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We give a new purely algebraic proof of the Baker–Campbell–Hausdorff theorem, which states that the homogeneous components of the formal expansion of log ( e A e B ) are Lie polynomials. Our proof is based on a recurrence formula for these components and a lemma that states that if under certain conditions a commutator of a non-commuting variable and a given polynomial is a Lie polynomial, then the given polynomial itself is a Lie polynomial.

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker–Campbell–Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler–Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

We study the Lie group structure of asymptotic symmetry groups in General Relativity from the viewpoint of infinite-dimensional geometry. To this end, we review the geometric definition of asymptotic simplicity and emptiness due to Penrose and the coordinate-wise definition of asymptotic flatness due to Bondi et al. Then we construct the Lie group structure of the Bondi--Metzner--Sachs (BMS) group and discuss its Lie theoretic properties. We find that the BMS group is regular in the sense of Milnor, but not real analytic. This motivates us to conjecture that it is not locally exponential. Finally, we verify the Trotter property as well as the commutator property. As an outlook, we comment on the situation of related asymptotic symmetry groups. In particular, the much more involved situation of the Newman--Unti group is highlighted, which will be studied in future work., 29 pages, article; minor revisions; version to appear in Classical and Quantum Gravity

In this paper we extend the notion of CP covers for groups to the field of Lie algebras, and show that despite the case of groups, all CP covers of a Lie algebra are isomorphic. Finally we show that CP covers of groups and Lie rings which are in Lazard correspondence, are in Lazard correspondence too, and the Bogomolov multipliers are isomorphic.

We consider a class of exponentials in the Weyl-Heisenberg algebra with exponents of type at most linear in coordinates and arbitrary functions of momenta. They are expressed in terms of normal ordering where coordinates stand to the left from momenta. Exponents appearing in normal ordered form satisfy differential equations with boundary conditions that could be solved perturbatively order by order. Two propositions are presented for the Weyl-Heisenberg algebra in 2 dimensions and their generalizations in higher dimensions. These results can be applied to arbitrary noncommutative spaces for construction of star products, coproducts of momenta and twist operators. They can also be related to the BCH formula.

Application of current and near-term quantum hardware to the electronic structure problem is highly limited by qubit counts, coherence times, and gate fidelities. To address these restrictions within the variational quantum eigensolver (VQE) framework, many recent contributions have suggested dressing the electronic Hamiltonian to include a part of electron correlation, leaving the rest to be accounted by VQE state preparation. We present a new dressing scheme that combines preservation of the Hamiltonian hermiticity and an exact quadratic truncation of the Baker-Campbell-Hausdorff expansion. The new transformation is constructed as the exponent of an involutory linear combination (ILC) of anti-commuting Pauli products. It incorporates important strong correlation effects in the dressed Hamiltonian and can be viewed as a classical preprocessing step alleviating the resource requirements of the subsequent VQE application. The assessment of the new computational scheme for electronic structure of the LiH, H$_2$O, and N$_2$ molecules shows significant increase in efficiency compared to conventional qubit coupled cluster dressings.

We give explicit formulae for differential graded Lie algebra (DGLA) models of 3-cells. In particular, for a cube and an $n$-faceted banana-shaped 3-cell with two vertices, $n$ edges each joining those two vertices and $n$ bi-gon 2-cells, we construct a model symmetric under the geometric symmetries of the cell fixing two antipodal vertices. The cube model is to be used in forthcoming work for discrete analogues of differential geometry on cubulated manifolds., 16 pages, 5 figures

We investigate some topics related to the celebrated Baker-Campbell-Hausdorff Theorem: a non-convergence result and prolongation issues. Given a Banach algebra $\mathcal{A}$ with identity $I$, and given $X,Y\in \mathcal{A}$, we study the relationship of different issues: the convergence of the BCH series $\sum_n Z_n(X,Y)$, the existence of a logarithm of $e^Xe^Y$, and the convergence of the Mercator-type series $\sum_n {(-1)^{n+1}}(e^Xe^Y-I)^n/n$ which provides a selected logarithm of $e^Xe^Y$. We fix general results and, by suitable matrix counterexamples, we show that various pathologies can occur, among which we provide a non-convergence result for the BCH series. This problem is related to some recent results, of interest in physics, on closed formulas for the BCH series: while the sum of the BCH series presents several non-convergence issues, these closed formulas can provide a prolongation for the BCH series when it is not convergent. On the other hand, we show by suitable counterexamples that an analytic prolongation of the BCH series can be singular even if the BCH series itself is convergent., Comment: 18pp Additional results and comments added

We give the normality, self-adjointness and spectral invariance of pseudo-differential operators on finite abelian groups. We also give a formula for the determinant of every element in a g...

