Sharp convergence domain of the Campbell–Hausdorff series over the real and complex numbers

The Campbell–Baker–Hausdorff (BCH) series expresses $\log(e^X e^Y)$ for noncommuting elements $X,Y$ of a Lie algebra as a series of iterated brackets, and its convergence near the origin is the analytic engine of local Lie theory. Serre's 'Lie Theory' chapter (Lecture Notes in Mathematics 1500) proves convergence over any complete valued field of characteristic zero on an explicit, dimension-dependent polydisc and records that the estimate is not sharp in the archimedean case, where the base field is $\mathbb R$ or $\mathbb C$. Later literature has settled the sharp cumulative convergence radius in associative normed settings: it is exactly $C_2=2.89847930...$ in general Banach algebras (Lakos), it is $\|X\|_2+\|Y\|_2<\pi$ with $\pi$ optimal for bounded Hilbert-space operators in the 2-norm (Casas–Murua), and the exact per-instance radius for complex matrices is effectively computable via eigenvalue collisions of $e^{\varepsilon X}e^{\varepsilon Y}$. In the normed Lie algebra setting, however, the natural home of the question for an abstract Lie algebra, only bounds are known ($2.93<C_2^{Lie}\le 5.4028570...$, with numerics suggesting about 5.15), and no explicit global (closed-form) description of the exact optimal convergence domain beyond cumulative norm balls — its shape and boundary — is known, even though the known per-ray eigenvalue-collision criterion determines that domain implicitly; the domain of the function represented by the series is known to differ genuinely from the convergence domain (analytic prolongation and monodromy phenomena). The problem is to determine this sharp archimedean convergence domain: the exact cumulative radius for normed real or complex Lie algebras, and an exact global description of the convergence domain for a fixed finite-dimensional real or complex Lie algebra.

Background

A Lie algebra is a vector space over a field equipped with an antisymmetric bilinear product $[x,y]$, the bracket, satisfying the Jacobi identity $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$; the basic example is the algebra of $n\times n$ matrices with $[X,Y]=XY-YX$. In an associative algebra the product of exponentials satisfies $e^X e^Y=e^{H(X,Y)}$ where $H$, the Campbell–Baker–Hausdorff (BCH) series, is a formal power series whose terms are iterated brackets (Dynkin's formula makes its coefficients explicit):

$$H(X,Y)=X+Y+[X,Y]/2+[X,[X,Y]]/12-[Y,[X,Y]]/12+\cdots$$

Because every term is a bracket, the series can be evaluated in any Lie algebra, not only in matrix algebras. Where it converges near the origin it defines a genuine local group law $X\circ Y=H(X,Y)$ with prescribed bracket, and this is the analytic engine behind Lie's Third Theorem (the statement that every finite-dimensional real or complex Lie algebra is the Lie algebra of a connected, simply connected Lie group): convergence of the BCH series reduces the integration of a Lie algebra to analysis.

The quantitative question is how large the convergence region is. Serre's chapter 'Lie Theory' in Lie Algebras and Lie Groups (1964 Harvard lectures; Lecture Notes in Mathematics 1500, second edition 1992, pp. 129–160) works over a complete valued field $k$ of characteristic zero — 'complete valued field' means a field with an absolute value $|\cdot|$ whose induced metric is complete; the archimedean cases are $k=\mathbb R$ and $k=\mathbb C$ with the usual absolute value, and the non-archimedean cases are $p$-adic fields, where the absolute value satisfies $|p|<1$. For a finite-dimensional Lie algebra $g$ of dimension $n$ over $k$, with basis $(e_i)$ and structure constants $c_{ij}^h$ defined by $[e_i,e_j]=\sum_h c_{ij}^h e_h$, the chapter proves (section 4, Theorem 2, by a Dynkin-style majorant argument) that the BCH series converges near the origin: after a rescaling of the basis that normalizes the moduli of the structure constants, the series converges on the polydisc of radius

$$R<\frac{a^3}{n^2}\cdot\frac{\sqrt2-1}{\sqrt2},$$

where a polydisc is the set on which every coordinate has modulus below the given bound, $a=1$ in the archimedean case, and $a=|p|^{1/(p-1)}$ in the $p$-adic case. The chapter records that this sufficient estimate is not sharp in the archimedean case, and the sharp size of the convergence region over $\mathbb R$ and $\mathbb C$ is left undetermined. The $p$-adic case is genuinely different, because there the factorial denominators of the BCH coefficients dominate the analysis.

