Translational symmetry

Pictures are called translation invariant in mathematics whose value does not change under a translation. More specifically is a functional translation invariant, when the value of the functional does not change when the function of a translation is subjected to displacement vector.

For example, any constant function is translationally invariant. A more interesting example is the Lebesgue integral. This clearly means its translation invariance, that the value of an integral does not change when the domain is displaced, as well as does not change the volume of a body by pure displacement in space.

General Definition: translation invariance in groups

More generally, it is possible to define translational invariance for group operations. Let X be a set with a transitive operation of a group G. Then induces

For each element g of G is an automorphism of X and thus an automorphism on each functorial construction of F ( X) on X. The G- invariants in F ( X) are called translation invariant.

For a group G and X = G can be carried

Define two G-spaces, the associated translation invariance is called left or Rechtsinvarianz.

For example, the Lie algebra is a Lie group is the space of left-invariant vector fields. A hair - measure on a topological group is also translation invariant. The Petersson scalar product on the upper half-plane is defined by a SL (2, R)- invariant measure.

Others

Translation invariant is also a stochastic function which is changed only to an additive ( or subtractive ) components. Here are the laws that are described with the function not affected. Only the center or scale values ​​change.