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We construct and study solitonic representations of the conformal net associated to some vacuum Positive Energy Representation (PER) of a loop group LG. For the corresponding solitonic states, we prove the Quantum Null Energy Condition (QNEC) and the Bekenstein Bound. As an intermediate result, we show that a Positive Energy Representation of a loop group LG can be extended to a PER of $$H^{s}(S^1,G)$$ H s ( S 1 , G ) for $$s>3/2$$ s > 3 / 2 , where G is any compact, simple and simply connected Lie group. We also show the existence of the exponential map of the semidirect product $$LG \rtimes R$$ L G ⋊ R , with R a one-parameter subgroup of $$\mathrm{Diff}_+(S^1)$$ Diff + ( S 1 ) , and we compute the adjoint action of $$H^{s+1}(S^1,G)$$ H s + 1 ( S 1 , G ) on the stress energy tensor.

Here we demonstrate some of the benefits of a novel parameterization of the Lie groups $\mathrm{Sp}(2,\mathbb{R}) ≅ \mathrm{SL}(2,\mathbb{R})$. Relying on the properties of the exponential map $\mathfrak{sl}(2,\mathbb{R}) \rightarrow \mathrm{SL}(2,\mathbb{R})$, we have found a few explicit formulas for the logarithm of the matrices in these groups. Additionally, the explicit analytic description of the ellipse representing their field of values is derived and this allows a direct graphical identification of various types.

This paper is made out of necessity as a doctoral student taking the exam from Lie groups. Using the literature suggested to me by the professor, I felt the need to, in addition to that literature, and since there was more superficial in that book with some remarks about the examples given in relation to the left group. I decided to try a little harder and collect as much literature as possible, both for the needs of me and the others who will take after me. Searching for literature in my mother tongue I could not find anything, in English as someone who comes from a small country like Montenegro, all I could find was through the internet. I decided to gather what I could find from the literature in my own way and to my observation and make this kind of work. The main content of this paper is to present the Lie group in the simplest way. Before and before I started writing or collecting about Lie groups, it was necessary to say something about groups and subgroups that are taught in basic studies in algebra. In them I cited several deficits and an example. The following content of the paper is related to Lie groups primarily concerning the definition of examples such as The General Linear Group GL(n, R), The Complex General Linear Group GL(n, C), The Special Linear Group SL(n, R)=SL(V), The Complex Special Linear Group SL(n, C), Unitary and Orthogonal Groups, Symplectic Group, The groups R*, C*, S1 and Rn and others. In addition, invariant vector fields and the exponential map and the lie algebra of a lie group. For me, this work has the significance of being useful to all who need it.

We introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the exponential map., v3: minor changes; to appear in Glasgow Mathematical Journal. v2: significant changes compared to v1; the result on quasi-isometry classification of csc nilpotent Lie groups removed due to a flaw in the proof

The objective of this chapter is to introduce the concepts of Lie groups and their Lie algebras. The Lie algebra \(\mathfrak {g}\) of a Lie group G is defined as the space of invariant vector fields (left or right, depending on choice), with bracket given by the Lie bracket of vector fields. The flows of invariant vector fields establish the exponential map \(\exp :\mathfrak {g}\rightarrow G\), which is the main link between \(\mathfrak {g}\) and G. These constructions make exhaustive use of results about vector fields on manifolds and their Lie brackets. A short collection of such results can be found in Appendix A. Adjoint representations are another tool for linking Lie groups and their Lie algebras. The formulas involving such representations are developed in this chapter. They are used along the whole theory.

We introduce a Frechet Lie group structure on the Riordan group. We give a description of the corresponding Lie algebra as a vector space of infinite lower triangular matrices. We describe a natural linear action induced on the Frechet space K N by any element in the Lie algebra. We relate this to a certain family of bivariate linear partial differential equations. We obtain the solutions of such equations using one-parameter groups in the Riordan group. We show how a particular semidirect product decomposition in the Riordan group is reflected in the Lie algebra. We study the stabilizer of a formal power series under the action induced by Riordan matrices. We get stabilizers in the fraction field K ( ( x ) ) using bi-infinite representations. We provide some examples. The main tool to get our results is the paper [18] where the Riordan group was described using inverse sequences of groups of finite matrices.

