Direct sum of topological groups

In mathematics, a topological group [math]\displaystyle{ G }[/math] is called the topological direct sum of two subgroups [math]\displaystyle{ H_1 }[/math] and [math]\displaystyle{ H_2 }[/math] if the map [math]\displaystyle{ \begin{align} H_1\times H_2 &\longrightarrow G \\ (h_1,h_2) &\longmapsto h_1 h_2 \end{align} }[/math] is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.

Definition

More generally, [math]\displaystyle{ G }[/math] is called the direct sum of a finite set of subgroups [math]\displaystyle{ H_1, \ldots, H_n }[/math] of the map [math]\displaystyle{ \begin{align} \prod^n_{i=1} H_i &\longrightarrow G \\ (h_i)_{i\in I} &\longmapsto h_1 h_2 \cdots h_n \end{align} }[/math] is a topological isomorphism.

If a topological group [math]\displaystyle{ G }[/math] is the topological direct sum of the family of subgroups [math]\displaystyle{ H_1, \ldots, H_n }[/math] then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family [math]\displaystyle{ H_i. }[/math]

Topological direct summands

Given a topological group [math]\displaystyle{ G, }[/math] we say that a subgroup [math]\displaystyle{ H }[/math] is a topological direct summand of [math]\displaystyle{ G }[/math] (or that splits topologically from [math]\displaystyle{ G }[/math]) if and only if there exist another subgroup [math]\displaystyle{ K \leq G }[/math] such that [math]\displaystyle{ G }[/math] is the direct sum of the subgroups [math]\displaystyle{ H }[/math] and [math]\displaystyle{ K. }[/math]

A the subgroup [math]\displaystyle{ H }[/math] is a topological direct summand if and only if the extension of topological groups [math]\displaystyle{ 0 \to H\stackrel{i}{{} \to {}} G\stackrel{\pi}{{} \to {}} G/H\to 0 }[/math] splits, where [math]\displaystyle{ i }[/math] is the natural inclusion and [math]\displaystyle{ \pi }[/math] is the natural projection.

Examples

Suppose that [math]\displaystyle{ G }[/math] is a locally compact abelian group that contains the unit circle [math]\displaystyle{ \mathbb{T} }[/math] as a subgroup. Then [math]\displaystyle{ \mathbb{T} }[/math] is a topological direct summand of [math]\displaystyle{ G. }[/math] The same assertion is true for the real numbers [math]\displaystyle{ \R }[/math]

See also

• Complemented subspace
• Direct sum – Operation in abstract algebra composing objects into "more complicated" objects
• Direct sum of modules – Operation in abstract algebra