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created: 2023-03-26T11:58
updated: 2024-07-02T20:20
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# Results on lattices

- Let $L$ be a unimodular lattice, $M \subset L$ a nondegenerate primitive sublattice, and $\alpha: M \rightarrow M, \beta: M^{\perp} \rightarrow M^{\perp}$ isometries. Then the isometry $(\alpha, \beta)$ of $M \perp M^{\perp}$ extends to $L$ if and only if the automorphisms $\tilde{\alpha}$ on $M^{\vee} / M$ induced by $\alpha$ and $\tilde{\beta}$ on $\left(M^{\perp}\right)^{\vee} / M^{\perp}$ induced by $\beta$ satisfy $j_M \circ \tilde{\alpha}=\tilde{\beta} \circ j_M$.[Lemma 1.1. (Barth, W., Peters, C.,  1982. Automorphisms of Enriques Surfaces.)](zotero://select/library/items/Q9AY2RNU)

- Let $M \subset L$ and $\alpha: M \rightarrow M$ be as above. If $\alpha$ extends to an isometry of $L$ restricting to $\pm \mathrm{id}$ on $M^{\perp}$, then this extension is unique. Such an extension exists if and only if $\alpha^{\vee}: M^{\vee} \rightarrow M^{\vee}$ induces $\pm \mathrm{id}$ on $M^{\vee} / M$. [Cor 1.2. (Barth, W., Peters, C.,  1982. Automorphisms of Enriques Surfaces.)](zotero://select/library/items/Q9AY2RNU)

- Niemeier lattices: 
  ![](attachments/Pasted%20image%2020230828192323.png)

![](2024-06-23.png)

![](2024-06-23-1.png)

![](2024-06-24-4.png)

![](2024-06-24-5.png)


![](2024-06-27.png)

![](2024-06-29.png)

![](2024-06-29-1.png)
![](2024-06-29-2.png)

![](2024-07-02.png)