Unimodular lattice

Unimodular lattice

In mathematics, a unimodular lattice is a lattice of discriminant 1 or −1.The "E"8 lattice and the Leech lattice are two famous examples.

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 "(ai, aj)", where the elements "ai" 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 "E"8 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 ("a"1,...,"a"m+n) and ("b"1,...,"b"m+n) given by

:"a"1"b"1+...+"a"m"b"m − "a"m+1"b"m+1 − ... − "a"m+n"b"m+n. In "R"m,n there is one odd unimodular lattice up to isomorphism, denoted by

:"Im,n",

which is given by all vectors ("a"1,...,"a"m+n)in "R"m,n with all the "ai" 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

:"IIm,n".

This is given by all vectors ("a"1,...,"a"m+n)in "R"m,n such that either all the "ai" are integers or they are all integersplus 1/2, and their sum is even. The lattice "II8,0" is the same as the "E8" lattice.

Positive definite unimodular lattices have been classified up to dimension 25. There is a unique example "In,0" in each dimension "n" less than 8, and two examples ("I8,0" and "II8,0") in dimension 8. The number of lattices increases moderately up to dimension 25 (where there are 665 of them), but beyond dimension 25 the Smith-Minkowski-Siegel mass formula implies that the number increases very rapidly with the dimension; for example, there are more than 80,000,000,000,000,000 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 norm2 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 bythe 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 dimension16 ("E82" and "II16,0"), and 24 in dimension 24, called the Niemeier lattices (examples:the Leech lattice, "II24,0", "II16,0+II8,0", "II8,03"). 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 dimension24 (the Leech lattice and the odd Leech lattice), and 0, 1, 3, 38 in dimensions25, 26, 27, 28. Beyond this the number increases very rapidly; there are at least 8000in dimension 29. In sufficiently high dimensions most unimodular lattices have no roots.

The only non-zero example of even positive definite unimodular lattices with noroots 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.

Beyond 32 dimensions, the numbers increase even more 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 simply connected oriented topological 4-manifoldis a unimodular lattice. Michael Freedman showedthat this lattice almost determines the manifold: there is a unique such 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 conjecturefor 4 dimensional topological manifolds.
Donaldson's theorem states 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

*citation|last=Bacher|first= Roland|last2= Venkov|first2= Boris |chapter-url=http://www-fourier.ujf-grenoble.fr/PREP/html/a332/a332.html
chapter=Réseaux entiers unimodulaires sans racine en dimension 27 et 28 (Unimodular integral lattices without roots in dimensions 27 and 28)
title=Réseaux euclidiens, designs sphériques et formes modulaires|pages= 212-267|series= Monogr. Enseign. Math.|volume= 37| Enseignement Math.|publication-place=Geneva|year= 2001|id= MathSciNet|id=1878751

* Conway and Sloane, "Sphere packings, lattices, and groups", ISBN 0-387-98585-9
* Milnor and Husemoller, "Symmetric bilinear forms" ISBN 0-387-06009-X
* J-P. Serre, "A course in Arithmetic", ISBN 0-387-90040-3
* Sloane's [http://www.research.att.com/%7Enjas/lattices/unimodular.html catalogue] of unimodular lattices.
* [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A005134 Number of unimodular lattices of given dimension]
* [http://arxiv.org/abs/math.NT/0012231 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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