Local Lie Theory in Quasi-Banach Lie Algebras: Convergence of the BCH Series and Geometric Implications

Amel Jadlaoui; Lotfi Souden; Marwa Ennaceur; Mohsen Ben Abdallah; Mondher Damak; Nassim Athmouni

arxiv: 2604.08265 · v1 · submitted 2026-04-09 · 🧮 math.FA · math.RA· math.RT· math.SP

Local Lie Theory in Quasi-Banach Lie Algebras: Convergence of the BCH Series and Geometric Implications

Nassim Athmouni , Mohsen Ben Abdallah , Mondher Damak , Marwa Ennaceur , Amel Jadlaoui , Lotfi Souden This is my paper

Pith reviewed 2026-05-10 17:38 UTC · model grok-4.3

classification 🧮 math.FA math.RAmath.RTmath.SP

keywords quasi-Banach Lie algebrasBaker-Campbell-Hausdorff serieslocal Lie groupsAoki-Rolewicz theoremconvergence of seriesweak Schatten idealsp-norm

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The pith

The Baker-Campbell-Hausdorff series converges near the origin in quasi-Banach Lie algebras with continuous brackets, defining a local Lie group via the exponential map.

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

The paper shows that for Lie algebras equipped with a quasi-norm that makes the bracket continuous, the Baker-Campbell-Hausdorff series converges in a small neighborhood of zero. This convergence allows the exponential map to turn the algebra into a local Lie group. The argument relies on the Aoki-Rolewicz theorem to replace the quasi-norm with an equivalent p-norm where standard completeness arguments apply. This extends local Lie theory to spaces that are not Banach, such as certain operator ideals with p less than 1. Numerical computations of the series coefficients indicate that the actual radius of convergence can be larger than theoretical bounds predict because of cancellations in the algebra.

Core claim

We prove that the BCH series converges in a neighborhood of the origin, provided the quasi-norm admits a continuous Lie bracket with finite continuity constant Cbracket. The proof relies on the Aoki-Rolewicz theorem to construct an equivalent p-norm satisfying p-subadditivity, enabling rigorous Cauchy-sequence arguments in the complete quasi-metric space. This yields a well-defined local Lie group structure via the exponential map, while the quasi-norm exponent p modifies metric properties but preserves the Lie algebraic structure.

What carries the argument

The convergence of the Baker-Campbell-Hausdorff series in the quasi-metric, established by reducing to an equivalent p-norm via the Aoki-Rolewicz theorem for Cauchy estimates.

If this is right

The exponential map defines a local Lie group structure on the algebra.
Classical combinatorial bounds on the convergence radius are conservative due to algebraic cancellations, allowing larger practical radii.
The theory applies directly to weak Schatten ideals L_{p,∞}(H) for 0 < p < 1.
The geometric deformation induced by p in (0,1] changes distances but keeps the underlying Lie operations intact.
When the quasi-norm is already a p-norm, the convergence radius improves to 1/(4 Cbracket) without extra factors.

Where Pith is reading between the lines

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

This framework may extend to other quasi-normed spaces where completeness holds but the triangle inequality is relaxed, such as certain Hardy space operator algebras.
Computing BCH coefficients via Hall-Lyndon bases could be used in numerical studies of non-associative structures beyond Lie algebras.
Further work could test whether the local group structure can be globalized under additional assumptions on the quasi-norm.
The results suggest that many infinite-dimensional geometric constructions relying on BCH formulas remain valid in quasi-Banach settings.

Load-bearing premise

The Lie bracket must be continuous with respect to the quasi-norm, with some finite continuity constant.

What would settle it

A counterexample would be a complete quasi-normed Lie algebra with continuous bracket in which the partial sums of the BCH series fail to form a Cauchy sequence for some arbitrarily small elements x and y.

Figures

Figures reproduced from arXiv: 2604.08265 by Amel Jadlaoui, Lotfi Souden, Marwa Ennaceur, Mohsen Ben Abdallah, Mondher Damak, Nassim Athmouni.

**Figure 1.** Figure 1: Logarithmic-scale comparison of BCH coefficient sums up to degree 20. The Lie-projected data Bn show approximately geometric decay; the Catalan bound grows exponentially, illustrating the conservatism of the combinatorial majorant. Note that the y-axis lower limit has been extended to 10−7 to display the small values of Bn for n ≥ 15 (compare [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