We address the problem of constructing the non-associative version of the Dynkin form of the Baker-Campbell-Hausdorff formula; that is, expressing $\log (\exp (x)\exp(y))$, where $x$ and $y$ are non-associative variables, in terms of the Shestakov-Umirbaev primitive operations. In particular, we obtain a recursive expression for the Magnus expansion of the Baker-Campbell-Hausdorff series and an explicit formula in degrees smaller than 5. Our main tool is a non-associative version of the Dynkin-Specht-Wever Lemma. A construction of Bernouilli numbers in terms of binary trees is also recovered.

The well-known Baker–Campbell–Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product $$\text {e}^X \text {e}^Y$$ can be expressed in terms of iterated commutators of X and Y. This paper provides a gentle introduction to Ecalle’s mould calculus and shows how it allows for a short proof of the above result, together with the classical Dynkin (Dokl Akad Nauk SSSR (NS) 57:323–326, 1947) explicit formula for the logarithm, as well as another formula recently obtained by Kimura (Theor Exp Phys 4:041A03, 2017) for the product of exponentials itself. We also analyse the relation between the two formulas and indicate their mould calculus generalization to a product of more exponentials.

An exact representation of the Baker–Campbell–Hausdorff formula as a power series in just one of the two variables is constructed. Closed form coefficients of this series are found in terms of hyperbolic functions, which contain all of the dependence on the second variable. It is argued that this exact series may then be truncated and be expected to give a good approximation to the full expansion if only the perturbative variable is small. This improves upon existing formulae, which require both to be small. Several different representations are provided and emphasis is given to the situation where one of the matrices is diagonal, where a particularly easy to use formula is obtained.

Motivated by the physical applications ofq-calculus and ofq-deformations, the aim of this paper is twofold. Firstly, we prove theq-deformed analogue of the celebrated theorem by Baker, Campbell, and Hausdorff for the product of two exponentials. We deal with theq-exponential functionexpq⁡(x)=∑n=0∞(xn/[n]q!), where[n]q=1+q+⋯+qn-1denotes, as usual, thenthq-integer. We prove that ifxandyare any noncommuting indeterminates, thenexpq⁡(x)expq⁡(y)=expq⁡(x+y+∑n=2∞Qn(x,y)), whereQn(x,y)is a sum of iteratedq-commutators ofxandy(on the right and on the left, possibly), where theq-commutator[y,x]q≔yx-qxyhas always the innermost position. When[y,x]q=0, this expansion is consistent with the known result by Schützenberger-Cigler:expq⁡(x)expq⁡(y)=expq⁡(x+y). Our result improves and clarifies some existing results in the literature. Secondly, we provide an algorithmic procedure for obtaining identities between iteratedq-commutators (of any length) ofxandy. These results can be used to obtain simplified presentation for the summands of theq-deformed Baker-Campbell-Hausdorff Formula.

The paper proposes, for the first time, a closed form of the Baker&ndash, Campbell&ndash, Hausdorff&ndash, Dynkin (BCHD) formula in the particular case of the Lie algebra of rigid body displacements. For this purpose, the structure of the Lie group of the rigid body displacements S E ( 3 ) and the properties of its Lie algebra se ( 3 ) are used. In addition, a new solution to this problem in dual Lie algebra of dual vectors is delivered using the isomorphism between the Lie group S E ( 3 ) and the Lie group of the orthogonal dual tensors.

We show that there are {\it 13 types} of commutator algebras leading to the new closed forms of the Baker-Campbell-Hausdorff (BCH) formula $$\exp(X)\exp(Y)\exp(Z)=\exp({AX+BZ+CY+DI}) \ , $$ derived in arXiv:1502.06589, JHEP {\bf 1505} (2015) 113. This includes, as a particular case, $\exp(X) \exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$. The algorithm exploits the associativity of the BCH formula and is based on the decomposition $\exp(X)\exp(Y)\exp(Z)=\exp(X)\exp({\alpha Y}) \exp({(1-\alpha) Y}) \exp(Z)$, with $\alpha$ fixed in such a way that it reduces to $\exp({\tilde X})\exp({\tilde Y})$, with $\tilde X$ and $\tilde Y$ satisfying the Van-Brunt and Visser condition $[\tilde X,\tilde Y]=\tilde u\tilde X+\tilde v\tilde Y+\tilde cI$. It turns out that $e^\alpha$ satisfies, in the generic case, an algebraic equation whose exponents depend on the parameters defining the commutator algebra. In nine {\it types} of commutator algebras, such an equation leads to rational solutions for $\alpha$. We find all the equations that characterize the solution of the above decomposition problem by combining it with the Jacobi identity., Comment: 15 pages. Typos corrected. JGP version