In Chapter 4 we defined the notion of a Lie group as a certain type of manifold embedded in \({\mathbb {R}}^N\), for some N ≥ 1. Now that we have the general concept of a manifold, we can define Lie groups in more generality. If every Lie group was a linear group (a group of matrices), then there would be no need for a more general definition. However, there are Lie groups that are not matrix groups, although it is not a trivial task to exhibit such groups and to prove that they are not matrix groups.

Multistep methods of BDF type are the method-of-choice in several industrial multibody system simulation packages. In the present paper, BDF is applied to constrained systems in nonlinear configuration spaces with Lie group structure that allows, e.g., a representation of multibody systems with large rotations without singularities. The k -step Lie group integrator BLieDF avoids order reduction by a slightly perturbed argument of the exponential map for representing the nonlinearity of the numerical flow in the configuration space without any time-consuming re-parametrization. This integrator is compared with multistep methods on Lie groups suggested by Faltinsen, Marthinsen and Munthe-Kaas (2001) and the advantages of the novel BLieDF integrator are shown.

In this work, we began to take forward kinematics of the Gough–Stewart (G-S) platform as an unconstrained optimization problem on the Lie group-structured manifold SE(3) instead of simply relaxing its intrinsic orthogonal constraint when algorithms are updated on six-dimensional local flat Euclidean space or adding extra unit norm constraint when orientation parts are parametrized by a unit quaternion. With this thought in mind, we construct two kinds of iterative problem-solving algorithms (Gauss–Newton (G-N) and Levenberg–Marquardt (L-M)) with mathematical tools from the Lie group and Lie algebra. Finally, a case study for a general G-S platform was carried out to compare these two kinds of algorithms on SE(3) with corresponding algorithms that updated on six-dimensional flat Euclidean space or seven-dimensional quaternion-based parametrization Euclidean space. Experiment results demonstrate that those algorithms on SE(3) behave better than others in convergence performance especially when the initial guess selection is near to branch solutions.

Jean-Marie Souriau extended Urbain Jean Joseph Leverrier algorithm to compute characteristic polynomial of a matrix in 1948. This Souriau algorithm could be used to compute exponential map of a matrix that is a challenge in Lie Group Machine Learning. Main property of Souriau Exponential Map numerical scheme is its scalability with highly parallelization.

In this work, the exponential and the Cayley maps, from the Lie algebra $\mathfrak{se(2)}$ of the planar motion group $SE(2)$, to the group itself are studied. The comparison between these maps on $SE(2)$ is given by using the Rodrigues vector. A three joint mechanism is discussed as an application.

We consider left-invariant optimal control problems on connected Lie groups such that generic stabilizer of the coadjoint action is connected and has dimension not more than 1. We introduce a construction for symmetries of the exponential map. These symmetries play a key role in investigation of optimality of extremal trajectories., 16 pages, 1 figures

This paper presents a generalized exponential formula for Forward and Differential Kinematics of open-chain multi-body systems with multi-degree-of-freedom, holonomic and nonholonomic joints. The notion of lower kinematic pair is revisited, and it is shown that the relative configuration manifolds of such joints are indeed Lie groups. Displacement subgroups, which correspond to different types of joints, are categorized accordingly, and it is proven that except for one class of displacement subgroups the exponential map is surjective. Screw joint parameters are defined to parameterize the relative configuration manifolds of displacement subgroups using the exponential map of Lie groups. For nonholonomic constraints the admissible screw joint speeds are introduced, and the Jacobian of the open-chain multi-body system is modified accordingly. Computational aspects of the developed formulation for Forward and Differential Kinematics of open-chain multi-body systems are explored by assigning coordinate frames to the initial configuration of the multi-body system, employing the matrix representation of SE(3) and choosing a basis for se(3). Finally, an example of a mobile manipulator mounted on a spacecraft, i.e., a six-degree-of-freedom moving base, elaborates the computational aspects.

We demonstrate the functional inverse of a Winter map, which is an analog of the exponential map, for Lie algebras over fields of prime characteristic.

The exponential map for Lie groups is studied and used to prove properties of Lie groups and the relation with Lie algebras.