read the original abstract

We develop a local Lie theory for Lie algebras equipped with a quasi-norm, i.e., complete topological vector spaces satisfying a relaxed triangle inequality $\|x+y\|\le \Ctri(\|x\|+\|y\|)$ with $\Ctri\ge 1$. We prove that the Baker--Campbell--Hausdorff (BCH) series converges in a neighborhood of the origin, provided the quasi-norm admits a continuous Lie bracket with finite continuity constant $\Cbracket$. The proof relies on the Aoki--Rolewicz theorem to construct an equivalent $p$-norm satisfying $p$-subadditivity, enabling rigorous Cauchy-sequence arguments in the complete quasi-metric space $(E, d_p)$. This yields a well-defined local Lie group structure via the exponential map. We analyze the geometric deformation induced by the quasi-norm exponent $p\in(0,1]$, showing that it modifies metric properties while preserving the underlying Lie algebraic structure. Numerical estimates of BCH coefficients up to degree $20$, with coefficients defined precisely via Hall--Lyndon basis projection, demonstrate that classical combinatorial bounds are conservative in the presence of algebraic cancellations, allowing significantly larger practical convergence radii in structured algebras. Applications include weak Schatten ideals $\mathcal{L}_{p,\infty}(H)$ for $0<p<1$ and certain Hardy-space operator algebras. \smallskip\noindent\textbf{Remark on the convergence radius.} The Catalan-majorant method yields convergence for $\|x\|+\|y\| < 1/(4\Cbracket)$; the additional factor $\Ctri$ appearing in the combined constant $\Ctotal = \Ctri\Cbracket$ is an artefact of switching to the $p$-norm to establish Cauchyness of partial sums. When the quasi-norm itself is directly a $p$-norm ($\Ctri=1$), no such penalty arises and the radius reduces to $1/(4\Cbracket)$.

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.

Desk Editor's Note private letter to a colleague

The paper shows BCH convergence holds in quasi-Banach Lie algebras under bracket continuity by reducing to an equivalent p-norm, with a clean radius estimate and some numerical checks on coefficient cancellations.

read the letter

The main result is that the Baker-Campbell-Hausdorff series converges in a neighborhood of zero in a quasi-Banach Lie algebra when the bracket is continuous with respect to the quasi-norm. This produces a local Lie group structure via the exponential map. They achieve this by applying the Aoki-Rolewicz theorem to obtain an equivalent p-norm, carrying out the usual Cauchy estimates there, and transferring the conclusion back. The radius is given explicitly as 1 over 4 times Cbracket, with the extra Ctri factor flagged as an artifact of the reduction step. When the quasi-norm is already a p-norm the penalty disappears. The numerical estimates on coefficients up to degree 20, using the Hall-Lyndon basis, illustrate that classical majorants are loose due to cancellations, which is a practical observation not present in the standard Banach literature. The geometric discussion of how the p-exponent deforms the metric while preserving the Lie algebra structure is straightforward but clarifies the setting. The argument itself is standard once the p-norm is available, and the completeness and continuity transfer without circularity. The soft spots are limited. The applications to weak Schatten ideals and certain Hardy-space operator algebras are mentioned but receive no concrete calculations or radius computations inside those spaces. The numerical work is general rather than specialized to the target algebras, so it supports the conservatism claim without giving sharp bounds for the listed applications. The geometric section stays descriptive and does not produce additional theorems. This is aimed at people working in functional analysis and infinite-dimensional Lie theory who need local group structures in p<1 or quasi-normed settings. A reader looking for a concrete extension of the classical theory will get direct value from the convergence theorem and the radius discussion. It deserves a serious referee because the reduction is explicit, the central claim is new in this context, and the supporting estimates are reproducible.

Referee Report

0 major / 3 minor

Summary. The manuscript develops local Lie theory for quasi-Banach Lie algebras (complete topological vector spaces with relaxed triangle inequality ||x+y|| ≤ Ctri(||x|| + ||y||), Ctri ≥ 1). It proves that the Baker-Campbell-Hausdorff series converges in a neighborhood of the origin whenever the Lie bracket is continuous with respect to the quasi-norm (finite continuity constant Cbracket). The argument invokes the Aoki-Rolewicz theorem to produce an equivalent p-norm, performs standard majorant/Cauchy estimates in the resulting complete p-metric space, and transfers the result back by topological equivalence. This yields a local Lie group structure via the exponential map. The paper further analyzes the metric deformation induced by the quasi-norm exponent p ∈ (0,1], supplies numerical BCH coefficient estimates up to degree 20 (via Hall-Lyndon basis), and indicates applications to weak Schatten ideals L_{p,∞}(H) and certain Hardy-space operator algebras. A remark notes that the Catalan-majorant radius is ||x|| + ||y|| < 1/(4 Cbracket) when Ctri = 1, with an extra Ctri factor appearing as an artefact in the general case.

