Niemeier lattice

In mathematics, a Niemeier lattice is one of the 24
positive definite even unimodular lattices of rank 24,
which were classified by. gave a simplified proof of the classification. mentions that he found more than 10 such lattices, but gives no further details. One example of a Niemeier lattice is the Leech lattice found in 1967.

Classification

Niemeier lattices are usually labelled by the Dynkin diagram of their
root lattice. Each Niemeier lattice can be constructed from its root lattice by adjoining elements known as glue vectors, as detailed in §16.1 of. The Dynkin diagrams associated with a Niemeier lattice have rank either 0 or 24, and all of their components have the same Coxeter number. There are exactly 24 Dynkin diagrams with these properties, and there turns out to be a unique Niemeier
lattice for each of these Dynkin diagrams.
The complete list of Niemeier lattices is given in the following table.
In the table,

Row	Lattice root system	Dynkin diagram	Coxeter number	G₀	G₁	G₂	G_∞
1	Leech lattice		0	1	2Co₁	1	Z²⁴
2	A₁²⁴		2	2²⁴	1	M₂₄	2¹²
3	A₂¹²		3	3!¹²	2	M₁₂	3⁶
4	A₃⁸		4	4!⁸	2	1344	4⁴
5	A₄⁶		5	5!⁶	2	120	5³
6	A₅⁴D₄		6	6!⁴	2	24	72
7	D₄⁶		6	⁶	3	720	4³
8	A₆⁴		7	7!⁴	2	12	7²
9	A₇²D₅²		8	8!²²	2	4	32
10	A₈³		9	9!³	2	6	27
11	A₉²D₆		10	10!²	2	2	20
12	D₆⁴		10	⁴	1	24	16
13	E₆⁴		12	⁴	2	24	9
14	A₁₁D₇E₆		12	12!	2	1	12
15	A₁₂²		13	13!²	2	2	13
16	D₈³		14	³	1	6	8
17	A₁₅D₉		16	16!	2	1	8
18	A₁₇E₇		18	18!	2	1	6
19	D₁₀E₇²		18	²	1	2	4
20	D₁₂²		22	²	1	2	4
21	A₂₄		25	25!	2	1	5
22	D₁₆E₈		30		1	1	2
23	E₈³		30	³	1	6	1
24	D₂₄		46	2²³24!	1	1	2

The neighborhood graph of the Niemeier lattices

If L is an odd unimodular lattice of dimension 8n and M its sublattice of even vectors, then M is contained in exactly 3 unimodular lattices, one of which is L and the other two of which are even. The Kneser neighborhood graph in 8n dimensions has a point for each even lattice, and a line joining two points for each odd 8n dimensional lattice with no norm 1 vectors, where the vertices of each line are the two even lattices associated to the odd lattice. There may be several lines between the same pair of vertices, and there may be lines from a vertex to itself. Kneser proved that this graph is always connected. In 8 dimensions it has one point and no lines, in 16 dimensions it has two points joined by one line, and in 24 dimensions it is the following graph:
Each point represents one of the 24 Niemeier lattices, and the lines joining them represent the 24 dimensional odd unimodular lattices with no norm 1 vectors. The number on the left is the Coxeter number of the Niemeier lattice. The red index number in the node indicates the row of the associated table above.
In 32 dimensions the neighborhood graph has more than a billion vertices.

Properties

Some of the Niemeier lattices are related to sporadic simple groups.
The Leech lattice is acted on by a double cover of the Conway group,
and the lattices A₁²⁴ and A₂¹²
are acted on by the Mathieu groups M₂₄ and M₁₂.
The Niemeier lattices, other than the Leech lattice, correspond to
the deep holes of the Leech lattice. This implies that the affine Dynkin diagrams of the Niemeier lattices can be seen inside the Leech lattice, when
two points of the Leech lattice are joined by no lines when they have distance
, by 1 line if they have distance,
and by a double line if they have distance.
Niemeier lattices also correspond to the 24 orbits of primitive norm zero vectors w of the even unimodular Lorentzian lattice II_25,1, where the Niemeier lattice corresponding to w is w^⊥/w.