SUMMARYA nonholonomic system subjected to external noise from the environment, or internal noise in its own actuators, will evolve in a stochastic manner described by an ensemble of trajectories. This ensemble of trajectories is equivalent to the solution of a Fokker–Planck equation that typically evolves on a Lie group. If the most likely state of such a system is to be estimated, and plans for subsequent motions from the current state are to be made so as to move the system to a desired state with high probability, then modeling how the probability density of the system evolves is critical. Methods for solving Fokker-Planck equations that evolve on Lie groups then become important. Such equations can be solved using the operational properties of group Fourier transforms in which irreducible unitary representation (IUR) matrices play a critical role. Therefore, we develop a simple approach for the numerical approximation of all the IUR matrices for two of the groups of most interest in robotics: the rotation group in three-dimensional space,SO(3), and the Euclidean motion group of the plane,SE(2). This approach uses the exponential mapping from the Lie algebras of these groups, and takes advantage of the sparse nature of the Lie algebra representation matrices. Other techniques for density estimation on groups are also explored. The computed densities are applied in the context of probabilistic path planning for kinematic cart in the plane and flexible needle steering in three-dimensional space. In these examples the injection of artificial noise into the computational models (rather than noise in the actual physical systems) serves as a tool to search the configuration spaces and plan paths. Finally, we illustrate how density estimation problems arise in the characterization of physical noise in orientational sensors such as gyroscopes.

Lai, W\"ustner and Chatterjee have shown in a variety of cases that the exponential map of a connected complex Lie group is surjective only if its center is connected. Here we prove this is true without restriction for any complex connected Lie group.

For any solvable Lie group whose exponential map $\exp_G\colon{\mathfrak g}\to G$ is bijective, we prove that the real rank of $C^*(G)$ is equal to $\dim({\mathfrak g}/[{\mathfrak g},{\mathfrak g}])$. We also indicate a proof of a similar formula for the stable rank of $C^*(G)$, as well as some estimates on the ideal generated by the projections in $C^*(G)$., 8 pages

In this paper we consider Newton iteration methods for solving nonlinear equations on matrix Lie groups. Recently, Owren and Welfert have proposed a method where the original nonlinear equation is transformed into a nonlinear equation on the Lie algebra of the group via the exponential map, thus Newton iteration methods may be applied. Based on this we suggest two improved variants of Newton iteration algorithm. One is that the exponential map would be approximated by Cayley map and give a Cayley version Newton iteration method for solving nonlinear equations on quadratic Lie groups, then we show that, the proposed method converges quadratically; Another is a variant of Newton type method with accelerated convergence and the numerical tests reported seem to support that it converges with cubically.

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is - ∞ .

Numerical integration methods based on the Lie group theoretic geometrical approach are applied to articulated multibody systems with rigid body displacements, belonging to the special Euclidean group SE(3), as a part of generalized coordinates. Three Lie group integrators, the Crouch-Grossman method, commutator-free method, and Munthe-Kaas method, are formulated for the equations of motion of articulated multibody systems. The proposed methods provide singularity-free integration, unlike the Euler-angle method, while approximated solutions always evolve on the underlying manifold structure, unlike the quaternion method. In implementing the methods, the exact closed-form expression of the differential of the exponential map and its inverse on SE(3) are formulated in order to save computations for its approximation up to finite terms. Numerical simulation results validate and compare the methods by checking energy and momentum conservation at every integrated system state.

Let G be a compact, connected Lie group endowed with a bi-invariant Riemannian metric. Let ρt be the heat kernel on G; that is, ρt is the fundamental solution to the heat equation on the group determined by the Laplace–Beltrami operator. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new family of functions on G. These functions, obtained from ρt and its derivatives, are the compact group analogs of the classical Hermite polynomials on \(\mathbb{R}^n \). Previous work of this author has established that these Hermite functions approach the classical Hermite polynomials on \(\mathfrak{g}\;\; = \;\;Lie(G)\) in the limit of small t, where the Hermite functions are viewed as functions on \(\mathfrak{g}\) via composition with the exponential map. The present work extends these results by showing that these Hermite functions can be expanded in an asymptotic series in powers of \(\sqrt t \). For symmetrized derivatives, it is shown that the terms with fractional powers of t vanish. Additionally, the asymptotic series for Hermite functions associated to powers of the Laplacian are computed explicitly. Remarkably, these asymptotic series terminate, yielding a polynomial in t.

