Character group

of a group

The group of all characters of (cf. Character of a group) with values in an Abelian group , under the operation

induced by the operation in . When , then

where are quasi-cyclic groups, one for each prime number . This group is algebraically compact (see Pure subgroup). If is Abelian, then is a divisible group if and only if is torsion free and it is a reduced group if and only if is periodic [4].

The character group of a topological group is the group of all continuous homomorphisms , equipped with the compact-open topology. It is a Hausdorff Abelian topological group. If is locally compact, then so is ; if is compact, then is discrete, and if is discrete, then is compact.

Examples of character groups:

for any finite discrete Abelian group .

With every continuous homomorphism of topological groups there is associated a homomorphism of the character groups . Here the correspondence , , is a contravariant functor from the category of topological groups into the category of Abelian topological groups. If the category is restricted to locally compact Abelian groups , then this functor determines an equivalence of that category and its dual category (see Pontryagin duality).

The character group of an algebraic group over a field is the group of all rational characters . If is an Abelian affine algebraic group, then generates the space (that is, is a basis in this space) if and only if is a diagonalizable algebraic group, i.e. is isomorphic to a closed subgroup of a certain torus . Here is a finitely generated Abelian group (without -torsion if ), and is the group algebra of over , which makes it possible to define a duality between the categories of diagonalizable groups and that of finitely generated Abelian groups (without -torsion if ), cf. [1]. When is a finite group (regarded as a -dimensional algebraic group) and , then this duality is the same as the classical duality of finite Abelian groups.

For any connected algebraic group , the group is torsion free. In particular, a diagonalizable group is a torus if and only if .

References

[1]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	S.A. Morris, "Pontryagin duality and the structure of locally compact Abelian groups" , London Math. Soc. Lecture Notes , 29 , Cambridge Univ. Press (1977) MR0442141 Zbl 0446.22006
[3]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[4]	L. Fuchs, "Infinite abelian groups" , 1 , Acad. Press (1970) MR0255673 Zbl 0209.05503
[5]	J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039

Comments

In the article above denotes the circle group. A periodic group is also called a torsion group. An Abelian group is reduced if it contains no non-trivial divisible subgroups.

Above, the phrase "character" is of course strictly used in its narrowest meaning of a (continuous) homomorphism and not in the sense of the character of some representation.

The character groups of many locally Abelian groups can be found in [a1].

References

[a1]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1 , Springer (1963) MR0156915 Zbl 0115.10603
[a2]	N. Bourbaki, "Elements of mathematics. Spectral theories" , Addison-Wesley (1977) (Translated from French) MR0583191 Zbl 1106.46004