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Concept
Unimodular lattice
Summary
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1.
The E8 lattice and the Leech lattice are two famous examples.
A lattice is a free abelian group of finite rank with a symmetric bilinear form (·, ·).
The lattice is integral if (·,·) takes integer values.
The dimension of a lattice is the same as its rank (as a Z-module).
The norm of a lattice element a is (a, a).
A lattice is positive definite if the norm of all nonzero elements is positive.
The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (ai, aj), where the elements ai form a basis for the lattice.
An integral lattice is unimodular if its determinant is 1 or −1.
A unimodular lattice is even or type II if all norms are even, otherwise odd or type I.
The minimum of a positive definite lattice is the lowest nonzero norm.
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.
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.
An integral lattice is unimodular if and only if its dual lattice is integral. Unimodular lattices are equal to their dual lattices, and for this reason, unimodular lattices are also known as self-dual.
Given a pair (m,n) of nonnegative integers, an even unimodular lattice of signature (m,n) exists if and only if m−n is divisible by 8, but an odd unimodular lattice of signature (m,n) always exists. In particular, even unimodular definite lattices only exist in dimension divisible by 8. Examples in all admissible signatures are given by the IIm,n and Im,n constructions, respectively.
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Related concepts (4)
E8 lattice
In mathematics, the E_8 lattice is a special lattice in R^8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E_8 root system. The norm of the E_8 lattice (divided by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite, even, unimodular lattice of rank 8. The existence of such a form was first shown by H.
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Ernst Witt in 1940. The Leech lattice Λ24 is the unique lattice in 24-dimensional Euclidean space, E24, with the following list of properties: It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1. It is even; i.e.
E8 manifold
DISPLAYTITLE:E8 manifold In mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice. The manifold was discovered by Michael Freedman in 1982. Rokhlin's theorem shows that it has no smooth structure (as does Donaldson's theorem), and in fact, combined with the work of Andrew Casson on the Casson invariant, this shows that the manifold is not even triangulable as a simplicial complex.