Lie Bracket Research Articles

Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection

For an oriented 2-dimensional manifold Σ of genus g with n boundary components, the space Cπ1(Σ)/[Cπ1(Σ),Cπ1(Σ)] carries the Goldman–Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded Lie bialgebra (under the natural filtration) is described by cyclic words in H1(Σ) and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [13] using Kontsevich integrals and in [2] using solutions of the Kashiwara–Vergne problem.In this note, we give an elementary proof of this isomorphism over C. It uses the Knizhnik–Zamolodchikov connection on C\\{z1,…zn}. We show that the isomorphism naturally depends on the complex structure on the surface. The proof of the isomorphism for Lie brackets is a version of the classical result by Hitchin [9]. Surprisingly, it turns out that a similar proof applies to cobrackets.Furthermore, we show that the formality isomorphism constructed in this note coincides with the one defined in [2] if one uses the solution of the Kashiwara–Vergne problem corresponding to the Knizhnik–Zamolodchikov associator.

Iterated Lie brackets for nonsmooth vector fields

If the vector fields $$f_1, f_2$$ are locally Lipschitz, the classical Lie bracket $$[f_1,f_2]$$ is defined only almost everywhere. However, it has been shown that, by means of a set-valued Lie bracket $$[f_1,f_2]_{set}$$ (which is defined everywhere), one can generalize classical results like the Commutativity theorem and Frobenius’ theorem, as well as a Chow–Rashevski’s theorem involving Lie brackets of degree 2 (we call ‘degree’ the number of vector fields contained in a formal bracket). As it might be expected, these results are consequences of the validity of an asymptotic formula similar to the one holding true in the regular case. Aiming to more advanced applications—say, a general Chow–Rashevski’s theorem or higher order conditions for optimal controls—we address here the problem of defining, for any $$m>2$$ and any formal bracket B of degree m, a Lie bracket $$B(f_1,\ldots , f_m)$$ corresponding to vector fields $$(f_1,\ldots , f_m)$$ lacking classical regularity requirements. A major complication consists in finding the right extension of the degree 2 bracket, namely a notion of bracket which admits an asymptotic formula. In fact, it is known that a mere iteration of the construction performed for the case $$m=2$$ is not compatible with the validity of an asymptotic formula. We overcome this difficulty by introducing a set-valued bracket $$x\mapsto B_{set}(f_1,\ldots , f_m)(x)$$ , defined at each point x as the convex hull of the set of limits along suitable d-tuples of sequences of points converging to x. The number d depends only on the formal bracket B and is here called the diff-degree of B. It counts the maximal order of differentiations involved in $$B(g_1,\ldots , g_m)$$ (for any smooth m-tuple of vector fields $$(g_1,\ldots , g_m)$$ ). The main result of the paper is an asymptotic formula valid for the bracket $$B_{set}(f_1,\ldots , f_m)(x)$$ .

Classical Affine W-Superalgebras via Generalized Drinfeld–Sokolov Reductions and Related Integrable Systems

The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel'd-Sokolov (D-S) reduction associated to a Lie superalgebra $g$ and its even nilpotent element $f$, and we find a new definition of the classical affine W-superalgebra $W(g,f,k)$ via the D-S reduction. This new construction allows us to find free generators of $W(g,f,k)$, as a differential superalgebra, and two independent Lie brackets on $W(g,f,k)/\partial W(g,f,k).$ Moreover, we describe super-Hamiltonian systems with the Poisson vertex algebras theory. A W-superalgebra with certain properties can be understood as an underlying differential superalgebra of a series of integrable super-Hamiltonian systems.

