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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.

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.

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 .

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.

There are two well-known ways of describing elements of the rotation group SO$(m)$. First, according to the Cartan-Dieudonn\'e theorem, every rotation matrix can be written as an even number of reflections. And second, they can also be expressed as the exponential of some anti-symmetric matrix. In this paper, we study similar descriptions of a group of rotations SO${}_0$ in the superspace setting. This group can be seen as the action of the functor of points of the orthosymplectic supergroup OSp$(m|2n)$ on a Grassmann algebra. While still being connected, the group SO${}_0$ is thus no longer compact. As a consequence, it cannot be fully described by just one action of the exponential map on its Lie algebra. Instead, we obtain an Iwasawa-type decomposition for this group in terms of three exponentials acting on three direct summands of the corresponding Lie algebra of supermatrices. At the same time, SO${}_0$ strictly contains the group generated by super-vector reflections. Therefore, its Lie algebra is isomorphic to a certain extension of the algebra of superbivectors. This means that the Spin group in this setting has to be seen as the group generated by the exponentials of the so-called extended superbivectors in order to cover SO${}_0$. We also study the actions of this Spin group on supervectors and provide a proper subset of it that is a double cover of SO${}_0$. Finally, we show that every fractional Fourier transform in n bosonic dimensions can be seen as an element of this spin group., Comment: 28 pages

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.

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

Let $G$ be a connected reductive algebraic group over a non-Archimedean local field $K$, and let $g$ be its Lie algebra. By a theorem of Harish-Chandra, if $K$ has characteristic zero, the Fourier transforms of orbital integrals are represented on the set of regular elements in $g(K)$ by locally constant functions, which, extended by zero to all of $g(K)$, are locally integrable. In this paper, we prove that these functions are in fact specializations of constructible motivic exponential functions. Combining this with the Transfer Principle for integrability [R. Cluckers, J. Gordon, I. Halupczok, "Transfer principles for integrability and boundedness conditions for motivic exponential functions", preprint arXiv:1111.4405], we obtain that Harish-Chandra's theorem holds also when $K$ is a non-Archimedean local field of sufficiently large positive characteristic. Under the hypothesis on the existence of the mock exponential map, this also implies local integrability of Harish-Chandra characters of admissible representations of $G(K)$, where $K$ is an equicharacteristic field of sufficiently large (depending on the root datum of $G$) characteristic., Comment: Compared to v2/v3: some proofs simplified, the main statement generalized; slightly reorganized. Regarding the automatically generated text overlap note: it overlaps with the Appendix B (which is part of arXiv:1208.1945) written by us; the appendix and this article cross-reference each other, and since the set-up is very similar, some overlap is unavoidable

A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra so ( 3 , 2 ) with quadratic Casimir equals to − 5 / 4 . As both are also bases of square-integrable functions, the universal enveloping algebra of so ( 3 , 2 ) is thus shown to be homomorphic to the space of linear operators acting on the L 2 functions defined on ( − 1 , 1 ) × Z and on the sphere S 2 , respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining in this way the “algebraic special functions” that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the L 2 functions is homomorphic to the universal enveloping algebra. The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group SO ( 3 , 2 ) .

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

For the p-adic group G = SL(2), we present results of the computations of the sums of the Bernstein projectors of a given depth. Motivation for the computations is based on a conversation with Roger Howe in August 2013. The computations are elementary, but they provide an expansion of the delta distribution \(\delta _{1_{G}}\) into an infinite sum of G-invariant locally integrable essentially compact distributions supported on the set of topologically unipotent elements. When these distributions are transferred, by the exponential map, to the Lie algebra, they give G-invariant distributions supported on the set of topologically nilpotent elements, whose Fourier transforms turn out to be characteristic functions of very natural G-domains. The computations in particular rely on the SL(2) discrete series character tables computed by Sally–Shalika in 1968. This new phenomenon for general rank has also been independently noticed in recent work of Bezrukavnikov, Kazhdan, and Varshavsky.

We show that a group of diffeomorphisms $\D$ on the open unit interval $I,$ equipped with the topology of uniform convergence on any compact set of the derivatives at any order, is non regular: the exponential map is not defined for some path of the Lie algebra. this result extends to the group of diffeomorphisms of finite dimensional, non compact manifold $M.$, Comment: Corrected version, better presentation

In this Note, we unveil homotopy-rich algebraic structures generated by the Atiyah classes relative to a Lie pair ( L , A ) of algebroids. In particular, we prove that the quotient L / A of such a pair admits an essentially canonical homotopy module structure over the Lie algebroid A, which we call Kapranov module.