The time-dependent Schr¨ odinger equation is solved for a linear potential using operational methods ; in particular, an extension of the Baker-Campbell-Hausdorff formula is exposed and used. Several initial conditions are considered. A closed form for the Wigner function is presented. The results can be extended to the propagation of an electromagnetic field in the paraxial approximation. odinger equation, linear potential, Zassenhaus formula, Baker-Campbell-Hausdorff formula, Wigner function

In this note we establish several basic properties of nilpotent Sabinin algebras. Namely, we show that nilpotent Sabinin algebras (1) can be integrated to produce nilpotent loops, (2) satisfy an analogue of the Ado theorem, (3) have nilpotent Lie envelopes. We also give a new set of axioms for Sabinin algebras. These axioms reflect the fact that a complementary subspace to a Lie subalgebra in a Lie algebra is a Sabinin algebra. Finally, we note that the non-associative version of the Jennings theorem produces a version of Ado theorem for loops whose commutator-associator filtration is of finite length., Comment: 17 pages, no figures

As an application of the BCH-formula, order conditions for splitting schemes are derived. The same conditions can be obtained by using non-commutative power series techniques and inspecting the coefficients of Lyndon–Shirshov words.

The famous Baker–Campbell–Hausdorff series defines a group composition on the set of the so called grouplike elements of a completed free Lie algebra as we can find in P. Cartier’s paper on the Campbell–Hausdorff formula in 1956. According to A. Baider and R.C. Churchill, we call this group the Campbell–Hausdorff group over the alphabet X of the free Lie algebra. By a certain coproduct and certain bialgebra endomorphisms of the free associative algebra over a set X which should be at most countably infinite, we get an alternative realisation of this Campbell–Hausdorff group. This realisation is much more comfortable to deal with than the classical one. The usual composition of endomorphisms gives a near-ring structure to this group. In this paper we consider a metric on the Campbell–Hausdorff near-ring over the alphabet N and the compatibility of this metric with the near-ring compositions and we make some topological remarks.

In this work we introduce the contact Heisenberg algebra which is the restriction of the Jacobi algebra on contact manifolds to the linear and constant functions. We give the exact expression of its corresponding Baker-Campbell-Hausdorff formula. We argue that this result is relevant to the quantization of contact systems., 8 pages, version 2 with major corrections and improved results

Let Lm,c be the free metabelian nilpotent of class c Lie algebra of rank m over a field K of characteristic 0. We describe the groups of inner and outer automorphisms of Lm,c. To obtain this result, we first describe the groups of inner and continuous outer automorphisms of the completion Fm with respect to the formal power series topology of the free metabelian Lie algebra Fm of rank m. © Taylor & Francis Group, LLC. Council for Higher Education The research of the second named author was partially supported by the Council of Higher Education (YÖK) in Turkey. He is also very grateful to the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences for the creative atmosphere and the warm hospitality during his visit when most of this project was carried out.

The Lazard correspondence establishes an equivalence of categories between p-groups of nilpotency class less than p and nilpotent Lie rings of the same class and order. The main tools used to achieve this are the Baker–Campbell–Hausdorff formula and its inverse formulae. Here we describe methods to compute the inverse Baker–Campbell–Hausdorff formulae. Using these we get an algorithm to compute the Lie ring structure of a p-group of class

We consider products of matrix exponentials under the assumption that the matrices span a nilpotent Lie algebra. In 1995, Gurvits conjectured that nilpotency implies that these products are, in some sense, simple. More precisely, there exists a uniform bound l such that any product can be represented as a product of no more than I matrix exponentials. This conjecture has important applications in the analysis of linear switched systems, as it is closely related to the problem of reachability using a uniformly bounded number of switches. It is also closely related to the concept of nice reachability for bilinear control systems. The conjecture is trivially true for the case of first-order nilpotency. Gurvits proved the conjecture for the case of second-order nilpotency using the Baker-Campbell-Hausdorff formula. We show that the conjecture is false for the third-order nilpotent case using an explicit counterexample. Yet, the underlying philosophy behind Gurvits' conjecture is valid in the case of third-order nilpotency. Namely, such systems do satisfy the following nice reachability property: any point in the reachable set can be reached using a piecewise constant control with no more than four switches. We show that even this form of finite reachability is no longer true for the case of fifth-order nilpotency.