Stable solutions of symmetric systems involving hypoelliptic operators

Let X and Y be two noncommuting vector fields in an open set Ω in a manifold M equipped with a sub-Riemannian structure. We examine stable solutions of the following symmetric systemΔXYui=Hi(u1,⋯,um)in Ω for 1≤i≤m, when the operator ΔXY is the Hörmander's operator given by ΔXY(⋅):=X(X⋅)+Y(Y⋅) and Hi∈C1(Rm). We prove the following identity for any w∈C2(Ω)|∇XYXw|2+|∇XYYw|2−|X|∇XYw||2−|Y|∇XYw||2={|∇XYw|2[A2+B2]in {|∇XYw|>0}∩Ω,0a.e.in {|∇XYw|=0}∩Ω, where A is the intrinsic curvature of the level sets of w and B is connected with the intrinsic normal and the intrinsic tangent direction to the level sets and also with the Lie bracket [X,Y]. We then apply this to establish a geometric Poincaré inequality for stable solutions of the above system for general vector fields X and Y. This inequality enables us to analyze the level sets of stable solutions. In addition, we provide certain reduction of dimensions results which can be regarded as counterparts of the classical De Giorgi type results. This is remarkable since the classical one-dimensional symmetry results do not hold for general vector fields. Our approaches can be applied, but not limited, to the Grushin vector fields X=(1,0) and Y=(0,x) in R2 and the Heisenberg vector fields X=(1,0,−y2) and Y=(0,1,x2) in R3 and their multidimensional extensions. These specific vector fields generate nonelliptic operators which are hypoelliptic.

Asymptotics for the normalized error of the Ninomiya–Victoir scheme

In Gerbi et al. (2016) we proved strong convergence with order 1∕2 of the Ninomiya–Victoir scheme XNV,η with time step T∕N to the solution X of the limiting SDE. In this paper we check that the normalized error defined by NX−XNV,η converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher random variables η. This result can be seen as a first step to adapt to the Ninomiya–Victoir scheme the central limit theorem of Lindeberg Feller type, derived in Ben Alaya and Kebaier (2015) for the multilevel Monte Carlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes. This suggests that the rate of convergence is greater than 1∕2 in this case and we actually prove strong convergence with order 1 and study the limit of the normalized error NX−XNV,η. The limiting SDE involves the Lie brackets between the Brownian vector fields and the Stratonovich drift vector field. When all the vector fields commute, the limit vanishes, which is consistent with the fact that the Ninomiya–Victoir scheme coincides with the solution to the SDE on the discretization grid.

Steering Control for a Rigid Body with two Torque Actuators using Adaptive Back Stepping

This paper presents a simple steering control algorithm for a rigid body model, which is a famous example of non-holonomic control systems with drift. The controllability Lie Algebra of a rigid body model contains Lie brackets of depth two. We propose a back-stepping-based adaptive controller design under the strict-feedback form. We analyze two cases for continuous steering. In the first case, the parameters of the model are assumed to be known while in the second case these are estimated by considering them unknown. This approach does not necessitate the conversion of the system model into a “chained form”, and thus does not rely on any special transformation techniques. The practical effectiveness of the controller is illustrated by numerical simulations and graceful stabilization.

Central extensions of the algebra of formal pseudo-differential symbols via Hochschild (co)homology and quadratic symplectic Lie algebras

We describe the space of central extensions of the associative algebra Ψn of formal pseudo-differential symbols in n≥1 independent variables using Hochschild (co)homology groups: we prove that the first Hochschild (co)homology group HH1(Ψn) is 2n-dimensional and we use this fact to calculate the first Lie (co)homology group HLie1(Ψn) of Ψn equipped with the Lie bracket induced by its associative algebra structure. As an application, we use our calculations to provide examples of infinite-dimensional quadratic symplectic Lie algebras.

Comparison morphisms between two projective resolutions of monomial algebras

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

Comparison morphisms between two projective resolutions of monomial algebras

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

Iterative Learning and Extremum Seeking for Repetitive Time-Varying Mappings

In this paper, we develop an extremum seeking control method integrated with iterative learning control to track a time-varying optimizer within finite time. The behavior of the extremum seeking system is analyzed via an approximating system - the modified Lie bracket system. The modified Lie bracket system is essentially an online integral-type iterative learning control law. The paper contributes to two fields, namely, iterative learning control and extremum seeking. First, an online integral type iterative learning control with a forgetting factor is proposed. Its convergence is analyzed via $k$-dependent (iteration- dependent) contraction mapping in a Banach space equipped with $\lambda$-norm. Second, the iterative learning extremum seeking system can be regarded as an iterative learning control with "control input disturbance." The tracking error of its modified Lie bracket system can be shown uniformly bounded in terms of iterations by selecting a sufficiently large frequency. Furthermore, it is shown that the tracking error will finally converge to a set, which is a $\lambda$-norm ball. Its center is the same with the limit solution of its corresponding "disturbance-free" system (the iterative learning control law); and its radius can be controlled by the frequency.