The modern literature reformulates the question with norms. Give $g$ a norm with bracket constant $\mu$, meaning $\|[x,y]\|\le\mu\|x\|\,\|y\|$ for all $x,y$; every finite-dimensional Lie algebra admits such a norm, the natural normalization is $\mu=1$, and in a normed associative algebra the bracket constant is $\mu=2$. The cumulative norm of a pair is $\|X\|+\|Y\|$. A Banach algebra is a complete normed associative algebra, and a normed Lie algebra (also called a Banach–Lie algebra when complete) is a real or complex Lie algebra with a norm satisfying $\|[x,y]\|\le\|x\|\,\|y\|$. Uniform convergence questions reduce to the cumulative convergence radius of a class: the largest $\rho$ such that the series converges absolutely for all admissible $X,Y$ with $\|X\|+\|Y\|<\rho$. The continuous analogue is the Magnus expansion: the solution of the linear initial-value problem $U'(t)=A(t)U(t)$, $U(0)=1$, has the form $U(t)=e^{\Omega(t)}$ where $\Omega(t)$ is given by a Lie series (the Magnus expansion) whose special case for piecewise-constant $A$ is the BCH series; its convergence domain controls the validity of exponential integrators in numerical analysis. The basic divergence mechanism is spectral: if the BCH series converged at $(X,Y)$ in a matrix algebra, its sum would be a logarithm of $e^X e^Y$, so any obstruction to logarithms — in particular an eigenvalue collision of $e^X e^Y$, meaning that it acquires a repeated eigenvalue in a nontrivial Jordan block, where the matrix logarithm ceases to be analytic — forces divergence.

In normed associative settings the sharp archimedean picture is now essentially complete. Convergence of the BCH series for cumulative norm below 2 goes back to the classical estimates and their refinements by Newman, So and Thompson (1989) and Moan and Oteo (2001), as reviewed by Lakos. Moan and Niesen (2008) showed that $\pi$ is the critical threshold for the Magnus expansion on a Hilbert space. Casas and Murua (2009) proved that the BCH series converges for all bounded operators on a Hilbert space of dimension at least 2 with $\|X\|_2+\|Y\|_2<\pi$, where $\|\cdot\|_2$ is the spectral (operator) norm, the largest singular value; they showed this generic domain is optimal, and that for complex $n\times n$ matrices the exact radius along each ray $(\varepsilon X,\varepsilon Y)$ is effectively computable: it is the modulus of the smallest $\varepsilon$ at which $e^{\varepsilon X}e^{\varepsilon Y}$ undergoes a non-extraneous eigenvalue collision. Lakos determined the exact cumulative radius in general Banach algebras, $C_2=2.89847930...$, with convergence below it and counterexamples on it, together with an exact finiteness dichotomy for the majorant series; he proved the analogous $\pi$ result for C*-algebras (norm-closed self-adjoint algebras of bounded operators) in the 2-norm, and constructed finite-dimensional counterexamples of BCH type for every cumulative norm above 2 (necessarily in unbounded dimension).

What remains open sits precisely where the source question lives. In the normed Lie setting — a real or complex Lie algebra with $\|[x,y]\|\le\|x\|\,\|y\|$, the natural setting for the BCH series of an abstract finite-dimensional Lie algebra as in the source — the convergence of the BCH–Dynkin series here was studied early in the infinite-dimensional case (Biagi and Bonfiglioli 2013), but no sharp constant is known, only the bounds $2.93<C_2^{Lie}\le 2v_{Mi}=5.4028570...$, where $v_{Mi}$ is a constant defined in that work, with the exact value open (Lakos, Part III, version 3 of 2025; numerics from coefficients up to degree 16 suggest about 5.15). And beyond cumulative norm balls the exact optimal domain is unknown even in closed form for matrix algebras: Biagi, Bonfiglioli and Matone (2020) showed that the domain of convergence of the BCH series and the domain of the function it represents genuinely differ — there are matrix pairs where the series converges but its analytic prolongation (analytic continuation of the sum) is singular, and matrix pairs where a closed-form expression for the BCH function extends the local group law beyond the region where the series itself converges, a monodromy-type phenomenon in which the analytic continuation becomes multivalued around the eigenvalue-collision singularities.

