Unimodular lattice
Encyclopedia
In mathematics
, a unimodular lattice is a lattice
of determinant 1 or −1.
The E8 lattice
and the Leech lattice
are two famous examples.
Write Rm,n for the m+n dimensional vector space
Rm+n with the inner product of
(a1,...,am+n) and (b1,...,bm+n) given by
In Rm,n there is one odd unimodular lattice up to isomorphism,
denoted by
which is given by all vectors (a1,...,am+n)
in Rm,n with all the ai integers.
There are no even unimodular lattices unless
in which case there is a unique example up to isomorphism, denoted by
This is given by all vectors (a1,...,am+n)
in Rm,n such that either all the ai are integers or they are all integers
plus 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 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 II16,0),
and 24 in dimension 24, called the Niemeier lattice
s (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 dimension
24 (the Leech lattice and the odd Leech lattice), and showed that there are 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 from 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.
of weight n/2.
If the lattice is odd the theta function has level 4.
is a unimodular lattice. Michael Freedman
showed that 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 conjecture
for 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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a unimodular lattice is a lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
of determinant 1 or −1.
The E8 lattice
E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...
and the Leech lattice
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...
are two famous examples.
Definitions
- A lattice is a free abelian groupFree abelian groupIn 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. Hence, free abelian groups over a basis B are...
of finite rankRank of an abelian groupIn mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
with an integral symmetric bilinear formSymmetric bilinear formA symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
(·,·). - A lattice is even if (a, a) is always even.
- The dimension of a lattice is the same as its rank (as a Z-moduleModule (mathematics)In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
). - 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 E8 latticeE8 latticeIn mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8...
, an even 8 dimensional lattice, - The Leech latticeLeech latticeIn mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...
, the 24 dimensional even unimodular lattice with no roots.
Classification
For indefinite lattices, the classification is easy to describe.Write Rm,n for the m+n dimensional vector space
Rm+n with the inner product of
(a1,...,am+n) and (b1,...,bm+n) given by
- a1b1+...+ambm − am+1bm+1 − ... − am+nbm+n.
In Rm,n there is one odd unimodular lattice up to isomorphism,
denoted by
- Im,n,
which is given by all vectors (a1,...,am+n)
in Rm,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 (a1,...,am+n)
in Rm,n such that either all the ai are integers or they are all integers
plus 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 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 II16,0),
and 24 in dimension 24, called the Niemeier lattice
Niemeier lattice
In mathematics, a Niemeier lattice is one of the 24positive definite even unimodular lattices of rank 24,which were classified by . gave a simplified proof of the classification. has a sentence mentioning that he found more than 10 such lattices, but gives no further details...
s (examples:
the Leech lattice
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...
, 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 dimension
24 (the Leech lattice and the odd Leech lattice), and showed that there are 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 from 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 lattice) | 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 (E82, D16+) | 0 |
| 17 | 9 | 0 | ||
| 18 | 13 | 0 | ||
| 19 | 16 | 0 | ||
| 20 | 28 | 0 | ||
| 21 | 40 | 0 | ||
| 22 | 68 | 0 | ||
| 23 | 117 | 1 (Shorter Leech lattice) | ||
| 24 | 273 | 1 (Odd Leech lattice) | 24 (Niemeier lattices) | 1 (Leech lattice) |
| 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 increase even more rapidly.
Properties
The theta function of an even unimodular positive definite lattice of dimension n is a level 1 modular formModular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
of weight n/2.
If the lattice is odd the theta function has level 4.
Applications
The second cohomology group of a closed simply connected oriented topological 4-manifold4-manifold
In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...
is a unimodular lattice. Michael Freedman
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
showed that 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 conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
for 4 dimensional topological manifolds. Donaldson's theorem
Donaldson's theorem
In mathematics, Donaldson's theorem states that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive definite, it can be diagonalized to the identity matrix...
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
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....
.