The Baker-Campbell-Hausdorff series computes the quantity \begin{equation*} Z(X,Y)=\ln\left( e^X e^Y \right) = \sum_{n=1}^\infty z_n(X,Y), \end{equation*} where $X$ and $Y$ are not necessarily commuting, in terms of homogeneous multinomials $z_n(X,Y)$ of degree $n$. (This is essentially equivalent to computing the so-called Goldberg coefficients.) The Baker-Campbell-Hausdorff series is a general purpose tool of wide applicability in mathematical physics, quantum physics, and many other fields. The Reinsch algorithm for the truncated series permits one to calculate up to some fixed order $N$ by using $(N+1)\times(N+1)$ matrices. We show how to further simplify the Reinsch algorithm, making implementation (in principle) utterly straightforward. This helps provide a deeper understanding of the Goldberg coefficients and their properties. For instance we establish strict bounds (and some equalities) on the number of non-zero Goldberg coefficients. Unfortunately, we shall see that the number of terms in the multinomial $z_n(X,Y)$ often grows very rapidly (in fact exponentially) with the degree $n$. We also present some closely related results for the symmetric product \begin{equation*} S(X,Y)=\ln\left( e^{X/2} e^Y e^{X/2} \right) = \sum_{n=1}^\infty s_n(X,Y). \end{equation*} Variations on these themes are straightforward. For instance, one can just as easily consider the series \begin{equation*} L(X,Y)=\ln\left( e^{X} e^Y e^{-X} e^{-Y}\right) = \sum_{n=1}^\infty \ell_n(X,Y). \end{equation*} This type of series is of interest, for instance, when considering parallel transport around a closed curve. Several other related series are investigated., v1: 23 pages. v2: Minor edits to fix some annoying typos; minor change of terminology; no scientific changes. v3: Now 21 pages plus 7 pages supplementary material. Some corrections, changes in presentation. This version more closely resembles the published article

The Baker-Campbell-Hausdorff formula is a general result for the quantity $Z(X,Y)=\ln( e^X e^Y )$, where $X$ and $Y$ are not necessarily commuting. For completely general commutation relations between $X$ and $Y$, (the free Lie algebra), the general result is somewhat unwieldy. However in specific physics applications the commutator $[X,Y]$, while non-zero, might often be relatively simple, which sometimes leads to explicit closed form results. We consider the special case $[X,Y] = u X + vY + cI$, and show that in this case the general result reduces to \[ Z(X,Y)=\ln( e^X e^Y ) = X+Y+ f(u,v) \; [X,Y]. \] Furthermore we explicitly evaluate the symmetric function $f(u,v)=f(v,u)$, demonstrating that \[ f(u,v) = {(u-v)e^{u+v}-(ue^u-ve^v)\over u v (e^u - e^v)}, \] and relate this to previously known results. For instance this result includes, but is considerably more general than, results obtained from either the Heisenberg commutator $[P,Q]=-i\hbar I$ or the creation-destruction commutator $[a,a^\dagger]=I$., V1: 5 pages. V2: 4 references added, some minor typos fixed, some discussion added. No change in conclusions. Now 6 pages. This version accepted for publication in Journal of Physics A: Mathematical and Theoretical

In this chapter, the stabilization of nonlinear impulsive systems under time-dependent switching control is investigated. In the open-loop approach, the switching rule is programmed in advance and the switched system is composed entirely of unstable subsystems. Sufficient conditions are found that establish the existence of stabilizing time-dependent switching rules using the Campbell–Baker–Hausdorff formula and Lyapunov stability theory.

This article presents basic notions of Lie theory in the context of matrix groups with goals of minimizing the required mathematical background and maximizing accessibility. It is structured with exercises that enhance the text and make the notes suitable for (part of) an introductory course at the upper level undergraduate or early graduate level. Indeed the notes were originally written as part of an introductory course to geometric control theory.