Problem Statement

Determine the sharp (optimal) domain of convergence of the Campbell–Baker–Hausdorff series in the archimedean case, that is, over the base fields $\mathbb R$ and $\mathbb C$, the question left open by the recorded non-sharpness of the sufficient polydisc estimate in Serre's 'Lie Theory' chapter. Two linked precise targets, in the modern norm-based reformulation of that question, constitute its currently unresolved content. Together the two targets resolve the common objective, the exact archimedean convergence behavior of the series.

(1) Exact cumulative radius in the normed Lie setting. Call a real or complex Lie algebra with a norm a normed Lie algebra if $\|[x,y]\|\le\|x\|\,\|y\|$ holds for all $x,y$ (the natural normalization of the source's structure-constant assumption). Define $C_2^{Lie}$ as the supremum of the numbers $\rho>0$ such that the BCH series converges absolutely at $(X,Y)$ for every normed real or complex Lie algebra and all $X,Y$ with $\|X\|+\|Y\|<\rho$. Determine the exact value of $C_2^{Lie}$, currently only bounded by $2.93<C_2^{Lie}\le 5.4028570...$: this requires a proof that the series converges absolutely for all admissible pairs with cumulative norm below the claimed value, together with explicit divergence witnesses — for every $\varepsilon>0$, a concrete normed real or complex Lie algebra, finite- or infinite-dimensional (given by structure constants, by matrices with the stated norm, or by a completed free Banach–Lie-type construction), and elements $X,Y$ whose cumulative norm lies strictly between the claimed value and the claimed value plus $\varepsilon$, at which the series provably diverges. Since $C_2^{Lie}$ is defined over the combined real-and-complex class, it equals the smaller of the two field-specific radii.

(2) Exact convergence domain beyond cumulative norm balls. For a fixed finite-dimensional real or complex Lie algebra with a fixed norm (equivalently, in the source's coordinate formulation with basis and structure constants fixed), give an exact global description of the set of pairs $(X,Y)$ at which the BCH series converges absolutely, together with its boundary behavior. The description sought is an explicit closed-form condition — a finite expression in the algebra's fixed data (a basis with structure constants and the norm) and $(X,Y)$ — that decides, for each pair, whether the series converges absolutely. This is how the description goes beyond the known effective per-ray criterion: the exact radius along each ray $(\varepsilon X,\varepsilon Y)$ is computable in matrix algebras by the Casas–Murua non-extraneous eigenvalue-collision criterion, so that criterion already fixes the domain ray by ray for any fixed finite-dimensional Lie algebra via a faithful matrix representation, but it is a per-pair computation rather than a single closed-form condition in the algebra's fixed data. Here 'boundary behavior' means a determination, for boundary points of the convergence set, of whether the BCH series converges absolutely or diverges there. The known prolongation and monodromy phenomena — the prolongation domain of the local group law $X\circ Y=H(X,Y)$ can strictly contain the convergence domain (closed-form prolongations of a divergent series) and can also fail to extend it (convergent series with singular prolongation), as the matrix examples of Biagi, Bonfiglioli and Matone show — are background context for the boundary structure, not an additional deliverable.

An answer must establish optimality — for each target, matching convergence and divergence behavior on the two sides of its claimed boundary. The question concerns the archimedean fields $\mathbb R$ and $\mathbb C$ only; the $p$-adic case, where the factorial denominators dominate and the source's estimate has a different character, is out of scope.

