Hausdorff Series Research Articles

A note on modified Hamiltonians for numerical integrations admitting an exact invariant

This paper aims to give some explanation and examples to show that numerical schemes applied to nonlinear problems can have convergent modified (or interpolating) Hamiltonians (MHs). It is pointed out, by means of specific and rather simple examples, that there are systems where the MH has a closed form. These systems are related to integrable mappings. For a special illustrative example given here the MHs, of the numerical approximations arising from the symplectic Euler and leap-frog methods, are presented in closed form. This MH is also given (in orders of the time step) by Yoshida's scheme (which uses in an essential way the Baker–Campbell–Hausdorff series). It is shown that all symplectic integrations of this example, which belong to the class presented by Yoshida, preserve the original energy of the system and admit closed form MHs. A second Hamiltonian example is also introduced. Upon application of the symplectic Euler method to this system, we obtain an integrable mapping which exactly conserves a biquadratic invariant. Hence it is also possible to derive a closed form for the MH in this case. In the light of these examples, the potential importance of integrable mappings to the field of symplectic integration is discussed.

Hopf algebra primitives in perturbation quantum field theory

The analysis of the combinatorics resulting from the perturbative expansion of the transition amplitude in quantum field theories, and the relation of this expansion to the Hausdorff series leads naturally to consider an infinite dimensional Lie subalgebra and the corresponding enveloping Hopf algebra, to which the elements of this series are associated. We show that in the context of these structures the power sum symmetric functionals of the perturbative expansion are Hopf primitives and that they are given by linear combinations of Hall polynomials, or diagrammatically by Hall trees. We show that each Hall tree corresponds to sums of Feynman diagrams each with the same number of vertices, external legs and loops. In addition, since the Lie subalgebra admits a derivation endomorphism, we also show that with respect to it these primitives are cyclic vectors generated by the free propagator, and thus provide a recursion relation by means of which the ( n+1)-vertex connected Green functions can be derived systematically from the n-vertex ones.

Hopf co-addition for free magma algebras and the non-associative Hausdorff series

Generalizations of the series exp and log to noncommutative non-associative and other types of algebras were considered by M. Lazard, and recently by V. Drensky and L. Gerritzen. There is a unique power series exp( x) in one non-associative variable x such that exp( x)exp( x)=exp(2 x), exp′(0)=1. We call the unique series H= H( x, y) in two non-associative variables satisfying exp( H)=exp( x)exp( y) the non-associative Hausdorff series, and we show that the homogeneous components H n of H are primitive elements with respect to the co-addition for non-associative variables. We describe the space of primitive elements for the co-addition in non-associative variables using Taylor expansion and a projector onto the algebra A 0 of constants for the partial derivations. By a theorem of Kurosh, A 0 is a free algebra. We describe a procedure to construct a free algebra basis consisting of primitive elements.

Taylor expansion of noncommutative power series with an application to the Hausdorff series

Taylor expansion of noncommutative power series with an application to the Hausdorff series

The Campbell- Hausdorff Series of Local Analytic Bruck Loops

The class of local analyitic Bruck loops (or equivalently K-loops) is strongly related to locally symmetric spaces. In particular, both have Lie triple systems as their tangent algebra. In this paper, we consider the existence and some properties of the Campbell-Hausdorff series of local analytic Bruck loops (K-loops). This formula can be used to determine the local symmetries of the associated symmetric space.

A simple expression for the terms in the Baker–Campbell–Hausdorff series

A simple expression is derived for the terms in the Baker–Campbell–Hausdorff series. One formulation of the result involves a finite number of operations with matrices of rational numbers. Generalizations are discussed.

Operator Formalism in the Theory of Effective Parameters of Plane Stratified Bianisotropic Structures

ABSTRACT A formalism based on the use of intrinsic representation of vectors and operators is employed for the analysis of effective properties of periodic plane stratified bianisotropic systems in a wide wave band. The effective tensors of permittivity, permeability and two pseudotensors of gyrotropy are found for the structures of the most general type in the long wavelength limit. While some interesting phenomena, form bianisotropy, for example, can not be explained within the framework of this approximation it is shown that in many instances the effects of form bianisotropy can be explained in terms of the theory of effective parameters. This is achieved by introducing suitable, for use in wide wave band, material tensors. The method employed is based on the approximate calculation of the characteristic matrix of the unit cell of the system with the help of Campbell Hausdorff series.