The ancient Greek mathematician Thales of Miletus devised a method to make large measurements like the heights of pyramids. The straightedge or ruler construction of multiplication can be traced back to this method of Thales. The ruler construction also exists for addition. In this article, we indicate how the comparison between these two constructions leads to Lie theory, a device created by Lie to reduce the multiplicative problems to the additive ones. It is hoped that this enhances the interest in these two themes.

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.

Analytic curves are classified w.r.t. their symmetry under a regular and separately analytic Lie group action on an analytic manifold. We show that an analytic curve is either exponential or splits into countably many analytic immersive curves, each of them discretely generated by the symmetry group (i.e., each such curve naturally decomposes into countably many symmetry free subcurves that are mutually and uniquely related by the Lie group action). We additionally extend the classification result to the analytic 1-submanifold case. Specifically, we show that an analytic 1-submanifold is either free or (exponential, i.e.) analytically diffeomorphic (via the exponential map) to the unit circle or an interval. The corresponding decomposition results in the free case are outlined in this paper, but proven in a separate one., Comment: 77 pages. Version as published in Differential Geometry and its Applications (up to additional presentational improvements)

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.

The purpose of this note is to describe certain Levi-flats in [O] more in detail, by showing that they are natural generalization of Nemirovski’s example in [N]. Let C be a compact Riemann surface, let L → C be a holomorphic line bundle, and let s be a meromorphic section of L whose zeros and poles are all simple. Let C ′ ⊂ C be the complement of the set of zeros and poles of s, let L′ ⊂ L be the complement of the zero section, and let R′ = R\{0}. Let a > 1, and let Z act on L′ by fiberwise multiplication by a for m ∈ Z. Then it is easy to see that the closure of R′s(C ′)/Z in L′/Z is a Levi-flat whose complement is Stein if C ′ = C (cf. [N]). This construction is immediately generalized to produce Levi-flats with Stein complements in higher dimension, by considering smooth ample divisors and the associated line bundles with canonical sections. On the other hand, since not all elliptic principal bundles arise as quotients of C∗-bundles, it may not be so immediate to see how one can generalize Nemirovski’s construction to the elliptic principal bundles. However, it is actually immediate if one regards d log s as a meromorphic connection of the bundle L′/Z as we shall see below. Let C be as above, let E0 be an elliptic curve, i.e. a compact 1-dimensional complex Lie group, and let E π → C be a principal E0-bundle. Let g be the Lie algebra of E0. We shall regard E0 as a quotient space of g by the exponential map. The kernel of the exponential map will be denoted by gZ and E0 will be identified with g/gZ in what follows. Given a closed subgroup Γ ⊂ E0 and a finite subset Σ ⊂ C, let Ω(Γ,Σ) be the set of meromorphic connections ω of E with at most simple poles in Σ

The Duflo map is a valuable tool for operator ordering in contexts in which Kirillov-Kostant brackets and their quantizations play a role. A priori, the Duflo map is only defined on the subspace of the symmetric algebra over a Lie algebra consisting of elements invariant under the adjoint action. Here we discuss extensions to the whole symmetric algebra, as well as their application to the calculation of Chern-Simons theory expectation values., 22 pages, 2 figures, matches published version

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.

In this paper we consider numerical methods for solving nonlinear equations on matrix Lie groups. Recently Owren and Welfert (Technical Report Numerics, No 3/1996, Norwegian University of Science and Technology, Trondheim, Norway, 1996) have proposed a method where the original nonlinear equation F(Y)=0 is transformed into a nonlinear equation on the Lie algebra of the group, thus Newton-type methods may be applied which require the evaluation of exponentials of matrices. Here the previous transformation will be performed by the Cayley approximant of the exponential map. This approach has the advantage that no exponentials of matrices are needed. The numerical tests reported in the last section seem to show that our approach is less expensive and provides a larger convergence region than the method of Owren and Welfert.

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 .

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 prove a convergence result for the Campbell–Baker–Hausdorff–Dynkin series in infinite-dimensional Banach–Lie algebras L. In the existing literature, this topic has been investigated when L is the Lie algebra Lie(G) of a finite-dimensional Lie group G (see [Blanes and Casas, 2004]) or of an infinite-dimensional Banach–Lie group G (see [Mérigot, 1974]). Indeed, one can obtain a suitable ODE, which follows from the well-behaved formulas for the differential of the Exponential Map of the Lie group G. The novelty of our approach is to derive this ODE in any infinite-dimensional Banach–Lie algebra, not necessarily associated to a Lie group, as a consequence of an analogous abstract ODE first obtained in the most natural algebraic setting: that of the formal power series in two commuting indeterminates over the free unital associative algebra generated by two non-commuting indeterminates.

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.