Deep compositing is an important practical tool in creating digital imagery, but there has been little theoretical analysis of the underlying mathematical operators. Motivated by finding a simple formulation of the merging operation on OpenEXR -style deep images, we show that the Porter-Duff over function is the operator of a Lie group. In its corresponding Lie algebra, the splitting and mixing functions that OpenEXR deep merging requires have a particularly simple form. Working in the Lie algebra, we present a novel, simple proof of the uniqueness of the mixing function. The Lie group structure has many more applications, including new, correct resampling algorithms for volumetric images with alpha channels, and a deep image compression technique that outperforms that of OpenEXR .

This paper presents the strapdown inertial navigation algorithm design by using the matrix Lie group SE(3) and its Lie algebra se(3). The direction cosine matrix and translation of the vehicle are described by a group element as a 4×4 matrix. The navigation differential equations are given by three group rate equations of the same form. By taking the identity of the exponential map, the group equations are pulled back to the linear Lie algebra space. A new numerical integration is developed to solve the equations in Lie algebra, and then map them to update the group elements. The overall process is taking a geometric description, avoiding the complicate analysis of kinematics. A variety of simulations are conducted, the results show that the new algorithm improves the accuracy.

A Lie group is called exponential if its exponential map is surjective. It is called weakly exponential if the exponential image is dense, which is equivalent to the connectivity of each of the Cartan subgroups (compare [11]). In the present paper the authors study exponential Lie groups which are of mixed type, i.e., neither solvable nor semisimple. Necessary conditions and also, for special mixed Lie groups, sufficient conditions are given for exponentiality. Several counter examples are provided showing that none of the conditions which have surfaced during the course of our investigation can work as necessary and sufficient ones. All conditions considered deal with centralizers of ad-nilpotent elements of the Lie algebra. For example, it is shown that if G is exponential, there is a rather large characteristic subgroup B which contains the nilradical, all Levi factors, and all maximal compactly embedded subgroups, which is also exponential. Moreover, this subgroup is also Mal’cev splittable so that one can apply earlier results on Mal’cev splittable exponential Lie groups, which characterize exponentiality of these Lie groups (also by conditions concerning the centralizers of ad-nilpotent elements).

Consider a differential equation y' = A(t,y)y, y(0) = y0 with y 0 ∈ G and A: R + × G → g, where g is a Lie algebra of the matricial Lie group G. Every B ∈ g can be mapped to G by the matrix exponential map exp (tB) with t ∈ R. Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation y n of the exact solution y(t n ), t n ∈ R + , by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value y 0 . This ensures that .y n ∈ G. When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of exp (tB) plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or polynomial approximants are unsuitable and we consider alternative techniques, whereby exp (tB) is approximated by a product of simpler exponentials. In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of g and G are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper.

Many discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and by the exploitation of redundancies inherent in a Lie-algebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation, and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of m-nary multilinear operations which display alternating symmetry and a ‘hierarchy condition’. We present explicit formulae for the dimension of graded subspaces of free hierarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Lie-algebraic setting.

We consider the construction of geometric integrators in the class of RKMK methods. Any differential equation in the form of an infinitesimal generator on a homogeneous space is shown to be locally equivalent to a differential equation on the Lie algebra corresponding to the Lie group acting on the homogeneous space. This way we obtain a distinction between the coordinate-free phrasing of the differential equation and the local coordinates used. In this paper we study methods based on arbitrary local coordinates on the Lie group manifold. By choosing the coordinates to be canonical coordinates of the first kind we obtain the original method of Munthe-Kaas [16]. Methods similar to the RKMK method are developed based on the different coordinatizations of the Lie group manifold, given by the Cayley transform, diagonal Pade approximants of the exponential map, canonical coordinates of the second kind, etc. Some numerical experiments are also given.

