Lattice (group)
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- See lattice for other meanings of this term, both within and without mathematics.
In mathematics, 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 considering all linear combinations with integral coefficients.
A simple example of a lattice in Rn is the subgroup Zn. A more complicated example is the Leech lattice, which is a lattice in R24. The period lattice in R2 is central to the study of elliptic functions, developed in nineteenth century mathematics; it generalises to higher dimensions in the theory of abelian functions.
A typical lattice Λ in Rn thus has the form
- <math>
\Lambda = \left\{ \sum_{i=1}^n a_i v_i \; | \; a_i \in\Bbb{Z} \right\} <math> where {v1, ..., vn} is a basis for Rn. Different bases can generate the same lattice, but the absolute value of the determinant of the vectors vi is uniquely determined by Λ, and is denoted by d(Λ). If one thinks of a lattice as dividing the whole of Rn into equal polyhedra (known as the fundamental region of the lattice), then d(Λ) is equal to the n-dimensional volume of this polyhedron. This is why d(Λ) is sometimes called the covolume of the lattice.
Minkowski's theorem relates the number d(Λ) and the volume of a symmetric convex set S to the number of lattice points contained in S. The number of lattice points contained in a polytope all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d(Λ) as well.
Lattice basis reduction is the problem of finding a short lattice basis. The Lenstra-Lenstra-Lovász lattice reduction algorithm (LLL) finds a short lattice basis in polynomial time; it has found numerous applications, particularly in public-key cryptography.
A lattice in Cn is a discrete subgroup of Cn which spans the 2n-dimensional real vector space Cn. For example, the Gaussian integers form a lattice in C.
Every lattice in Rn is a free abelian group of rank n; every lattice in Cn is a free abelian group of rank 2n.
This concept is used in materials science, in which a lattice is a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal.
It also occurs in computational physics, in which a lattice is an n-dimensional geometrical structure of sites, connected by bonds, which represent positions which may be occupied by atoms, molecules, electrons, spins, etc. For an article dealing with the formal representation of such structures see Lattice Geometries (http://www.hermetic.ch/compsci/lattgeom.htm). Quite general lattice models are used in physics.
In Lie groups
More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that G/Γ is of finite measure, for the measure on it inherited from Haar measure on G (left-invariant, or right-invariant - the definition is independent of that choice). That will certainly be the case when G/Γ is compact, but that sufficient condition is not necessary, as is shown by the case of the modular group in SL2(R), which is a lattice but where the quotient isn't compact (it has cusps). There are general results stating the existence of lattices in Lie groups.