# Character group

2010 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

The character group of a group $G$ is the group $X(G) = \def\Hom{\textrm{Hom}}\Hom(G,A)$ of all characters of $G$ (cf. Character of a group) with values in an Abelian group $A$, under the operation

$$\def\a{\alpha}\def\b{\beta}(\a\b)(g) = \a(g)\b(g),\quad g\in G,\quad \a,\b\in X(G),$$ induced by the operation in $A$. When $A=T = \{ z\in\C \;|\ \ |z| =1\}$, then

$$X(G) \simeq \prod_p \Hom(G,\Z(p^\infty)),$$ where $\Z(p^\infty)$ are quasi-cyclic groups, one for each prime number $p$. This group is algebraically compact (see Pure submodule). If $G$ is Abelian, then $X(G)$ is a divisible group if and only if $G$ is torsion free and it is a reduced group if and only if $G$ is a torsion group [Fu].

The character group of a topological group $G$ is the group $X(G)$ of all continuous homomorphisms $G\to T$, equipped with the compact-open topology. It is a Hausdorff Abelian topological group. If $G$ is locally compact, then so is $X(G)$; if $G$ is compact, then $X(G)$ is discrete, and if $G$ is discrete, then $X(G)$ is compact.

Examples of character groups:

$$X(T)\simeq \Z,\quad X(\Z)\simeq T,\quad X(\R) \simeq \R,\quad X(G)\simeq G$$ for any finite discrete Abelian group $G$.

With every continuous homomorphism of topological groups $\def\phi{\varphi}\phi:G\to H$ there is associated a homomorphism of the character groups $\phi^*:X(H)\to X(G)$. Here the correspondence $G\mapsto X(G)$, $\phi\mapsto\phi^*$, 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 $G$, then this functor determines an equivalence of that category and its dual category (see Pontryagin duality).

The character group of an algebraic group $G$ over a field $K$ is the group $X(G)$ of all rational characters $\def\G{\mathbb{G}}G\to K^* = \G_m$. If $X(G)$ is an Abelian affine algebraic group, then $K[G]$ generates the space $G$ (that is, is a basis in this space) if and only if $G$ is a diagonalizable algebraic group, i.e. is isomorphic to a closed subgroup of a certain algebraic torus $\G_m^s$. Here $X(G)$ is a finitely generated Abelian group (without $p$-torsion if $\def\char{\textrm{char}\;}\char K = p > 0$), and $K[G]$ is the group algebra of $X(G)$ over $K$, which makes it possible to define a duality between the categories of diagonalizable groups and that of finitely generated Abelian groups (without $p$-torsion if $\char K = p > 0$), cf. [Bo]. When $G$ is a finite group (regarded as a $0$-dimensional algebraic group) and $\char K = 0$, then this duality is the same as the classical duality of finite Abelian groups.

For any connected algebraic group $G$, the group $X(G)$ is torsion free. In particular, a diagonalizable group $G$ is a torus if and only if $X(G)\simeq \Z^s$.

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