Unimodular lattice

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In mathematics, a unimodular lattice is a lattice of discriminant 1 or −1. The E₈ lattice and the Leech lattice are two famous examples.

Contents

1 Definitions
2 Examples
3 Classification
4 Properties
5 Applications
6 References

Definitions

A lattice is a free abelian group of finite rank with an integral symmetric bilinear form (·,·).
A lattice is even if (a, a) is always even.
The dimension of a lattice is the same as its rank (as a Z-module).
A lattice is positive definite if (a, a) is always positive for non-zero a.
The discriminant of a lattice is the determinant of the matrix with entries (a_i, a_j), where the elements a_i form a basis for the lattice.
A lattice is unimodular if its discriminant is 1 or −1.
Lattices are often embedded in a real vector space with a symmetric bilinear form. The lattice is positive definite, Lorentzian, and so on if its vector space is.
The signature of a lattice is the signature of the form on the vector space.

Examples

The three most important examples of unimodular lattices are:

The lattice Z, in one dimension.
The E₈ lattice, an even 8 dimensional lattice,
The Leech lattice, the 24 dimensional even unimodular lattice with no roots.

Classification

For indefinite lattices, the classification is easy to describe. Write R^m,n for the m+n dimensional vector space R^m+n with the inner product of (a₁,...,a_m+n) and (b₁,...,b_m+n) given by

a₁b₁+...+a_mb_m − a_m+1b_m+1 − ... − a_m+nb_m+n.

In R^m,n there is one odd unimodular lattice up to isomorphism, denoted by

I_m,n,

which is given by all vectors (a₁,...,a_m+n) in R^m,n with all the a_i integers.

There are no even unimodular lattices unless

m − n is divisible by 8,

in which case there is a unique example up to isomorphism, denoted by

II_m,n.

This is given by all vectors (a₁,...,a_m+n) in R^m,n such that either all the a_i are integers or they are all integers plus 1/2, and their sum is even. The lattice II_8,0 is the same as the E₈ lattice.

Positive definite unimodular lattices have been classified up to dimension 25. There is a unique example I_n,0 in each dimension n less than 8, and two examples (I_8,0 and II_8,0) in dimension 8. The number of lattices increases moderately up to dimension 25 (where there are 665 of them), but beyond dimension 25 the number increases very rapidly with the dimension; for example, there are more than 80000000000000000 in dimension 32.

In some sense unimodular lattices up to dimension 9 are controlled by E₈, and up to dimension 25 they are controlled by the Leech lattice, and this accounts for their unusually good behavior in these dimensions. For example, the Dynkin diagram of the norm 2 vectors of unimodular lattices in dimension up to 25 can be naturally identified with a configuration of vectors in the Leech lattice. The wild increase in numbers beyond 25 dimensions might be attributed to the fact that these lattices are no longer controlled by the Leech lattice.

Even positive definite unimodular lattice exist only in dimensions divisible by 8. There is one in dimension 8 (the E₈ lattice), two in dimension 16 (E₈² and II_16,0), and 24 in dimension 24, called the Niemeier lattices (examples: the Leech lattice, II_24,0, II_16,0+II_8,0, II_8,0³). Beyond 24 dimensions the number increases very rapidly; in 32 dimensions there are more than a billion of them.

Unimodular lattices with no roots (vectors of norm 1 or 2) have been classified up to dimension 28. There are none of dimension less than 23 (other than the zero lattice!). There is one in dimension 23 (called the short Leech lattice), two in dimension 24 (the Leech lattice and the odd Leech lattice), and 0, 1, 3, 38 in dimensions 25, 26, 27, 28. Beyond this the number increases very rapidly; there are at least 8000 in dimension 29. In sufficiently high dimensions most unimodular lattices have no roots.

The only non-zero example of even positive definite unimodular lattices with no roots in dimension less than 32 is the Leech lattice in dimension 24. In dimension 32 there are more than ten million examples, and above dimension 32 the number increases very rapidly.

The following table gives the numbers of (or lower bounds for) even or odd unimodular lattices in various dimensions, and shows the very rapid growth starting shortly after dimension 24.

Dimension Odd lattices Odd lattices, no roots Even lattices Even lattices, no roots

0 0 0 1 1

1 1 0

2 1 0

3 1 0

4 1 0

5 1 0

6 1 0

7 1 0

8 1 0 1 (E₈) 0

9 2 0

10 2 0

11 2 0

12 3 0

13 3 0

14 4 0

15 5 0

16 6 0 2 0

17 9 0

18 13 0

19 16 0

20 28 0

21 40 0

22 68 0

23 117 1

24 273 1 24 (Niemeier) 1 (Leech)

25 665 0

26 ≥2307 1

27 ≥14179 3

28 ≥327972 38

29 ≥37938009 ≥8900

30 ≥20169641025 ≥82000000

31 ≥5000000000000 ≥800000000000

32 ≥80000000000000000 ≥10000000000000000 ≥1160000000 ≥10900000

Dimension	Odd lattices	Odd lattices, no roots	Even lattices	Even lattices, no roots
0	0	0	1	1
1	1	0
2	1	0
3	1	0
4	1	0
5	1	0
6	1	0
7	1	0
8	1	0	1 (E₈)	0
9	2	0
10	2	0
11	2	0
12	3	0
13	3	0
14	4	0
15	5	0
16	6	0	2	0
17	9	0
18	13	0
19	16	0
20	28	0
21	40	0
22	68	0
23	117	1
24	273	1	24 (Niemeier)	1 (Leech)
25	665	0
26	≥2307	1
27	≥14179	3
28	≥327972	38
29	≥37938009	≥8900
30	≥20169641025	≥82000000
31	≥5000000000000	≥800000000000
32	≥80000000000000000	≥10000000000000000	≥1160000000	≥10900000

Beyond 32 dimensions, the numbers continue to increase very rapidly.

Properties

The theta function of an even unimodular positive definite lattice of dimension n is a level 1 modular form of weight n/2. If the lattice is odd the theta function has level 4.

Applications

The second cohomology group of a compact orientable topological 4-manifold is a unimodular lattice. When the manifold is simply connected, Michael Freedman showed that this lattice almost determines the manifold: there is a unique manifold for each even unimodular lattice, and exactly two for each odd unimodular lattice. In particular if we take the lattice to be 0, this implies the Poincaré conjecture for 4 dimensional topological manifolds. Simon Donaldson proved that if the manifold is smooth and the lattice is positive definite, then it must be a sum of copies of Z, so most of these manifolds have no smooth structure.

References

Conway and Sloane, Sphere packings, lattices, and groups, ISBN 0387985859
Milnor and Husemoller, Symmetric bilinear forms ISBN 038706009X
J-P. Serre, A course in Arithmetic, ISBN 0387900403
Sloane's [catalogue] of unimodular lattices.
[Number of unimodular lattices of given dimension]
[A mass formula for unimodular lattices with no roots] Oliver King Mathematics of Computation, vol. 72, no. 242 (2003), 839-863.

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