On the dearth of coproducts in the category of locally compact groups

2021/07/25 by Alexandru Chirvăsitu, Chirvasitu, Alexandru
Mathematics · #18A30 #22D05 #22D12 #22E46 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2107.11796

openalex publication_date 2021/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that a family of at least two non-trivial, almost-connected locally compact groups cannot have a coproduct in the category of locally compact groups if at least one of the groups is connected; this confirms the intuition that coproducts in said category are rather hard to come by, save for the usual ones in the category of discrete groups. Along the way we also prove a number of auxiliary results on characteristic indices of locally compact or Lie groups as defined by Iwasawa: that characteristic indices can only decrease when passing to semisimple closed Lie subgroups, and also along dense-image morphisms.

Citations

• Large sets of generating tuples for Lie groups

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