Decompositions of the automorphism group of a locally compact abelian group

Details Decompositions of the automorphism group of a locally compact abelian group Decompositions of the automorphism group of a locally compact abelian group

Publishing date

2011-10-10

URL

arxiv.org/abs/1110.1923v1

Science

Mathematics

Field

General Topology

Additional info

11 pages, 1 figure. First version of paper, comments are appreciated

Type

Scientific paper

Abstract

It is well known that every locally compact abelian group L can be decomposed as L_1 \oplus R^n, where L_1 contains a compact-open subgroup. In this paper, we use this decomposition to study the topological group Aut(L) of automorphisms of L, equipped with the g-topology. We show that Aut(L) is topologically isomorphic to a matrix group with entries from Aut(L_1), Hom(L_1, R^n), Hom(R^n, L_1), and GL_n(R), respectively. It is also shown that the algebraic portion of the decomposition is not specific to locally compact abelian groups, but is also true for objects with a well-behaved decomposition in an additive category with kernels.

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