Let V be a real or complex vector space with a non-degenerate symmetric bilinear form B, and let q: ∧(V)→Cl(V) be the quantization map. The space q(∧2(V)) is a Lie subalgebra under commutation in the Clifford algebra. This subspace is canonically isomorphic to the orthogonal Lie algebra of V, and the restriction of the exponential map for the Clifford algebra is identified with the exponential map for the spin group. One of the problems addressed in this chapter is to give explicit formulas for these Clifford exponentials, in terms of exterior algebra exponentials of the corresponding elements in ∧2(V). These questions will be studied using the spin representation for the vector space V ∗⊕V, with bilinear form given by the pairing. One of the outcomes of this discussion is the construction of a remarkable ∧(V)-valued function on the orthogonal Lie algebra, which will play a role in our discussion of the Duflo theorem in Chapter 7.

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 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.

The topic of this paper is the relationship between characters of irreducible supercuspidal representations of the p-adic unramified 3 x 3 unitary group and Fourier transforms of invariant measures on elliptic adjoint orbits in the Lie algebra. We prove that most supercuspidal representations have the property that, on some neighbourhood of zero, the character composed with the exponential map coincides with the formal degree of the representation times the Fourier transform of a measure on one elliptic orbit. For the remainder, a linear combination of the Fourier transforms of measures on two elliptic orbits must be taken. As a consequence of these relations between characters and Fourier transforms, the coefficients in the local character expansions are expressed in terms of values of Shalika germs. By calculating which of the values of the Shalika germs associated to regular nilpotent orbits are nonzero, we determine which irreducible supercuspidal representations have Whittaker models. Finally, the coefficients in the local character expansions of three families of supercuspidal representations are computed.

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.

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.

The canonical decomposition of a Lorentz algebra element into a sum of orthogonal simple (decomposable) Lorentz bivectors is defined and discussed. This decomposition on the Lie algebra level leads to a natural decomposition of a proper orthochronous Lorentz transformation into a product of commuting Lorentz transformations, each of which is the exponential of a simple bivector. While this later result is known, we present novel formulas that are independent of the form of the Lorentz metric chosen. As an application of our methods, we obtain an alternative method of deriving the formulas for the exponential and logarithm for Lorentz transformations.

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 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.

We present Lie group integrators for nonlinear stochastic differential equations with non-commutative vector fields whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action, and subsequently pull back the flow to the corresponding Lie algebra via the exponential map. We construct an approximation to the stochastic flow in the Lie algebra via closed operations and then push back to the Lie group and then to the manifold, thus ensuring our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas methods after their deterministic counterparts. We also present stochastic Lie group integration schemes based on Castell--Gaines methods. These involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential Lie series. They become stochastic Lie group integrator schemes if we use Munthe-Kaas methods as the underlying ordinary differential integrator. Further, we show that some Castell--Gaines methods are uniformly more accurate than the corresponding stochastic Taylor schemes. Lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes., Comment: 20 pages, 4 figures

The second author gave a formula for the elements of the enveloping algebra of a Lie superalgebra defined by Gorelik under an appropriate unimodularity assumption. We show that this formula is a particular case of a formula for the Jacobian of the exponential map of a symmetric superspace., Comment: Accepted for publication in Journal of Algebra, special volume in honour of E.B. Vinberg (32 pages)

Let $\g$ be a finite dimensional complex Lie algebra and $��\subset \g$ a Lie subalgebra equipped with the structure of a factorizable quasitriangular Lie bialgebra. Consider the Lie group $\Exp ��$ with the Semenov-Tjan-Shansky Poisson bracket as a Poisson Lie manifold for the double Lie bialgebra $\D��$. Let $\Nc_��(0)\subset ��$ be an open domain parameterizing a neighborhood of the identity in $\Exp ��$ by the exponential map. We present dynamical $r$-matrices with values in $\g\wedge \g$ over the Poisson Lie base manifold $\Nc_��(0)$., AMS LaTeX, 8 pages

The inventors of Lie groups and Lie algebras (starting with Lie!) regarded Lie groups as groups of symmetries of various topological or geometric objects. Lie algebras were viewed as the “infinitesimal transformations” associated with the symmetries in the Lie group. For example, the group SO(n) of rotations is the group of orientation-preserving isometries of the Euclidean space \( \mathbb{E}^n \) . The Lie algebra so (n,ℝ) consisting of real skew symmetric n × n matrices is the corresponding set of infinitesimal rotations. The geometric link between a Lie group and its Lie algebra is the fact that the Lie algebra can be viewed as the tangent space to the Lie group at the identity. There is a map from the tangent space to the Lie group, called the exponential map. The Lie algebra can be considered as a linearization of the Lie group (near the identity element), and the exponential map provides the “delinearization,” i.e., it takes us back to the Lie algebra. These concepts have a concrete realization in the case of groups of matrices, and for this reason we begin by studying the behavior of the exponential maps on matrices.