Unique Normal Form for a Class of Three-Dimensional Nilpotent Vector Fields

In this paper, the unique normal form for a class of three-dimensional nilpotent vector fields with symmetry is investigated. The key technique used is a combination of multiple Lie brackets, linear grading function, new notations of block matrices and first integral of the linear part of the given vector field which avoids complicated calculations. The first and the second order normal forms are computed successively, and the uniqueness of the second order normal form is proved under certain conditions.

Distributed Optimization over Directed Graphs with the help of Lie Brackets

This work addresses the problem of solving distributed optimization problems over directed graphs. Using techniques from geometric control theory, a novel methodology is proposed to design continuous-time distributed algorithms that approximate the solutions to the saddle-point algorithms over directed graphs. Graph-theoretic conditions are established under which the proposed methodology provides a distributed solution.

Extremum Seeking for Time-Varying Functions using Lie Bracket Approximations

The paper presents a control algorithm that steers a system to an extremum point of a time-varying function. The proposed extremum seeking law depends on values of the cost function only and can be implemented without knowing analytical expression of this function. By extending the Lie brackets approximation method, we prove the local and semi-global practical uniform asymptotic stability for time-varying extremum seeking problems. For this purpose, we consider an auxiliary non-autonomous system of differential equations and propose asymptotic stability conditions for a family of invariant sets. The obtained control algorithm ensures the motion of a system in a neighborhood of the curve where the cost function takes its minimal values. The dependence of the radius of this neighborhood on the bounds of the derivative of a time-varying function is shown.

Extremum seeking for dynamic maps using Lie brackets and singular perturbations

We introduce a framework for the analysis of extremum seeking systems for dynamic maps based on Lie bracket approximations and singular perturbation theory. Using the introduced framework, we provide an extremum seeking regulator for a unicycle and prove its stability properties.

From Koopman\u2013von Neumann theory to quantum theory

Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function S^{(K)}(q,p,t) whose physical meaning is unknown. We show that a different S(q, p, t), with well-defined physical meaning, may be introduced without destroying the attractive “quantum-like” mathematical features of the KvN theory. This new S(q, p, t) is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schrödinger’s equation if functions on phase space are reduced to functions on configuration space. This new kind of “quantization” does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing frac{partial }{partial p} by 0 and p by frac{hbar }{imath } frac{partial }{partial q}, thus providing an explanation for the common quantization rules.

Gerstenhaber Algebra Structure on the Hochschild Cohomology of Quadratic String Algebras

We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH∗(A) when A is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell’s resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell’s resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.

Functional Itô calculus, path-dependence and the computation of Greeks

Dupire’s functional Itô calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional Itô calculus framework. Namely, we consider the Lie bracket of the space and time functional derivatives, which we use to classify functionals accordingly to their degree of path-dependence. We then revisit the problem of efficient numerical computation of Greeks for path-dependent derivatives using integration by parts techniques. Special attention is paid to path-dependent functionals with zero Lie bracket, called locally weakly path-dependent functionals in our classification. Hence, we derive the weighted-expectation formulas for their Greeks. In the more general case of fully path-dependent functionals, we show that, equipped with the functional Itô calculus, we are able to analyze the effect of the Lie bracket on the computation of Greeks. Moreover, we are also able to consider the more general dynamics of path-dependent volatility. These were not achieved using Malliavin calculus.

Quantifying the brain's sheet structure with normalized convolution.