Consider three elementary results from the preceding chapters of this book: (1) Every matrix Lie group G has a Lie algebra \(\mathfrak{g}\). (2) A continuous homomorphism \(\Phi\) between matrix Lie groups G and H gives rise to a Lie algebra homomorphism \(\phi : \mathfrak{g} \rightarrow \mathfrak{h}\). (3) If G and H are matrix Lie groups and H is a subgroup of G, then the Lie algebra \(\mathfrak{h}\) of H is a subalgebra of the Lie algebra \(\mathfrak{g}\) of G. Each of these results goes in the “easy” direction, from a group notion to an associated Lie algebra notion. In this chapter, we attempt to go in the “hard” direction, from the Lie algebra to the Lie group. We will investigate three questions relating to the preceding three theorems.

We present an approach that allows performing computations related to the Baker-Campbell-Haussdorff (BCH) formula and its generalizations in an arbitrary Hall basis, using labeled rooted trees. In particular, we provide explicit formulas (given in terms of the structure of certain labeled rooted trees) of the continuous BCH formula. We develop a rewriting algorithm (based on labeled rooted trees) in the dual Poincare-Birkhoff-Witt (PBW) basis associated to an arbitrary Hall set, that allows handling Lie series, exponentials of Lie series, and related series written in the PBW basis. At the end of the paper we show that our approach is actually based on an explicit description of an epimorphism ν of Hopf algebras from the commutative Hopf algebra of labeled rooted trees to the shuffle Hopf algebra and its kernel ker ν.

In this paper, we consider a queueing system with postservice activity. During the time when the server is engaged in the postservice activity (wrap-up time), the waiting customer, if any, cannot receive his or her service. This type of queueing system has been used to model automatic call distribution (ACD) systems. We consider the waiting time distribution of the queueing system. Using the Markovian point process that can be expressed by the so-called Markovian arrival process (MAP), we derive the waiting time distribution in terms of the representing matrices of a particular MAP. Then we apply the Baker-Hausdorff lemma to the matrices and derive the conditional waiting time distribution in closed form by exploiting the specific structure of the matrices. As a byproduct, we give an explicit solution of the number of arrivals for the MAP.

We estimate Grobner-Shirshov bases for the Lie algebra An given arbitrary orders of generators (nodes of a Dynkin graph). Previously, the Grobner-Shirshov basis was computed in [1] for the particular case where nodes of the Dynkin graph are ordered successively.

In this paper, we consider the counting process for a class of Markovian arrival processes (MAPs). We assume that the representing matrices in such MAPs are expanded in terms of matrix representations of the standard generators in the Lie algebra of the special linear group. The primary purpose of this paper is to construct an explicit solution of the time-dependent distribution and factorial moments of the number of arrival events in (0,t] of the counting process for this class of MAPs. Our construction relies on the Baker--Hausdorff lemma and the specific structure of the representing matrices. To investigate the efficiency of CPU usage with the explicit solution, we have conducted numerical experiments on computing the time-dependent distribution of the counting process through the explicit solution and uniformization-based method. We show that the CPU times required to compute the time-dependent distribution of the number of arrival events in (0,t] through the explicit solution have little sensitivity to t, while the consumption of CPU times with the uniformization-based method becomes greater as t increases. For illustrative purposes, we present a system performance analysis of a queueing system for possible use in automatic call distribution (ACD) systems. As an application of the explicit solution, we use it to express the waiting time distribution of the queueing system. Some numerical examples are also given with comparisons to computer simulations.

We present an extrapolation algorithm for the integration of differential equations in Lie groups which is a suitable generalization of the well-known GBS-algorithm for ODEs. Sufficiently accurate approximations to the BCH-formula are required to reach a given order. We give such approximations with a minimized number of commutators.

The aim of this paper is to obtain an explicit expression, instead of using a recursive method, for the nth term coefficient of the generalized Baker–Campbell–Hausdorff–Dynkin (gBCHD) formula. The gBCHD formula has been applied to control theory, specially to nonholonomic motion planning.

We prove that there exists no maximal BCH-invertible subalgebra of P(n) that can be imagined as and called skew with respect to Z -gradation of P(n).

