Lattice (group)
In geometry and group theory, a lattice in the real coordinate space
Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set.
More abstractly, a lattice can be described as a free abelian group of dimension
, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way.
A lattice may be viewed as a regular tiling of a space by a primitive cell.
Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory.
They also arise in applied mathematics in connection with coding theory, in percolation theory to study connectivity arising from small-scale interactions, cryptography because of conjectured computational hardness of several lattice problems, and are used in various ways in the physical sciences.
For instance, in materials science and solid-state physics, a lattice is a synonym for the framework of a crystalline structure, a 3-dimensional array of regularly spaced points coinciding in special cases with the atom or molecule positions in a crystal.
A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translation of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previous sense.
is central to the study of elliptic functions, developed in nineteenth century mathematics; it generalizes to higher dimensions in the theory of abelian functions.
into equal polyhedra (copies of an n-dimensional parallelepiped, known as the fundamental region of the lattice), then d(
) 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.
For example, the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL) has been used in the cryptanalysis of many public-key encryption schemes,[2] and many lattice-based cryptographic schemes are known to be secure under the assumption that certain lattice problems are computationally difficult.
[3] There are five 2D lattice types as given by the crystallographic restriction theorem.
For example, below the hexagonal/triangular lattice is given twice, with full 6-fold and a half 3-fold reflectional symmetry.
If the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. For the classification of a given lattice, start with one point and take a nearest second point.
(Not logically equivalent but in the case of lattices giving the same result is just "Choose a point for which the larger of the two is least".)
In a rhombic lattice, the shortest distance may either be a diagonal or a side of the rhombus, i.e., the line segment connecting the first two points may or may not be one of the equal sides of the isosceles triangle.
This ensures that p and q themselves are integer linear combinations of the other two vectors.
Each pair p, q defines a parallelogram, all with the same area, the magnitude of the cross product.
Up to size and orientation, a pair can be represented by their quotient.
Equivalence in the sense of generating the same lattice is represented by the modular group:
represents choosing a different side of the triangle as reference side 0–1, which in general implies changing the scaling of the lattice, and rotating it.
Each "curved triangle" in the image contains for each 2D lattice shape one complex number, the grey area is a canonical representation, corresponding to the classification above, with 0 and 1 two lattice points that are closest to each other; duplication is avoided by including only half of the boundary.
The rectangular lattices are at the imaginary axis, and the remaining area represents the parallelogrammatic lattices, with the mirror image of a parallelogram represented by the mirror image in the imaginary axis.
More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that the quotient 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.
this concept can be generalized to any finite-dimensional vector space over any field.
- the unit group of elements in R with multiplicative inverses) then the lattices generated by these bases will be isomorphic since T induces an isomorphism between the two lattices.
