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Direct sum of topological groups

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In mathematics, a topological group G is called the topological direct sum of two subgroups H1 and H2 if the map

Contents

H 1 × H 1 ⟶ G ( h 1 , h 2 ) ⟼ h 1 h 2

is a topological isomorphism.

More generally, G is called the direct sum of a finite set of subgroups H i , i = 1 , … , n of the map

∏ i = 1 n H i ⟶ G ( h i ) i ∈ I ⟼ h 1 h 2 ⋯ h n

Note that if a topological group G is the topological direct sum of the family of subgroups H i then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family H i .

Topological direct summands

Given a topological group G, we say that a subgroup H is a topological direct summand of G (or that splits topologically form G ) if and only if there exist another subgroup K ≤ G such that G is the direct sum of the subgroups H and K.

A the subgroup H is a topological direct summand if and only if the extension of topological groups

0 → H → i G → π G / H → 0

splits, where i is the natural inclusion and π is the natural projection.

Examples

  • Suppose that G is a locally compact abelian group that contains the unit circle T as a subgroup. Then T is a topological direct summand of G. The same assertion is true for the real numbers R

    • References

    Direct sum of topological groups Wikipedia
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