We derive a closed formula for the Baker–Campbell–Hausdorff series expansion in the case of complex matrices. For arbitrary matrices A and B, and a matrix Z such that , our result expresses Z as a linear combination of A and B, their commutator , and the identity matrix I. The coefficients in this linear combination are functions of the traces and determinants of A and B, and the trace of their product. The derivation proceeds purely via algebraic manipulations of the given matrices and their products, making use of relations developed here, based on the Cayley–Hamilton theorem, as well as a characterization of the consequences of and/or its determinant being zero or otherwise. As a corollary of our main result we also derive a closed formula for the Zassenhaus expansion. We apply our results to several special cases, most notably the parametrization of the product of two matrices and a verification of the recent result of Van-Brunt and Visser (2015 J. Phys. A: Math. Theor. 48 225207) for complex matrices, in this latter case deriving also the related Zassenhaus formula which turns out to be quite simple. We then show that this simple formula should be valid for all matrices and operators.

We find local closed-form expression for the Baker—Campbell—Hausdorff formula in the Lie algebra so(3), and interpret the formula geometrically in terms of rotation vectors in ℝ3.

Motivated by the search for solutions of the quantum Yang–Baxter equation, an algebraic theory of quantum stochastic product integrals is developed. The product integrators are formal power series in an indeterminate h whose coefficients are elements of the Lie algebra L labelling the usual integrators of a many-dimensional quantum stochastic calculus. The product integrals are also formal power series in h, whose coefficients are finite iterated additive stochastic integrals which act on the exponential domain in the Fock space of the calculus and which represent elements of the universal enveloping algebra U of L. They obey a multiplication rule suggested by the quantum Ito product formula, and are characterized among all such formal power series by a grouplike property.

An exponential representation of the Jordanian matrix quantum group GLh (2) is constructed in explicit form by the generators of the classical Lie algebra sl * (2,C ).

Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the Baker–Campbell–Hausdorff formula and the recently developed Lie group methods for integration of differential equations on manifolds. This paper is concerned with complexity and optimization of such computations in the general case where the Lie algebra is free , i.e. no specific assumptions are made about its structure. It is shown how transformations applied to the original variables of a problem yield elements of a graded free Lie algebra whose homogeneous subspaces are of much smaller dimension than the original ungraded one. This can lead to substantial reduction of the number of commutator computations. Witt's formula for counting commutators in a free Lie algebra is generalized to the case of a general grading. This provides good bounds on the complexity. The interplay between symbolic and numerical computations is also discussed, exemplified by the new MATLAB toolbox ‘DIFFMAN’

In the present paper we generalize the lifting formulas from [Sh] and obtain the formula for the (2n+2l-1)-cocycle on the LIe algebra of differential operators on a n-dimensional space for arbitrary n and l., Comment: 10 pages, 8 Postscript figures, LaTeX2e

The functional Ito formula, in the form $df({\bf \Lambda})=f({\bf \Lambda} +d{\bf \Lambda})-f({\bf \Lambda} )$ associated with a quantum (non-commutative) stochastic calculus. Here is an element of the universal enveloping algebra is given a meaning using the coproduct structure of even though the individual terms of this expression have no meaning. The Ito formula is equivalent to a chaotic expansion formula for which is found explicitly. 1991 Mathematics Subject Classification: primary 81S25; secondary 60H05; tertiary 18B25.

It is proved that for a Banach–Lie algebra L the Baker–Campbell–Hausdorff series converges for any pair a , b ∈ L if and only if for each c ∈ L the adjoint operator a dc is quasi-nilpotent.

A new scheme for deriving compact expressions of the logarithm of the exponential product is proposed and it is applied to several exponential product formulas. A generalization of the Dynkin–Specht–Wever (DSW) theorem on free Lie elements is given, and it is used to study the relation between the traditional method (based on the DSW theorem) and the present new scheme. The concept of the operator functional derivative is also proposed, and it is applied to ordered exponentials, such as time-evolution operators for time-dependent Hamiltonians.

After the torch of Anders Kock [Taylor series calculus for ring objects of line type, Journal of Pure and Applied Algebra, 12 (1978), 271-293], we will establish the Baker-Campbell-Hausdorff formula as well as the Zassenhaus formula in the theory of Lie groups.

This paper has two parts. The first part is a review and extension of the methods of integration of Leibniz algebras into Lie racks, including as new feature a new way of integrating 2-cocycles (see Lemma 3.9). In the second part, we use the local integration of a Leibniz algebra h using a Baker-Campbell-Hausdorff type formula in order to deformation quantize its linear dual h^*. More precisely, we define a natural rack product on the set of exponential functions which extends to a rack action on C^{\infty}(h^*)., Comment: 37 pages, added explicit computation of the first term of the deformation, i.e. the bracket which is to be quantized