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We prove an implicit function theorem for nilpotent (not necessarily associative or Lie) algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of the classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras. [ABSTRACT FROM AUTHOR]

Based on the concept of ontological states and their dynamical evolution by permutations, as assumed in the Cellular Automaton Interpretation (CAI) of quantum mechanics, we address the issue of whether quantum-classical hybrids can be described consistently in this framework. We consider chains of 'classical' two-state Ising spins and their discrete deterministic dynamics as an ontological model with an unitary evolution operator generated by pair-exchange interactions. A simple error mechanism is identified, which turns them into quantum mechanical objects: chains of qubits. Consequently, an interaction between a quantum mechanical and a 'classical' chain can be introduced and its consequences for this quantum-classical hybrid can be studied. We found that such hybrid character of composites, generally, does not persist under interactions and, therefore, cannot be upheld consistently, or even as a fundamental notion à la Kopenhagen interpretation, within CAI. [ABSTRACT FROM AUTHOR]

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations e σ i in the spin representation (a.k.a. Lorentz rotors) in terms of their generators σ i : ln (e σ 1 e σ 2 ) = tanh − 1 tanh σ 1 + tanh σ 2 + 1 2 [ tanh σ 1 , tanh σ 2 ] 1 + 1 2 { tanh σ 1 , tanh σ 2 } . This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension ≤ 4 , naturally generalizing Rodrigues' formula for rotations in ℝ 3 . In particular, it applies to Lorentz rotors within the framework of Hestenes' spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex 2 × 2 matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3 -velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle. [ABSTRACT FROM AUTHOR]

We introduce some general tools to design exact splitting methods to compute numerically semigroups generated by inhomogeneous quadratic differential operators. More precisely, we factorize these semigroups as products of semigroups that can be approximated efficiently, using, for example, pseudo-spectral methods. We highlight the efficiency of these new methods on the examples of the magnetic linear Schrödinger equations with quadratic potentials, some transport equations and some Fokker–Planck equations. [ABSTRACT FROM AUTHOR]

We get compact expressions for the Baker–Campbell–Hausdorff series Z = log (e X e Y) in terms of right-nested commutators. The reduction in the number of terms originates from two facts: (i) we use as a starting point an explicit expression directly involving independent commutators and (ii) we derive a complete set of identities arising among right-nested commutators. The procedure allows us to obtain the series with fewer terms than when expressed in the classical Hall basis at least up to terms of grade 10. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Based on the concept of ontological states and their dynamical evolution by permutations, as assumed in the Cellular Automaton Interpretation (CAI) of quantum mechanics, we address the issue of whether quantum-classical hybrids can be described consistently in this framework. We consider chains of ‘classical’ two-state Ising spins and their discrete deterministic dynamics as an ontological model with an unitary evolution operator generated by pair-exchange interactions. A simple error mechanism is identified, which turns them into quantum mechanical objects: chains of qubits. Consequently, an interaction between a quantum mechanical and a ‘classical’ chain can be introduced and its consequences for this quantum-classical hybrid can be studied. We found that such hybrid character of composites, generally, does not persist under interactions and, therefore, cannot be upheld consistently, or even as a fundamental notion à la Kopenhagen interpretation, within CAI.

The dynamics-from-permutations of classical Ising spins is generalized here for an arbitrarily long chain. This serves as an ontological model with discrete dynamics generated by pairwise exchange interactions defining the unitary update operator. The model incorporates a finite signal velocity and resembles in many aspects a discrete free field theory. We deduce the corresponding Hamiltonian operator and show that it generates an exact terminating Baker–Campbell–Hausdorff formula. Motivation for this study is provided by the Cellular Automaton Interpretation of Quantum Mechanics. We find that our ontological model, which is classical and deterministic, appears as if of quantum mechanical kind in an appropriate formal description. However, it is striking that (in principle arbitrarily) small deformations of the model turn it into a genuine quantum theory. This supports the view that quantum mechanics stems from an epistemic approach handling physical phenomena. [ABSTRACT FROM AUTHOR]

The dynamics-from-permutations of classical Ising spins are studied for a chain of four spins. We obtain the Hamiltonian operator which is equivalent to the unitary permutation matrix that encodes assumed pairwise exchange interactions. It is shown how this can be summarized by an exact terminating Baker-Campbell-Hausdorff formula, which relates the Hamiltonian to a product of exponentiated two-spin exchange permutations. We briefly comment upon physical motivation and implications of this study. [ABSTRACT FROM AUTHOR]

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 group of pseudo-differential operators on a finite abelian group. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of CP covers for groups to the class of Lie algebras, and show that despite the case of groups, all CP covers of a Lie algebra are isomorphic. Moreover we show that CP covers of groups and Lie rings which are in Lazard correspondence, are in Lazard correspondence too, and the Bogomolov multipliers of the group and the Lie ring are isomorphic. [ABSTRACT FROM AUTHOR]

In order to prepare for the introduction of dynamical many-body and, eventually, field theoretical models, we show here that quantum mechanical exchange interactions in a three-spin chain can emerge from the deterministic dynamics of three classical Ising spins. States of the latter form an ontological basis, which will be discussed with reference to the ontology proposed in the Cellular Automaton Interpretation of Quantum Mechanics by 't Hooft. Our result illustrates a new Baker–Campbell–Hausdorff (BCH) formula with terminating series expansion. [ABSTRACT FROM AUTHOR]

