In mathematics, a unimodular lattice is a lattice of discriminant 1 or −1. The E₈ lattice and the Leech lattice are two famous examples. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... See lattice for other meanings of this term, both within and without mathematics. ... In mathematics, a polynomial P(T) has a discriminant, which is a polynomial function of its coefficients, and discriminates the case of a multiple root (for which the graph of P(x) would touch the x-axis). ... In mathematics, E8 is the name of a Lie group and also its Lie algebra . ... In mathematics, the Leech lattice is a lattice Λ in R24 discovered John Leech ( 16 (1964), 657--682). ...

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.

In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ... In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space one would need to contain it; or alternatively how large a free abelian group it can contain as a subgroup. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ... In abstract algebra, a module is a generalization of a vector space. ... In mathematics, signature can refer to The signature of a permutation is ±1 according to whether it is an even/odd permutation. ...

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.

In mathematics, E8 is the name of a Lie group and also its Lie algebra . ... In mathematics, the Leech lattice is a lattice Λ in R24 discovered John Leech ( 16 (1964), 657--682). ...

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. See also Simple Lie group. ...

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. In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by Niemeier. ... In mathematics, the Leech lattice is a lattice Λ in R24 discovered John Leech ( 16 (1964), 657--682). ...

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.

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. In mathematics, theta functions are special functions of several complex variables. ... A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ...

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. In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... Several specialized usages of the terms compact and compactness exist. ... This article or section should be merged with Orientable manifold. ... In mathematics, a differentiable manifold is a topological space that looks locally like the Euclidean space Rn, and the Euclidean space indeed provides the simplest example of a manifold. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, USA) is a mathematician at Microsoft Research. ... In mathematics, the Poincaré conjecture is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds. ... Simon Kirwan Donaldson, born in Cambridge in 1957, is a mathematician famous for his work on exotic four-dimensional spaces in differential geometry using instantons, and the discovery of new differential invariants. ... Smooth could mean many things, including: Smooth function, a function that is infinitely differentiable, used in calculus and topology. ... In mathematics, a differentiable manifold is a topological space that looks locally like the Euclidean space Rn, and the Euclidean space indeed provides the simplest example of a manifold. ...

Further reading

• 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