Lattice (group)

In mathematics grids are periodic amounts in a sense. See, inter alia, application in group theory, geometry, and at Approximationsfragestellungen.

The individual elements of a lattice are called grid points or lattice vectors.

Lattice in Euclidean space

There are linearly independent vectors of the Euclidean vector space. Then called

A lattice with basis of rank. The matrix formed from the vectors is called a basis matrix of. The base is not fixed by the grid. However, each base has the same rank. As a subgroup of the additive group of a free abelian group of rank.

The limited amount

Is called basic or fundamental stitch stitch from. It extends in one -dimensional subspace

And forms is a fairly open -dimensional parallelepiped.

Through the grating is defined at an equivalence ratio as follows:

Each element of is to exactly one element from the basic stitch equivalent. Each equivalence class has exactly one representative in the basic stitch.

There with no one to. Since the interest so happens only in the subspace, and this is isomorphic to, most authors only consider the case of equality (grid with full rank ).

In this case, the entire mesh with the shape of the base mesh are cradled. However, forms are of interest that are not parallelepiped. This is called a fundamental region.

A grid is full, if the dot product is an integer for all. If, in addition, it is called the grid straight.

Examples:

Lattice in the complex plane

By perceives the complex plane as a real vector space, one can speak of grids in; they are free abelian groups of rank 2 play a central role in the theory of elliptic functions and elliptic curves.

More generally, a natural number, so are grid in real -dimensional space in relation to complex tori and abelian varieties.

Lattice in Lie groups

A subgroup of a topological group is called discrete, if for every open surroundings with

There.

If a locally compact group with Haarschem measure is then called a discrete subgroup of a lattice if the quotient finite volume (with respect to the Haar measure ) has.

A lattice is called uniform or kokompakt if is compact.

Lattice in Lie groups play an important role in Thurston's Geometrisierungsprogramm.

Examples

  • Be the belonging to the base matrix lattice of rank 2 is then.
  • Be. Then the basic stitch of the -dimensional hypercube, and it applies, for example.
  • The ring of Gaussian numbers is a lattice in.
  • The Ring of Hurwitzquaternionen is a grid in the skew field of quaternions.

Gitterdiskriminante

One parameter for classification of grids is the Gitterdiskriminante. It is calculated as the volume of the basic stitch.

In lattices in Euclidean space with the base matrix this corresponds to the formula

As an invariant of the value of the Gitterdiskriminante is independent of the chosen basis.

Lattice reduction

The lattice reduction, the problem is to be calculated from a given grating base, a base having certain properties, such as a base with short, nearly orthogonal vectors. The LLL algorithm ( according to Lenstra, Lenstra and Lovász ) calculated in polynomial time a so-called LLL - reduced basis, with which one obtains provably short lattice vectors. In fact the length of the first vector of a LLL reduced base is close to the length of the shortest non-trivial grating vector.

The LLL algorithm has found many applications in cryptanalysis of asymmetric encryption methods such as RSA cryptosystem and the Merkle -Hellman cryptosystem.

Rank of an abelian group Fundamental domain Locally compact space Lenstra–Lenstra–Lovász lattice basis reduction algorithm Public-key cryptography Jacques Stern Hendrik Lenstra Bravais lattice Richard Dedekind Order (ring theory) Latticework
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