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) |
[2] | S.A. Morris, "Pontryagin duality and the structure of locally compact Abelian groups" , London Math. Soc. Lecture Notes , 29 , Cambridge Univ. Press (1977) |
[3] | L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) |
[4] | L. Fuchs, "Infinite abelian groups" , 1 , Acad. Press (1970) |
[5] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) |
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) |
[a2] | N. Bourbaki, "Elements of mathematics. Spectral theories" , Addison-Wesley (1977) (Translated from French) |
Character group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Character_group&oldid=13518