---
created: 2024-06-23T23:09
updated: 2025-01-13T15:11
---
References:
- Kondo, 
	- ![](K3%20Surfaces%20-%20Shigeyuki%20Kondo.pdf#page=23)
- 

- [ ] The category of lattices
	- [ ] Bilinear forms
		- [ ] Gram matrix and its associated matrix
		- [ ] Symmetric
		- [ ] Skew-symmetric
		- [ ] Symplectic
		- [ ] Nondegenerate
		- [ ] Perfect pairing
	- [ ] Quadratic forms
		- [ ] Polarization identity
	- [ ] Associated discriminant forms
	- [ ] Lattice
	- [ ] Examples of lattices
		- [ ] $\gens{n}$
		- [ ] $U$
		- [ ] $A_n$, in particular $A_2$
		- [ ] $D_n$
		- [ ] $E_n$
		- [ ] $\rm{I}_{p, q}$ and $\rm{II}_{p, q}$
	- [ ] Twisting lattices
		- [ ] Examples: $A_2$, $U(2)$
	- [ ] Morphisms of lattices
		- [ ] Embeddings
		- [ ] Sublattices
		- [ ] Isometries
	- [ ] Orthogonal group
	- [ ] Orthogonality of vectors
	- [ ] Orthogonal direct sums
	- [ ] Primitive embeddings
		- [ ] The diagonal embedding $L(2) \injects L\oplus L$.
- [ ] Special types of lattices
	- [ ] Integral
	- [ ] Even and odd
	- [ ] Unimodular
	- [ ] $p$-elementary
	- [ ] $n$-reflective and reflective
- [ ] Auxiliary structures
	- [ ] Orthogonal complement
	- [ ] Primitive sublattices
	- [ ] Quotients
	- [ ] Overlattice
	- [ ] Dual lattice
	- [ ] Discriminant group
	- [ ] Discriminant form
- [ ] Invariants
	- [ ] Rank
	- [ ] Definite
	- [ ] Hyperbolic
	- [ ] Signature
	- [ ] Index
	- [ ] Volume
	- [ ] Self-intersections
	- [ ] Root sublattices
	- [ ] Divisibility of vectors
	- [ ] Theta function
- [ ] Isotropic sublattices
- [ ] Genus
- [ ] 2-elementary lattices
	- [ ] $(r,a,\delta)_{n_+}$ notation
- [ ] Results
	- [ ] ![](2024-07-05.png)
	- [ ] ![](2024-07-05-1.png)
	- [ ] ![](2024-07-05-2.png)
	- [ ] ![](2024-07-05-3.png)
	- [ ] ![](2024-07-05-4.png)
	- [ ] ![](2024-07-05-5.png)
	- [ ] ![](2024-07-05-6.png)
	- [ ] ![](2024-07-05-7.png)
	- [ ] ![](2024-07-05-8.png) ![](2024-07-05-9.png)
	- [ ] ![](2024-07-05-10.png)
	- [ ] ![](2024-07-05-11.png)