We give a recursive algorithm to compute the multivariable Zassenhaus formula e X 1 + X 2 + ⋯ + X n = e X 1 e X 2 ... e X n ∏ k = 2 ∞ e W k and derive an effective recursion formula of Wk. [ABSTRACT FROM AUTHOR]

We show that there are 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 ( A X + B Z + C Y + D I ) , derived in Matone (2015). 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 ( α Y ) exp ( ( 1 − α ) Y ) exp ( Z ) , with α fixed in such a way that it reduces to exp ( X ̃ ) exp ( Y ̃ ) , with X ̃ and Y ̃ satisfying the Van-Brunt and Visser condition [ X ̃ , Y ̃ ] = u ̃ X ̃ + v ̃ Y ̃ + c ̃ I . It turns out that e α satisfies, in the generic case, an algebraic equation whose exponents depend on the parameters defining the commutator algebra. In nine types of commutator algebras, such an equation leads to rational solutions for α . We find all the equations that characterize the solution of the above decomposition problem by combining it with the Jacobi identity. [ABSTRACT FROM AUTHOR]

In this paper 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 the Ado theorem for loops whose commutator–associator filtration is of finite length. [ABSTRACT FROM AUTHOR]

The usual time-ordering operation and the corresponding time-ordered exponential play a fundamental role in physics and applied mathematics. In this work, we study a new approach to the understanding of time-ordering relying on recent progress made in the context of enveloping algebras of pre-Lie algebras. Various general formulas for pre-Lie and Rota-Baxter algebras are obtained in the process. Among others, we recover the noncommutative analog of the classical Bohnenblust-Spitzer formula, and get explicit formulae for operator products of time-ordered exponentials. [ABSTRACT FROM AUTHOR]

The disentangled form of unitary operators is an indispensable tool for physical applications such as the study of squeezing properties or the time evolution of quantum systems. Here we derive a closed form disentanglement for the most general element of group ISp(2, R), whose generating Lie algebra is obtained by joining the Heisenberg-Weyl algebra to su(1,1). We attain the disentanglement formula resorting to an extension of the Truax method and check our findings through an independent factorization approach, based on the use of displacement operators. As a result we obtain a new form of factorized squeezing operators, whose action on the light vacuum state is calculated. • Analytical disentanglement of a general element of group ISp(2,R). • Group element as product of exponentials of the six algebra generators. • Extension to algebra isp(2,R) of the Truax method. • Check with a further method based on displacement operator. • Construction of generalized squeezing operator and squeezed states. [ABSTRACT FROM AUTHOR]

Let Lm,c he the free metaheUan 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 und continuous outer automorphisms of the completion Fm with respect to the format power series topology of the free metabelian Lie algebra Fm of rank m. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, LK(G), grad(ℓ)(LK(G)), grad(g)(exp LK(G)), and LK(G). Let c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let Fn(c) be the relatively free group of rank n in c. We prove that LK(Fn(c)) is relatively free in some variety of nilpotent Lie algebras, and LK(Fn(c)) ≅ LK(Fn(c)) ≅ grad(ℓ)(LK(Fn(c))) ≅ grad(g)(exp LK(Fn(c))) as Lie algebras in a natural way. Furthermore, Fn(c) is a Magnus nilpotent group. Let G1 and G2 be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G1 and G2 are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over  of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker-Campbell-Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ≅ L(H) ≅ L(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that LK and LK are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that L(G) is not isomorphic to L(G) as Lie algebras. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie group structure. Our point of departure is the decomposition of Lie algebra as the semidirect product of two Lie subspaces and an application of the Baker–Campbell–Hausdorff formula. Our results extend the results in Iserles and Zanna (2005) , Zanna and Munthe-Kaas(2001/02)  to a range of Lie groups: the Lie group of all solid motions in Euclidean space, the Lorentz Lie group of all solid motions in Minkowski space and the group of all invertible (upper) triangular matrices. In our method, the matrix exponential group can be computed by a less computational cost and is more accurate than the current methods. In addition, by this method the approximated matrix exponential belongs to the corresponding Lie group. [Copyright &y& Elsevier]

The Magnus series is an infinite series which arises in the study of linear ordinary differential equations. If the series converges, then the matrix exponential of the sum equals the fundamental solution of the differential equation. The question considered in this paper is: When does the series converge? The main result establishes a sufficient condition for convergence, which improves on several earlier results. [ABSTRACT FROM AUTHOR]

A closed expression to the Baker–Campbell–Hausdorff (B-C-H) formula in SO(4) is given by making use of the magic matrix by Makhlin. As far as we know this is the first nontrivial example on (semi–) simple Lie groups summing up all terms in the B-C-H expansion. [ABSTRACT FROM AUTHOR]

Two-forms in Minkowski space-time may be considered as generators of Lorentz transformations. Here, the covariant and general expression for the composition law (Baker–Campbell–Hausdorff formula) of two Lorentz transformations in terms of their generators is obtained. For simplicity, the expression is first obtained for complex generators, then translated to real ones. Every generator has two essential eigenvalues and two invariant (two–)planes; the eigenvalues and the invariant planes of the Baker–Campbell–Hausdorff composition of two generators are also obtained. [ABSTRACT FROM AUTHOR]

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). [Copyright &y& Elsevier]

We find a 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 R 3. [ABSTRACT FROM AUTHOR]