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Explicit Baker--Campbell--Hausdorff Radii in Special Banach--Malcev Algebras of Shifts

Nassim Athmouni

TL;DR

This work addresses the analytic convergence of the Baker--Campbell--Hausdorff series in special Banach--Malcev algebras of shifts by embedding these algebras into Banach alternative algebras and imposing a bilinear bracket bound $\ \|[x,y]\|\le B\|x\|\|y\|$. It proves that the BCH series converges absolutely when $B(\|x\|+\|y\|)<1/(4K)$, with $K$ bounding the absolute BCH coefficients, via Catalan-number majorization, and identifies 1/(4KB) as a geometric analyticity radius for the induced Moufang loop. The paper then computes $B$ for a variety of models (operator-norm, exponential/polynomial weights, damped shifts, and the Zorn split-octonionic algebra), revealing explicit radii $\rho=1/(4KB)$ (often $1/(8K)$ in non-Lie cases) and highlighting the special assumption's central role. It also connects these analytic bounds to geometric interpretations and numerical stability of BCH-type integrators, and discusses refinements via spectral analysis and C*-algebra perspectives. Overall, the results provide the first explicit convergence criteria for BCH in non-associative Banach settings and offer a quantitative lens on non-associative symmetry models in mathematics and physics.

Abstract

We establish explicit convergence radii for the Baker--Campbell--Hausdorff (BCH) series in special Banach--Malcev algebras of shifts-those embeddable into a Banach alternative algebra. Under the continuity estimate $\|[x,y]\|\leq B\|x\|\|y\|$, the series converges absolutely whenever $B(\|x\|+\|y\|)<1/(4K)$, where $K\geq1$ bounds the absolute BCH coefficients. The constant $1/(4K)$ stems from a Catalan-number majorization and is sharp in the exponential-weight model. We compute $B$ explicitly for operator, exponential, polynomial, damped, and tree-like shift algebras, including the non-Lie split-octonionic (Zorn) algebra ($B=2$, $ρ=1/(8K)$). All results require the speciality assumption; the framework does not apply to general Malcev algebras. Geometrically, $ρ=1/(4KB)$ is the analyticity radius of the induced Moufang loop; numerically, it governs stability of BCH-type integrators.

Explicit Baker--Campbell--Hausdorff Radii in Special Banach--Malcev Algebras of Shifts

TL;DR

This work addresses the analytic convergence of the Baker--Campbell--Hausdorff series in special Banach--Malcev algebras of shifts by embedding these algebras into Banach alternative algebras and imposing a bilinear bracket bound . It proves that the BCH series converges absolutely when , with bounding the absolute BCH coefficients, via Catalan-number majorization, and identifies 1/(4KB) as a geometric analyticity radius for the induced Moufang loop. The paper then computes for a variety of models (operator-norm, exponential/polynomial weights, damped shifts, and the Zorn split-octonionic algebra), revealing explicit radii (often in non-Lie cases) and highlighting the special assumption's central role. It also connects these analytic bounds to geometric interpretations and numerical stability of BCH-type integrators, and discusses refinements via spectral analysis and C*-algebra perspectives. Overall, the results provide the first explicit convergence criteria for BCH in non-associative Banach settings and offer a quantitative lens on non-associative symmetry models in mathematics and physics.

Abstract

We establish explicit convergence radii for the Baker--Campbell--Hausdorff (BCH) series in special Banach--Malcev algebras of shifts-those embeddable into a Banach alternative algebra. Under the continuity estimate , the series converges absolutely whenever , where bounds the absolute BCH coefficients. The constant stems from a Catalan-number majorization and is sharp in the exponential-weight model. We compute explicitly for operator, exponential, polynomial, damped, and tree-like shift algebras, including the non-Lie split-octonionic (Zorn) algebra (, ). All results require the speciality assumption; the framework does not apply to general Malcev algebras. Geometrically, is the analyticity radius of the induced Moufang loop; numerically, it governs stability of BCH-type integrators.

Paper Structure

This paper contains 22 sections, 7 theorems, 74 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Functional Framework and Banach--Malcev Setting
  3. Convergence of the BCH Series
  4. Catalan Majorization and the General Convergence Theorem
  5. Toward a non-special framework?
  6. Explicit Computations of the Bracket Constant
  7. Bracket Constant Estimates in Weighted Shift Algebras
  8. Physical implications in exceptional symmetry models
  9. Refined convergence analysis and numerical validation
  10. Spectral refinement of the constant $B$
  11. Connection with C*-algebra theory
  12. Combinatorial, Geometric, and Numerical Interpretations
  13. Split-octonionic (Zorn) shift Malcev algebra (genuinely non-Lie).
  14. Technical Lemmas and Explicit Example
  15. A useful inequality
  16. ...and 7 more sections

Key Result

Lemma 2.2

Let $(w_n)_{n\ge0}$ be a weight sequence such that the induced bracket constants satisfy $B<\infty$ in eq:Bconst. Assume furthermore that the weight sequence grows at most exponentially (e.g. $w_n \le C\,\alpha^n$ for some $C,\alpha>0$). Then $(\mathfrak g,\|\cdot\|)$ is a Banach--Malcev algebra: it

Figures (6)

  • Figure 1: Full binary trees with $n=2,3,4$ leaves and their corresponding nested commutators $[x,y]$, $[[x,y],x]$, $[[[x,y],x],y]$.
  • Figure 2: The optimal analytic domain (red diamond) and the inscribed ball (blue circle) for $\rho = 1/(4KB)$.
  • Figure 3: Geometric domain of analyticity of the BCH product $x * y$. The radius $\rho = \frac{1}{4KB}$ acts as the local injectivity radius of the Moufang loop.
  • Figure 4: Numerical decay of the truncated BCH error following the Catalan bound. The analytic and numerical radii coincide at $r=\frac{1}{4KB}$.
  • Figure 5: Bracket-induced propagation in the weighted shift algebra: each commutator $[S^m,S^n]$ reaches degree $m+n$. Multiple pairs hit the same target (e.g. $S^4$ from $[S^1,S^3]$ and $[S^2,S^2]$).
  • ...and 1 more figures

Theorems & Definitions (37)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1: Arborescent majorization
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3: On the constant $K$
  • Corollary 3.4
  • Remark 3.5
  • ...and 27 more