Decompositions of the automorphism group of a locally compact abelian group

TL;DR

This paper investigates the structure of automorphism groups of locally compact abelian groups by decomposing them into matrix groups, revealing their topological and algebraic properties.

Contribution

It provides a detailed decomposition of Aut(L) into matrix groups based on the group's structure, extending the approach to objects in additive categories.

Findings

Aut(L) is topologically isomorphic to a matrix group with specific entries.

The algebraic decomposition applies beyond locally compact abelian groups.

The paper characterizes the topological and algebraic structure of automorphism groups.

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.