝑉$ be an $𝑛$ -dimensional space vector space over $𝕂$ . Let $𝐿 \leq 𝑉$ be a $ℤ$ -submodule spanning $𝑉$ . The following are equivalent¹ #m/thm/topology

$𝐿$ is a complete lattice subgroup of $𝑉$ ;
$𝐿$ is generated by $𝑛$ elements;
$𝐿 ≅ ℤ 𝑛$ in $ℤ 𝖬 𝗈 𝖽$ .

Proof

Suppose $𝐿 \leq ℤ 𝑉$ is discrete.

Then $𝐿$ is closed. For if $𝑈$ is an isolating neighbourhood of 0, then

𝑈' = ⋂ 𝑥 \in 𝑈 (𝑥 - (-)) - 1 𝑈

is an open neighbourhood of $0$ such that $𝑈' \subseteq 𝑈$ and the difference of any elements of $𝑈'$ lies in $𝑈$ . If there were an $𝑥 \notin 𝐿$ such that $𝑥 \in C l 𝐿$ , then there would be a two distinct elements $𝑙 1, 𝑙 2 \in 𝑥 + 𝑈'$ such that $0 \neq 𝑙 1 - 𝑙 2 \in 𝑈' - 𝑈' \subseteq 𝑈$ , so $0$ is not isolated in $𝑈$ , a contradiction.

Now let $B = {𝑢 𝑖} 𝑛 𝑖 = 1 \subseteq 𝐿$ be a $𝕂$ -basis of $𝑉$ , and let $𝐿 0 = s p a n ℤ B \leq ℤ 𝐿$ . We will show that the Lagrange index $| 𝐿 / 𝐿 0 |$ is finite. Let $[𝑙 𝑖] \in 𝐿 / 𝐿 0$ for $𝑖 \in 𝐼$ be a complete system of representatives for each coset. Letting

Φ 0 = s p a n [0, 1) B

(this is an abuse of notation but the meaning is clear) we have

𝑙 𝑖 = 𝜇 𝑖 + 𝑙 0 𝑖, 𝜇 𝑖 \in Φ 0, 𝑙 0 𝑖 \in 𝐿 0 \leq ℤ 𝑉

where

𝜇 𝑖 = 𝑙 𝑖 - 𝑙 0 𝑖 \in 𝐿

lie discretely in the bounded set $Φ 0$ . Since $𝐿 \cap C l (Φ 0)$ is compact and discrete, and thus finite, it follows $𝐿 \cap Φ 0$ is finite, so the $𝜇 𝑖$ are finite and thus $𝑞 = | 𝐿 / 𝐿 0 |$ is finite.

It follows $𝑞 𝐿 \leq ℤ 𝐿 0$ , whence

𝐿 \leq ℤ 1 𝑞 𝐿 0 = s p a n ℤ (1 𝑞 B)

implying $𝐿$ possesses a $ℤ$ -basis of length less than $𝑛$ .

Lattice subgroup

Classical lattice

See also

Footnotes