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arxiv: 2310.15564 · v2 · submitted 2023-10-24 · 🧮 math.GR

Cartan subgroups in connected locally compact groups

Arunava Mandal , Riddhi Shah This is my paper

Pith reviewed 2026-05-24 06:59 UTC · model grok-4.3

classification 🧮 math.GR
keywords Cartan subgroupsconnected locally compact groupsLevi decompositionweak exponentialitypro-Lie algebraspower mapsmaximal torinilpotent subgroups
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The pith

Connected locally compact groups possess Cartan subgroups whose connectedness is equivalent to the group's weak exponentiality.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the classical Cartan subgroup notion from Lie groups to the setting of connected locally compact groups by proposing a specific definition that differs from Chevalley's general version. It proves existence of these subgroups, derives a Levi decomposition extending prior results for Lie groups, shows that quotients inherit Cartan subgroups as images, and links them to centralizers of maximal tori in the radical. The work further equates the density of power map images with surjectivity on Cartan subgroups and proves that weak exponentiality holds precisely when every Cartan subgroup is connected. A reader would care because the results supply concrete structural invariants for analyzing decompositions and exponential behavior in groups that are not necessarily Lie.

Core claim

Cartan subgroups are defined in connected locally compact groups to extend the Lie group case. Their existence is proved, the definition is justified by subsequent properties, quotients have Cartan subgroups as images of ambient ones, a Levi decomposition holds, the centralizer of any maximal torus in the radical is connected with its Cartan subgroups being those of the whole group, every Cartan subgroup arises this way, corresponding Cartan subalgebras in pro-Lie algebras are nilpotent, power map image density is characterized by surjectivity on all Cartan subgroups, and weak exponentiality of the group is equivalent to all its Cartan subgroups being connected.

What carries the argument

Cartan subgroup, defined to extend the Lie-group notion while differing from Chevalley's general definition, serving as the maximal nilpotent subgroup whose properties control decompositions and exponentiality.

If this is right

  • Cartan subgroups of any quotient are exactly the images of Cartan subgroups from the original group.
  • Every Cartan subgroup arises as the Cartan subgroup of the centralizer of a maximal torus in the radical.
  • Cartan subalgebras defined by Hofmann and Morris in the pro-Lie algebra of such a group coincide with those from the Cartan subgroups and are nilpotent.
  • The image of a power map is dense if and only if the map is surjective on every Cartan subgroup.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence may provide a practical test for weak exponentiality by inspecting only the Cartan subgroups rather than the whole group.
  • The Levi decomposition could be used to reduce questions about general connected locally compact groups to questions about their semisimple and radical parts separately.
  • The correspondence with pro-Lie algebras suggests the results may apply to inverse limits of Lie groups in a uniform way.

Load-bearing premise

The chosen definition of Cartan subgroup correctly captures the structural features needed for the existence, decomposition, and equivalence results to hold in connected locally compact groups.

What would settle it

A concrete connected locally compact group in which no subgroups satisfy the paper's definition of Cartan subgroup, or one where all Cartan subgroups are connected yet the group fails to be weakly exponential.

read the original abstract

We define Cartan subgroups in connected locally compact groups, which extends the classical notion of Cartan subgroups in Lie groups. We prove their existence and justify our choice of the definition which differs from the one given by Chevalley on general groups. Apart from proving some properties of Cartan subgroups, we show that the Cartan subgroups of the quotient groups are precisely the images of Cartan subgroups of the ambient group. We establish the so-called `Levi' decomposition of Cartan subgroups which extends W\"ustner's decomposition theorem and our earlier results for Lie groups. We also show that the centraliser of any maximal torus of the radical is connected and its Cartan subgroups are also Cartan subgroups of the ambient group; moreover, every Cartan subgroup arises this way. We prove that Cartan subalgebras defined by Hofmann and Morris in pro-Lie algebras are the same as those corresponding to Cartan subgroups in case of pro-Lie algebras of connected locally compact groups, and that they are nilpotent. We characterise density of the image of a power map in a connected locally compact group in terms of its surjectivity on all Cartan subgroups, and show that weak exponentiality of the group is equivalent to the condition that all its Cartan subgroups are connected.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript defines Cartan subgroups in connected locally compact groups, extending the Lie-group notion, and proves their existence while justifying the definition against Chevalley's. It establishes that Cartan subgroups of quotients are images of those in the ambient group, proves a Levi decomposition extending Wüstner's theorem, shows that the centralizer of any maximal torus in the radical is connected with its Cartan subgroups coinciding with those of the group (and conversely), identifies these with the nilpotent Hofmann-Morris Cartan subalgebras in the pro-Lie case, characterizes density of the power-map image via surjectivity on all Cartan subgroups, and proves that weak exponentiality is equivalent to connectedness of all Cartan subgroups.

Significance. If the derivations hold, the work supplies a coherent extension of Cartan theory to connected locally compact groups, including structural results (Levi decomposition, centralizer properties) and equivalences that link subgroup connectedness to global exponentiality and power-map behavior. These provide concrete tools for analyzing topological groups beyond the Lie setting and build directly on prior results for Lie groups and pro-Lie algebras.

minor comments (3)
  1. [Abstract] The abstract lists results in a single dense sentence; splitting into separate statements would improve readability.
  2. Verify that all references to prior work (Wüstner, Hofmann-Morris, Chevalley) appear in the bibliography with consistent formatting.
  3. Notation for the power map and its image density should be introduced explicitly in the first section where it appears, rather than assumed from context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our work and the positive assessment. The recommendation for minor revision is noted. No specific major comments appear in the report, so we have no points requiring point-by-point response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivations are from group axioms and local compactness

full rationale

The paper introduces a definition of Cartan subgroups tailored to connected locally compact groups, then derives existence, the Levi decomposition, nilpotency of corresponding subalgebras, and equivalences for power maps and weak exponentiality directly from the axioms of topological groups, local compactness, and connectedness. No equation or central claim reduces by construction to a fitted parameter, a self-referential definition, or an unverified self-citation chain; the cited prior results (Wüstner, Hofmann-Morris, and the authors' Lie-group work) function as external extensions rather than load-bearing premises that collapse the new claims. The justification for the definition differing from Chevalley's is precisely the verification that the listed properties hold under the stated hypotheses, which is standard non-circular practice in pure mathematics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard axioms of topological groups, connectedness, and local compactness; no numerical parameters are fitted and no new entities are postulated beyond the defined subgroups.

axioms (1)
  • domain assumption Connected locally compact groups admit the structures and operations used in the definitions and proofs (standard topological group axioms plus local compactness and connectedness).
    Invoked throughout to guarantee existence and the listed properties.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Twisted conjugacy classes in Lie groups

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    The paper gives necessary and sufficient conditions for infinite Reidemeister numbers under twisted conjugacy for connected solvable or compactly generated nilpotent Lie groups and proves the topological R_∞-property ...

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · cited by 1 Pith paper

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