Everything about Lattice Group totally explained
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
integral coefficients. 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, 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 a
crystalline structure, a 3-dimensional array of regularly spaced points coinciding with the
atom or
molecule positions in a
crystal. More generally,
lattice models are studied in
physics, often by the techniques of
computational physics.
Symmetry considerations and examples
A lattice is the
symmetry group of discrete
translational symmetry in
n directions. A pattern with this lattice of translational symmetry can't have more, but may have less symmetry than the lattice itself.
A lattice in the sense of a 3-
dimensional array of regularly spaced points coinciding with for example the
atom or
molecule positions in a
crystal, or more generally, the orbit of a
group action under translational symmetry, is a translate of the translation lattice: a
coset, which need not contain the origin, and therefore need not be a lattice in the previous sense.
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.
Dividing space according to a lattice
A typical lattice Λ in
Rn thus has the form
» - the
unit group of elements in
with multiplicative inverses) then the lattices generated by these bases will be isomorphic since
induces an
isomorphism between the two lattices.
Important cases occur in number theory with
K a
p-adic field and
R the
p-adic integers.
Further Information
Get more info on 'Lattice Group'.
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