References

1. Jean-Pierre Serre, Lie Theory, in: Lie Algebras and Lie Groups (1964 lectures given at Harvard University), Lecture Notes in Mathematics 1500, second edition, pp. 129–160, Springer, 1992. DOI: 10.1007/978-3-540-70634-2_12, https://doi.org/10.1007/978-3-540-70634-2_12
2. Fernando Casas and Ander Murua, An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications, Journal of Mathematical Physics 50 (2009) 033513. DOI: 10.1063/1.3078418, arXiv: 0810.2656, https://arxiv.org/abs/0810.2656
3. Gyula Lakos, Convergence estimates for the Magnus expansion I. Banach algebras, arXiv: 1709.01791 (version 5, 2025). https://arxiv.org/abs/1709.01791
4. Gyula Lakos, Convergence estimates for the Magnus expansion II. C*-algebras, arXiv: 1910.03328 (version 4, 2025). https://arxiv.org/abs/1910.03328
5. Gyula Lakos, Convergence estimates for the Magnus expansion III. Banach–Lie algebras, arXiv: 1910.03330 (version 3, 2025). https://arxiv.org/abs/1910.03330
6. Gyula Lakos, Convergence estimates for the Magnus expansion IE. Finite dimensional Banach algebras, arXiv: 2412.07886 (version 1, 2024). https://arxiv.org/abs/2412.07886
7. Stefano Biagi, Andrea Bonfiglioli and Marco Matone, On the Baker–Campbell–Hausdorff Theorem: non-convergence and prolongation issues, Linear and Multilinear Algebra 68 (2020) 1310–1328. DOI: 10.1080/03081087.2018.1540534, arXiv: 1805.10089, https://arxiv.org/abs/1805.10089
8. Stefano Biagi and Andrea Bonfiglioli, On the convergence of the Campbell–Baker–Hausdorff–Dynkin series in infinite-dimensional Banach–Lie algebras, Linear and Multilinear Algebra 62 (2014) 1591–1615. DOI: 10.1080/03081087.2013.839674, https://doi.org/10.1080/03081087.2013.839674
9. Morris Newman, Wasin So and Robert C. Thompson, Convergence domains for the Campbell-Baker-Hausdorff formula, Linear and Multilinear Algebra 24 (1989) 301–310. DOI: 10.1080/03081088908817923, https://doi.org/10.1080/03081088908817923
10. P. C. Moan and J. A. Oteo, Convergence of the exponential Lie series, Journal of Mathematical Physics 42 (2001) 501–508. DOI: 10.1063/1.1330198, https://doi.org/10.1063/1.1330198
11. Per Christian Moan and Jitse Niesen, Convergence of the Magnus Series, Foundations of Computational Mathematics 8 (2008) 291–301. DOI: 10.1007/s10208-007-9010-0, https://doi.org/10.1007/s10208-007-9010-0
12. Sergio Blanes and Fernando Casas, On the convergence and optimization of the Baker–Campbell–Hausdorff formula, Linear Algebra and its Applications 378 (2004) 135–158. DOI: 10.1016/j.laa.2003.09.010, https://doi.org/10.1016/j.laa.2003.09.010
13. W. T. van Est and Th. J. Korthagen, Non-Enlargible Lie Algebras, Indagationes Mathematicae 67 (1964) 15–31. DOI: 10.1016/S1385-7258(64)50003-7, https://doi.org/10.1016/S1385-7258(64)50003-7
14. Karl-Hermann Neeb, Towards a Lie theory of locally convex groups, Japanese Journal of Mathematics 1 (2006) 291–468. DOI: 10.1007/s11537-006-0606-y, https://doi.org/10.1007/s11537-006-0606-y
15. Helge Glöckner, Fundamentals of direct limit Lie theory, Compositio Mathematica 141 (2005) 1551–1577. DOI: 10.1112/s0010437x05001491, https://arxiv.org/abs/math/0403093
16. Christoph Wockel, Categorified central extensions, étale Lie 2-groups and Lie's Third Theorem for locally exponential Lie algebras, Advances in Mathematics 228 (2011) 2218–2257. DOI: 10.1016/j.aim.2011.07.003, arXiv: 0812.1673, https://arxiv.org/abs/0812.1673
17. Christoph Wockel and Chenchang Zhu, Integrating central extensions of Lie algebras via Lie 2-groups, Journal of the European Mathematical Society 18 (2016) 1273–1320. DOI: 10.4171/jems/613, https://doi.org/10.4171/jems/613
18. Christopher L. Rogers and Jesse Wolfson, Lie's Third Theorem for Lie ∞-Algebras, arXiv: 2409.08957 (first posted 2024; version 2, 2026). https://arxiv.org/abs/2409.08957
19. Nassim Athmouni, Explicit Baker–Campbell–Hausdorff Radii in Special Banach–Malcev Algebras of Shifts, arXiv: 2511.06357 (version 2, 2025). https://arxiv.org/abs/2511.06357
20. Nassim Athmouni, Mohsen Ben Abdallah, Mondher Damak, Marwa Ennaceur, Amel Jadlaoui and Lotfi Souden, Local Lie Theory in Quasi-Banach Lie Algebras: Convergence of the BCH Series and Geometric Implications, arXiv: 2604.08265 (version 1, 2026). https://arxiv.org/abs/2604.08265