The hypothesis that brain pathways form 2D sheet-like structures layered in 3D as "pages of a book" has been a topic of debate in the recent literature. This hypothesis was mainly supported by a qualitative evaluation of "path neighborhoods" reconstructed with diffusion MRI (dMRI) tractography. Notwithstanding the potentially important implications of the sheet structure hypothesis for our understanding of brain structure and development, it is still considered controversial by many for lack of quantitative analysis. A means to quantify sheet structure is therefore necessary to reliably investigate its occurrence in the brain. Previous work has proposed the Lie bracket as a quantitative indicator of sheet structure, which could be computed by reconstructing path neighborhoods from the peak orientations of dMRI orientation density functions. Robust estimation of the Lie bracket, however, is challenging due to high noise levels and missing peak orientations. We propose a novel method to estimate the Lie bracket that does not involve the reconstruction of path neighborhoods with tractography. This method requires the computation of derivatives of the fiber peak orientations, for which we adopt an approach called normalized convolution. With simulations and experimental data we show that the new approach is more robust with respect to missing peaks and noise. We also demonstrate that the method is able to quantify to what extent sheet structure is supported for dMRI data of different species, acquired with different scanners, diffusion weightings, dMRI sampling schemes, and spatial resolutions. The proposed method can also be used with directional data derived from other techniques than dMRI, which will facilitate further validation of the existence of sheet structure.

A characterization of the disk by eigenfunction of the Steklov eigenvalue

Suppose M is homeomorphic to 2-dimensional sphere and let \(f_i\), \(1\le i\le 3\), be the first eigenfunctions of the Laplacian on M. Cheng proved (Proc Am Math Soc 55(2):379–381, 1976) that if \(\sum _{i=1}^3 f_i^2\) is a constant, then M is isometric to a sphere of constant curvature. In view of the similarities between eigenvalues of the Laplacian and Steklov eigenvalue, we study eigenfuction of the first nonzero Steklov eigenvalue of a 2-dimensional compact manifold with boundary \(\partial M\). Suppose M is a domain equipped with the flat metric g, and let f be an eigenfunction of the first nonzero Steklov’s eigenvalue. We prove that if \(\nabla _g f\) is parallel along \(\partial M\), the Lie bracket of the vector field orthogonal to \(\nabla _g f\) and the tangent to \(\partial M\) is zero, and \(|\nabla _g f|^2\) is constant in M, then (M, g) is isometric to the disk equipped with the flat metric. We also prove that if the eigenfunctions \(f_1, f_2\) corresponding to the first nonzero Steklov eigenvalue satisfy that \(f_1^2+f_2^2\) is constant on \(\partial M\) and \(\varphi _*(g)\) is a Riemannian metric conformal to the flat metric of \(D^2\) where \(\varphi =(f_1,f_2)\), then (M, g) is isometric to the disk equipped with the flat metric. In another direction, we prove that a simply connected domain in the hyperbolic space \(\mathbb {H}^3\) such that its Steklov eigenvalues are the same as a geodesic ball must be isometric to the geodesic ball.

Feedback Stabilization of Nonholonomic Drift-Free Systems Using Adaptive Integral Sliding Mode Control

This article presents a novel stabilizing control algorithm for nonholonomic drift-free systems. The control algorithm is based on adaptive integral sliding mode control technique. In order to utilize the benefit of integral sliding mode control, extended Lie bracket system is used as a nominal system which can easily be asymptotically stabilized. Firstly the original nonholonomic drift-free system is augmented by adding its missing Lie brackets and some unknown adaptive parameters. Secondly the controller and the adaptive laws are designed in such a way that the behaviour of the augmented system is similar to that of the nominal system on the sliding surface and the addition of missing Lie brackets in the original system can be compensated adaptively. The proposed method is applied on two nonholonomic drift-free systems including the Brockett’s system and the hopping robot in flight phase. The controllability Lie Algebra of the Brockett’s system has Lie brackets of depth one, whereas the hopping robot model in flight phase contains Lie brackets of depth one and two. The effectiveness of the proposed technique is verified through simulation studies.