Pontryagin duality

The Pontryagin duality (after Lev Semenovich Pontryagin ) is a mathematical term from the harmonic analysis. A locally compact abelian group is another locally compact abelian group assigned as the dual group, such that the dual group of the dual group is the starting group back. This construction plays an important role in the abstract Fourier transform theory and structure of locally compact abelian groups.

Pontryagin duality

The circle is a compact group with multiplication as group link.

If G is a locally compact abelian group, ie a continuous homomorphism a character of G. The dual group of G is the set of all characters of G. The multiplication is an abelian group, and the topology of compact convergence makes it a locally compact group, ie whose topology is locally compact to a topological group.

Is a continuous homomorphism, so is also a continuous homomorphism which to dual homomorphism.

Examples

• The characters of the residue class group have the form, where. It applies if, and so.
• Each character has the form of a. One identifies with n, then.
• The group has the characters, with. The assignment provides.
• With the addition of a link and the Euclidean topology is a locally compact abelian group. Each character has the shape of a. So you identify with z, one has first as quantities. This applies to all and the picture is a homeomorphism, so you have abelian as locally compact groups.

Products of groups

If G and H locally compact abelian groups, their Cartesian product. Then define a character, if one sets. In this way one obtains a Gruppenhomöomorphismus.

This gives you many more examples:

• For any finite abelian group G, for such is a finite product of groups of the form (see: last generated abelian group).
• ,,

Duality theorem of Pontryagin

One has a natural map. The set of Pontryagin states that this figure is always a topological group isomorphism. This justifies the term dual group of G, since by the above theorem can be recovered by re- dual group G of Education.

Relationships between group and dual group

Due to the Pontryagin duality one expects a set of relationships between a locally compact abelian group G and its dual group. Here you will find relations between algebraic and topological properties. An example, the following applies:

• G is discrete is compact.
• G is compact is discrete.

For a compact group the following are equivalent:

• G is connected.
• G is divisible.
• Is torsion.

Another related property leads to the following equivalence:

• A compact group G is totally disconnected, if a divisible group.

A continuous homomorphism is called strict, if the picture is open, ie the image of every open set is relatively open in the image of. A sequence of homomorphisms is called strict if every homomorphism is strict. Finally Identifies the one-element group 1 and comply, then the following theorem holds:

• Let be a sequence of continuous homomorphisms between locally compact abelian groups. Then the following statements are equivalent:   is a strict and exact sequence.
• Is a strict and exact sequence.

From this it draws more conclusions:

• A continuous homomorphism if and only strictly if strict.
• Is a closed subgroup, then. This is the dual to the inclusion mapping.

Compact generated groups

The Pontryagin duality is an important tool in the structure theory for locally compact abelian groups. A locally compact group is called compact generated when there is a compact subset of G which generates G as a group. A discrete group is exact then generates compact if it is finitely generated.

For a locally compact abelian group are equivalent:

• G is generated compact.
• Where and K is a compact group.
• Where and D is a discrete group.

Addition: Here are the numbers m and n are uniquely determined by G and K is the largest compact subgroup of G.

Gelfand transform

As explained in the article harmonic analysis, the dual group of a locally compact abelian group G in the Gelfand transform of the convolution algebra over G occurs.

Pontryagin duality functor as

The Pontryagin duality, that is, the assignments described above and of locally compact abelian groups and continuous homomorphisms, apparently is a contravariant functor. The two fold applying this functor leads to the identical functor (more precisely, to a natural equivalence to the identical functor ).