Let p t ( x ) be the (Gaussian) heat kernel on R n at time t . The classical Hermite polynomials at time t may be defined by a Rodriguez formula, given by H α (− x , t ) p t ( x )= αp t ( x ), where α is a constant coefficient differential operator on R n . Recent work of Gross (1993) and Hijab (1994) has led to the study of a new class of functions on a general compact Lie group, G . In analogy with the R n case, these “Hermite functions” on G are obtained by the same formula, wherein p t ( x ) is now the heat kernel on the group, − x is replaced by x −1 , and α is a right invariant differential operator. Let g be the Lie algebra of G . Composing a Hermite function on G with the exponential map produces a family of functions on g . We prove that these functions, scaled appropriately in t , approach the classical Hermite polynomials at time 1 as t tends to 0, both uniformly on compact subsets of g and in L p ( g , μ ), where 1⩽ p μ is a Gaussian measure on g . Similar theorems are established when G is replaced by G / K , where K is some closed, connected subgroup of G .

In this article we would like to present a geometric approach to model the velocity-level behaviour of the free-floating space manipulator system. Namely we will incorporate some concepts from the theory of Lie groups which have already proved useful in the kinematic and dynamic modelling of fixed-base manipulators. The treatment presented in this article allows obtaining general and coordinate-free kinematic expressions which relate Cartesian velocities of rigid bodies with joint-space velocities. Moreover due to the exponential map and product of exponentials formula we were able to obtain closed form expressions for derivatives of system’s equations with respect to the joint angles.

This chapter is intended to give a summary of Lie groups and Lie algebras for computer graphics, including an example from interpolations and blending of motions and deformation. In animation and filmmaking procedure, we want to get a smooth transition of the drawing of given starting and ending point (the drawings at these points are called key frame). This problem can be understood as an interpolation, so that various approach have been proposed and used in Computer Graphics. After the seminal work so-called ARAP (as-rigid-as-possible), serious properties of matrix groups have been taken into account both in theoretical and in computational point of view. To understand and develop these properties, we here employ a Lie theoretic approach.

We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier-Saito theorem for solvable Lie groups. We then give a geometric characterization of the (strongly) exponential solvable symmetric spaces as those spaces for which every triangle admits a unique double triangle. This work is motivated by Weinstein's quantization by groupoids program applied to symmetric spaces., Some corrections and typos. To appear in International Mathematics Research Notices

In this chapter we introduce the exponential map of a Lie group, which is a canonical smooth map from the Lie algebra into the group, mapping lines through the origin in the Lie algebra to one-parameter subgroups. As our first application, we prove the closed subgroup theorem, which says that every topologically closed subgroup of a Lie group is actually an embedded Lie subgroup. Next we prove a higher-dimensional generalization of the fundamental theorem on flows: if G is a simply connected Lie group, then any Lie algebra homomorphism from its Lie algebra into the set of complete vector fields on a smooth manifold M generates a smooth action of G on M. Using this theorem, we prove that there is a one-to-one correspondence between isomorphism classes of finite-dimensional Lie algebras and isomorphism classes of simply connected Lie groups. At the end of the chapter, we show that connected normal subgroups of a Lie group correspond to ideals in its Lie algebra.

An important theorem of Cartan asserts that any two maximal tori in a compact Lie group are conjugate. There are different ways of proving this. We will deduce it from the surjectivity of the exponential map, which we will prove by showing that a geodesic between the origin and an arbitrary point of the group has the form \(t\mapsto {\mathrm{e}}^{tX}\) for some X in the Lie algebra.

The problem of describing or determining the image of the exponential map $ \exp :\mathfrak{g}\rightarrow G$ of a Lie group $G$ is important and it has many applications. If the group $G$ is compact, then it is well-known that the exponential map is surjective, hence the exponential image is $G$. In this case the problem is reduced to the computation of the exponential and the formulas strongly depend on the group $G$. In this paper we discuss the generalization of Rodrigues formulas for computing the exponential map of the special orthogonal group ${\rm SO}(n) $, which is compact, and of the special Euclidean group ${\rm SE}(n)$, which is not compact but its exponential map is surjective, in the case $ n\geq 4$.

The exponential map, introduced for closed Lie subgroups of \(\mathrm{GL}(n, \mathbb{C})\) in Chap. 5, can be defined for a general Lie group G as a map Lie(G) → G.

