Local Lie Theory in Quasi-Banach Lie Algebras: Convergence of the BCH Series and Geometric Implications
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2026
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| author | Athmouni, Nassim Abdallah, Mohsen Ben Damak, Mondher Ennaceur, Marwa Jadlaoui, Amel Souden, Lotfi |
| author_facet | Athmouni, Nassim Abdallah, Mohsen Ben Damak, Mondher Ennaceur, Marwa Jadlaoui, Amel Souden, Lotfi |
| contents | 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 $1/(4\Cbracket)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_08265 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Lie Theory in Quasi-Banach Lie Algebras: Convergence of the BCH Series and Geometric Implications Athmouni, Nassim Abdallah, Mohsen Ben Damak, Mondher Ennaceur, Marwa Jadlaoui, Amel Souden, Lotfi Functional Analysis Rings and Algebras Representation Theory Spectral Theory 17B05, 22E65, 46A16, 46H70, 65L05 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 $1/(4\Cbracket)$. |
| title | Local Lie Theory in Quasi-Banach Lie Algebras: Convergence of the BCH Series and Geometric Implications |
| topic | Functional Analysis Rings and Algebras Representation Theory Spectral Theory 17B05, 22E65, 46A16, 46H70, 65L05 |
| url | https://arxiv.org/abs/2604.08265 |