Quasinilpotent Banach–Lie Algebras Are Baker–Campbell–Hausdorff

It is proved that for a Banach–Lie algebraLthe Baker–Campbell–Hausdorff series converges for any paira, b∈Lif and only if for eachc∈Lthe adjoint operatoradcis quasi-nilpotent.

Relations among Lie-series transformations and isomorphisms between free Lie algebras

We study the subgroup generated by the exponentials of formal Lie series. We show three different ways to represent elements of this subgroup. These elements induce Lie-series transformations. Relations among these family of transformations furnish algorithms of composition. Starting from the Lazard elimination theorem and the Witt's formula, we show isomorphisms between some submodules of free Lie algebras. Combining different results, we also show that the homogeneous terms of the Hausdorff series H( a,b) freely generate the free Lie algebra L( a,b) without a line.

Some properties of the campbell baker hausdorff series

Some convergence properties for the Campbell-Baker-Hausdorff series for the logarithm of a product of exponentials are established.

Dynkin’s method of computing the terms of the Baker–Campbell–Hausdorff series

The infinite series for log(exp X exp Y) for noncommuting X and Y is expressible in terms of iterated commutators of X and Y except for the linear term X+Y. Dynkin derived an explicit expression for the terms as a sum of iterated commutators over a certain set of sequences. This paper presents a practical algorithm for applying Dynkin’s formula and gives several illustrative examples.

The generalization of the binomial theorem

As is well known, the binomial theorem is a classical mathematical relation that can be straightforwardly proved by induction or through a Taylor expansion, albeit it remains valid as long as [A,B]=0. In order to generalize such an important equation to cases where [A,B]≠0, an algebraic approach based on Cauchy’s integral theorem in conjunction with the Baker–Campbell–Hausdorff series is presented that allows a partial extension of the binomial theorem when the commutator [A,B]=c, where c is a constant. Some useful applications of the new proposed generalized binomial formula, such as energy eigenvalues and matrix elements of power, exponential, Gaussian, and arbitrary f(x̂) functions in the one-dimensional harmonic oscillator representation are given. The results here obtained prove to be consistent in comparison to other analytical methods.

On Sophus Lie's fundamental theorem

Sophus Lie's third theorem asserts that for every finite dimensional Lie algebra there is a local Lie group with the given algebra as tangent algebra at the identity. In particular, each subalgebra of the Lie algebra of a local Lie group determines a local subgroup whose Lie algebra is the given one. Recent interest in subsemigroups and local subsemigroups of Lie groups motivate the investigation to which extent local semigroups in Lie groups are determined by infinitesimal generators. It is known that the set L( S) of tangent vectors at the identity of a local semigroup S with identity in a (local) Lie group G is an additively closed convex subset of the Lie algebra L( G) which is topologically closed and contains 0; its largest vector subspace H is a subalgebra and all automorphisms of L( G) of the form e ad x with x ϵ H leave L( S) invariant as a whole (with (ad x)( y) = [ x, y]). We call an additively and topologically closed convex subset W containing 0 in a Lie algebra L a Lie wedge if it satisfies e ad x W = W for all x ϵ W ∩ t- W. In this paper we show that Lie wedges are precisely the infinitesimal generators of local semigroups in Lie groups. More precisely, we show that in a Banach Lie algebra L for every Lie wedge W whose edge W ∩ − W has a closed vector space complement in L, there is an open ball B around 0 in L such that the Cambell Hausdorff series x ∗ y = x + y + 1 2 [x, y] + ··· converges absolutely for x, y ϵ B and there is a subset S ⊂ B with (i) 0 ϵ S, (ii) (S ∗ S) ∩ B ⊂S, and L(S) = W .