The Lie group structure of the Lorentz group is explored. Its generators and its Lie algebra are exhibited, via the study of infinitesimal Lorentz transformations. The exponential map is introduced and it is shown that the study of the Lorentz group can be reduced to that of its Lie algebra. Finally, the link between the restricted Lorentz group and the special linear group \(\mathrm{SL}(2, \mathbb{C})\) is established via the spinor map. The Lie algebras of these two groups are shown to be identical (up to some isomorphism).

We propose a variational procedure for the recovery of internal variables, in effect extending them from integration points to the entire domain. The objective is to perform the recovery with minimum error and at the same time guarantee that the internal variables remain in their admissible spaces. The minimization of the error is achieved by a three-field finite element formulation. The fields in the formulation are the deformation mapping, the target or mapped internal variables and a Lagrange multiplier that enforces the equality between the source and target internal variables. This formulation leads to an $$L_2$$ L 2 projection that minimizes the distance between the source and target internal variables as measured in the $$L_2$$ L 2 norm of the internal variable space. To ensure that the target internal variables remain in their original space, their interpolation is performed by recourse to Lie groups, which allows for direct polynomial interpolation of the corresponding Lie algebras by means of the logarithmic map. Once the Lie algebras are interpolated, the mapped variables are recovered by the exponential map, thus guaranteeing that they remain in the appropriate space.

In this paper, a 2D shape registration algorithm for noisy data is established by combining the Iterative Closest Point (ICP) method, Expectation Maximization (EM) method, and Lie Group representation. First, the problem is formulated by a minimization problem with two sets of variables: the point-to-point correspondence, and the transformation (i.e., rotation, scaling and translation) between two data sets. The conventional way for solving this model is by iterating alternatively the following two steps: 1) having the transformation fixed, solve the correspondence, and 2) having the correspondence fixed, solve the transformation. In our approach, to enhance the robustness, the EM algorithm is introduced to find the correspondence by a probability which covers the relationship of all points, instead of one-to-one closest correspondence in ICP. Meanwhile, Lie group is used to parameterize transformation, i.e., in the iteration, the rotation, scaling and translation are all elements within respective Lie groups, and we use the element of Lie algebra to represent that of Lie group near the identity via exponential map. This forms a unified framework for registration algorithms. Then, transformation is estimated by solving a quadratic programming. The experimental result in 2D shape registration demonstrates that, compared with Lie-ICP, our algorithm is robuster and more accurate.

For a simply connected solvable Lie group we specify the structure of the product, the inverse and the exponential map expressed in suitable coordinates (canonical coordinates of the second kind), and point out that in these coordinates the product and inverse are expressed entirely in terms of polynomials, exponential functions and trigonometric functions. We devise algorithms for computing the product, the inverse and the exponential map.

where the right-hand side is an absolutely convergent series, involving higher-order brackets of X and Y, and (1.2) is valid for all small X and Y. For an exposition of this and any other facts Soncer3ing real Lie groups see [l]. The reason that (1.1) holds is that if X and Y commute so do tX and tY for real t so that by (1.2), exp’ tX mexp t Y = expt(X + Y) for small t, since all the other terms in the formula are zero. But then, by the identity theorem for real analytic functions, this holds for ali t. Taking t = 1 gives the result. Now, since the log inverts exp locally at the origin, and X and Y are small in (1.2), it follows that an absolutely convergent series of these brackets is also small. Hence

We present a general method to obtain a closed, finite formula for the exponential map from the Lie algebra to the Lie group, for the defining representation of the orthogonal groups. Our method is based on the Hamilton-Cayley theorem and some special properties of the generators of the orthogonal group, and is also independent of the metric. We present an explicit formula for the exponential of generators of the $SO_+(p,q)$ groups, with $p+q = 6$, in particular we are dealing with the conformal group $SO_+(2,4)$, which is homomorphic to the $SU(2,2)$ group. This result is needed in the generalization of U(1) gauge transformations to spin gauge transformations, where the exponential plays an essential role. We also present some new expressions for the coefficients of the secular equation of a matrix.