Significance. If the central convergence claim holds, the work extends the classical BCH theorem and local Lie-group construction from Banach to quasi-Banach settings, which is relevant for non-locally convex spaces that appear in operator theory and harmonic analysis. The explicit use of Aoki-Rolewicz to enable rigorous Cauchy-sequence arguments, together with the numerical demonstration that combinatorial bounds are conservative due to algebraic cancellations, are concrete strengths. The applications to L_{p,∞} and Hardy-space algebras suggest potential for further geometric and analytic developments in these spaces.

minor comments (3)

Remark on the convergence radius: while the artefact status of the Ctri factor is correctly identified, the main theorem statement would benefit from an explicit (even if conservative) radius expressed directly in the original quasi-norm constants Ctri and Cbracket, rather than leaving the reader to reconstruct it from the p-norm reduction.
The notation for the combined constant Ctotal = Ctri Cbracket is introduced only in the remark; defining it once in the introduction or in the statement of the main convergence theorem would improve readability.
The numerical section on BCH coefficients up to degree 20 should include a brief description of the precise recursive or combinatorial formula used to compute the coefficients via the Hall-Lyndon basis, to allow independent verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance, and recommendation for minor revision. No specific major comments or requested changes were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external theorem

full rationale

The central proof reduces the quasi-norm case to an equivalent p-norm via the external Aoki-Rolewicz theorem, then applies standard majorant estimates and Cauchy arguments for the BCH series (independent of the paper's own equations). Convergence yields the local group structure by the usual exponential-map construction. No step is self-definitional, no fitted input is relabeled as prediction, and no load-bearing claim rests on self-citation. Numerical coefficient estimates are supplementary and do not enter the existence proof.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The result rests on the standard completeness axiom for quasi-metric spaces and the domain assumption of a continuous bracket; the Aoki-Rolewicz theorem is imported from prior literature. No new entities are introduced and the constants Cbracket and Ctri are assumptions rather than fitted values.

free parameters (2)

Cbracket
Continuity constant of the Lie bracket, required to be finite but not derived from other data.
Ctri
Quasi-triangle constant appearing in the definition of the quasi-norm.

axioms (2)

domain assumption The space is complete with respect to the quasi-metric d induced by the quasi-norm.
Invoked to guarantee that Cauchy sequences converge to an element of the space.
domain assumption The Lie bracket is continuous with respect to the quasi-norm, with finite constant Cbracket.
Explicitly stated as the hypothesis that enables the convergence proof.

pith-pipeline@v0.9.0 · 5686 in / 1414 out tokens · 97737 ms · 2026-05-10T17:38:09.054100+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Albiac, J

F. Albiac, J. L. Ansorena, and P. Berná, Bases in quasi-Banach spaces of analytic functions,Rev. Mat. Complut. 36(2023), 1–30

work page 2023

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Aoki, Locally bounded linear topological spaces,Proc

T. Aoki, Locally bounded linear topological spaces,Proc. Imp. Acad. Tokyo18(1942), 588–594

work page 1942

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Athmouni, Explicit Baker–Campbell–Hausdorff Radii in Special Banach–Malcev Algebras of Shifts, arXiv:2511.06357v2[math.RA], 2025

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work page arXiv 2025

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Athmouni, Local Rigidity of Quasi–Lie Brackets on Quaternionic Banach Modules and Applications to Nonlinear PDEs,J

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work page 2026

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Athmouni and R

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work page 2018

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F. Harrathi, S. Mabrouk, N. Nawel, and S. D. Silvestrov, Bialgebras, Manin triples of Malcev–Poisson algebras and post–Malcev–Poisson algebras,arXiv:2502.19709[math.RA], 2025

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K. H. Hofmann and S. A. Morris,The Lie Theory of Connected Pro-Lie Groups, EMS Tracts in Mathematics, Vol. 2, European Mathematical Society, Zürich, 2007

work page 2007

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work page 2003

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N. J. Kalton and F. A. Sukochev, Symmetric function spaces and quasi-Banach algebras,J. Funct. Anal.281 (2021), no. 5, 109082

work page 2021

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Neeb and H

K.-H. Neeb and H. Salmasian, Positive energy representations of double extensions of Hilbert–Lie algebras,J. Reine Angew. Math.765(2020), 1–45

work page 2020

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Reutenauer,Free Lie Algebras, London Mathematical Society Monographs, New Series, Vol

C. Reutenauer,Free Lie Algebras, London Mathematical Society Monographs, New Series, Vol. 7, Oxford University Press, 1993

work page 1993

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Rolewicz, On a certain class of linear metric spaces,Bull

S. Rolewicz, On a certain class of linear metric spaces,Bull. Acad. Polon. Sci. Cl. III5(1957), 471–473. N. Athmouni: Université de Gafsa, Campus Universitaire, 2112 Gafsa, Tunisia Email address:nassim.athmouni@fsgf.u-gafsa.tn M. Ben Abdallah: Université de Sf ax, Route de la Soukra km 4, B.P. 802, 3038 Sf ax, Tunisia Email address:mohsenbenabdallah@gmail...

work page 1957