Given a singular foliation, we attach an "essential isotropy" group to each of its leaves, and show that its discreteness is the integrability obstruction of a natural Lie algebroid over the leaf. We show that a condition ensuring discreteness is the local surjectivity of a transversal exponential map associated with the maximal ideal of vector fields prescribed to be tangent to the foliation. The essential isotropy group is also shown to control the smoothness of the holonomy cover of the leaf (the associated fiber of the holonomy groupoid), as well as the {smoothness} of the associated isotropy group. Namely, the (topological) closeness of the essential isotropy group is a necessary and sufficient condition for the holonomy cover to be a smooth (finite-dimensional) manifold and the isotropy group to be a Lie group. These results are useful towards understanding the normal form of a singular foliation around a compact leaf. At the end of this article we briefly outline work of ours on this normal form, to be presented in a subsequent paper.

We consider the periodic μDP equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection ∇ on the Frechet Lie group Diff∞(\(\mathbb{S}^1\)) of all smooth and orientation-preserving diffeomorphisms of the circle \(\mathbb{S}^1\,=\,\mathbb{R}/\mathbb{Z}\). On the Lie algebra C∞(\(\mathbb{S}^1\)) of Diff∞(\(\mathbb{S}^1\)), this connection is canonically given by the sum of the Lie bracket and a bilinear operator. For smooth initial data, we show the short time existence of a smooth solution of μDP which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by ∇ is a smooth local diffeomorphism of a neighbourhood of zero in C∞(\(\mathbb{S}^1\)) onto a neighbourhood of the unit element in Diff∞(\(\mathbb{S}^1\)). Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Frechet space C∞(\(\mathbb{S}^1\)), and a sharp spatial regularity result for the geodesic flow.

A sequence of infinitely generated p‐adic Banach superalgebras and their locally convex projective and inductive limits were constructed; their spectral properties are characterized herein and by studying the algebras of matrices constructed with these superalgebras an exponential map and its inverse can be defined. Some examples of superalgebra with pathological spectral properties are discussed.

Known parametrizations of rotations are derived from the Lie group theoretical point of view considering the two groups ${\rm SO}(3)$ and ${\rm SU}(2)$. The concept of coordinates of the first and second kind for these groups is used to derive the axis and angle as well as the three-angle description of rotation matrices. With the homomorphism of the two groups the Euler parameter description arises from the axis and angle description of ${\rm SU}(2) $. Due to the topology of $ {\rm SO}(3)$ any three-angle description gives only a local parametrization like Euler angles such that the mapping from their time derivatives to the algebra $\frak{so}(3)$, i.e., to the angular velocity tensor, exhibits singularities. All these parametrizations are based on the generation of the respective group by the $\exp$ map from their algebras. Alternatively the Cayley transformation also maps algebra elements to group elements. This fact is well know on ${\rm SO}(3)$ and yields a representation of rotation matrices in terms for Rodrigues parameter, which is, however, not continuous. Generalizing this transformation to ${\rm SU}(2) $ allows for a singularity-free description of all rotations, which does not contain transcendental functions. While in the considered range the exponential map is of class $ C^\infty $ the $\mathrm{cay}$ map on ${\rm SU}(2) $ is only of class $ C^1$ and on ${\rm SO}(3)$ it is not even continuous. Simulation results exemplify the resultant numerical benefits for the simulation of rigid body dynamics. The problem caused by a lack of a continuous transformation from generalized accelerations to angular accelerations can be avoided for rigid body motions using the Bolzmann-Hamel equations.

A direct limit\(G = \mathop {\lim }\limits_ \to G_\alpha\) of (finite-dimensional) Lie groups has Lie algebra\(\mathfrak{g} = \mathop {\lim }\limits_ \to \mathfrak{g}_\alpha\) and exponential map exp G : g→G. BothG and g carry natural topologies.G is a topological group, and g is a topological Lie algebra with a natural structure of real analytic manifold. In this Letter, we show how a special growth condition, natural in certain physical settings and satisfied by the usual direct limits of classical groups, ensures thatG carries an analytic group structure such that exp G is a diffeomorphism from a certain open neighborhood of 0∈g onto an open neighborhood of 1 G ∈G. In the course of the argument, one sees that the structure sheaf for this analytic group structure coincides with the direct limit\(\mathop {\lim }\limits_ \to\)Cω(Gα) of the sheaves of germs of analytic functions on theGα.