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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 work is to compare several approaches to the process of renormalisation in the context of rough differential equations using the substitution bialgebra on rooted trees known from backward error analysis of $B$-series. For this purpose, we present a so-called arborification of the Hoffman--Ihara theory of quasi-shuffle algebra automorphisms. The latter are induced by formal power series, which can be seen to be special cases of the cointeraction of two Hopf algebra structures on rooted forests. In particular, the arborification of Hoffman's exponential map, which defines a Hopf algebra isomorphism between the shuffle and quasi-shuffle Hopf algebra, leads to a canonical renormalisation that coincides with Marcus' canonical extension for semimartingale driving signals. This is contrasted with the canonical geometric rough path of Hairer and Kelly by means of a recursive formula defined in terms of the coaction of the substitution bialgebra.

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

In this paper we discuss the construction of probability distributions on the group \(\text {SL}(2,\mathbb {C})\) and the Mobius group using the exponential map. In particular, we describe the injectivity and surjectivity domains of the exponential map and provide its Jacobian determinant. We also show that on \(\text {SL}(2,\mathbb {C})\) and the Mobius group, there are no isotropic distributions in the group sense.

A theory of local convexity for a second order differential equation (sode) on a Lie algebroid is developed. The particular case when the sode is homogeneous quadratic is extensively discussed., JCM acknowledges financial support from the IUMA (University of Zaragoza, Spain) and the Spanish Ministry of Science and Innovation under grant PGC2018-098265-B-C32. D. Martín de Diego acknowledges financial support from the Spanish Ministry of Science and Innovation, under grant PID2019-106715GB-C21 and from the Spanish Ministry of Science and Innovation, through the \"Severo Ochoa Programme for Centres of Excellence in R&D\" (CEX2019-000904-S). EMF acknowledges nancial support from the Spanish Ministry of Science and Innovation under grant PGC2018-098265-B-C31.

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 consider a possible framework to categorify the exponential map exp(-f) given the categorification of a generator f of $\frak{sl}_2$ by Lauda. In this setup the Taylor expansions of exp(-f) and exp(f) turn into complexes built out of categorified divided powers of f. Hom spaces between tensor powers of categorified f are given by diagrammatics combining nilHecke algebra relations with those for a additional "short strand" generator. The proposed framework is only an approximation to categorification of exponentiation, because the functors categorifying exp(f) and exp(-f) are not invertible., 27 pages

We give an algebraic description of the structure of the analytic universal cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we classify the models of theories of ‘universal covers’ of rigid divisible commutative finite Morley rank groups.

The exponential of a matrix plays an important role in the theory of Lie groups. The main purpose of this paper is to examine matrix groups over the split quaternions and the exponential map from their Lie algebras into the groups. Since the set of split quaternions is a noncommutative algebra, the way of computing the exponential of a matrix over the split quaternions is more difficult than calculating the exponential of a matrix over the real or complex numbers. Therefore, we give a method of finding exponential of a split quaternion matrix by its complex adjoint matrix.

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.

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

We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over $p$-adic rings extends uniquely to a cohomology theory for varieties over $p$-adic fields that satisfies $h$-descent. This new cohomology - syntomic cohomology - is a Bloch-Ogus cohomology theory, admits period map to \'etale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on the \'etale side. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soul\'e's \'etale regulators land in the potentially semistable Selmer groups. Our construction of syntomic cohomology is based on new ideas and techniques developed by Beilinson and Bhatt in their recent work on $p$-adic comparison, Comment: Final version; to appear in Algebra and Number Theory

We show that any element of the special linear group $SL_2(R)$ is a product of two exponentials if the ring $R$ is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to $SL_2(\mathbb{C})$ is not surjective. Our result extends to the linear group $GL_2(R)$., Comment: 9 pages

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.

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

In Cartier (Fonctions polylogarithmes, nombres polyzetas et groupes pro-unipotents. Sem. BOURBAKI, 53eme 2000–2001, no. 885), Pierre Cartier conjectured that for any non-commutative formal power series Φ on X={x 0,x 1} with coefficients in a $\mathbb{Q}$ -extension, A, subjected to some suitable conditions, there exists a unique algebra homomorphism φ from the $\mathbb{Q}$ -algebra generated by the convergent polyzetas to A such that Φ is computed from the Φ KZ Drinfel’d associator by applying φ to each coefficient. We prove that φ exists and that it is a free Lie exponential map over X. Moreover, we give a complete description of the kernel of ζ and draw some consequences about the arithmetical nature of the Euler constant and about an algebraic structure of the polyzetas.

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.

We define a class of derivations on the algebra of the non-commutative power series in two variables and study the structure of the Lie algebra generated by them. After detecting the corresponding automorphisms under the exponential map explicitly, we clarify the structure of the automorphism group by means of the Hausdorff group associated to the Lie algebra.

Recently we have used the Carlitz exponential map to define a finitely generated submodule of the Carlitz module having the right properties to be a function field analogue of the group of units in a number field. Similarly, we constructed a finite module analogous to the class group of a number field. In this short note more algebraic constructions of these "unit" and "class" modules are given and they are related to Ext modules in the category of shtukas., 9 pages

In this paper we define a p-adic analogue of the Borel regulator for the K-theory of p-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this p-adic regulator to the Bloch–Kato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups. We also show that the Soulé regulator is induced by continuous and even analytic classes.

The problem of finding the correct asymptotic rate of approximation by polynomial loops in dependence of the smoothness of the elements of a loop group seems not well-understood in general. For matrix Lie groups such as SU(N), it can be viewed as a problem of nonlinearly constrained trigonometric approximation. Motivated by applications to optical FIR filter design and control, we present some initial results for the case of SU(N)-loops, N>1. In particular, using representations via the exponential map and ideas from splitting methods, we prove that the best approximation of an SU(N)-loop belonging to a Hoelder-Zygmund class Lip_alpha with alpha>1/2 by a polynomial SU(N)-loop of degree n is of the order O(n^{-\alpha/(1+\alpha)}) as n tends to infinity. Although this approximation rate is not considered final (and can be improved in special cases), to our knowledge it is the first general, nontrivial result of this type., Comment: 14 pages, still submitted to J. Approx. Th. typos corrected, part of the proof of Lemma 4 concerning the auxiliary statement on page 8 rewritten in a clearer way

We show that every source connected Lie groupoid always has global bisections through any given point. This bisection can be chosen to be the multiplication of some exponentials as close as possible to a prescribed curve. The existence of bisections through more than one prescribed point is also discussed. We give some interesting applications of these results.

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.

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)

This is a paper with two aims. First, we show that the map from $\mathbb{Z}/p\mathbb{Z}$ to itself defined by exponentiation $x\to m^x$ has few $3$-cycles -- that is to say, the number of cycles of length three is $o(p)$. This improves on previous bounds. Our second objective is to contribute to an ongoing discussion on how to find a non-sofic group. In particular, we show that, if the Higman group were sofic, there would be a map from $\mathbb{Z}/p\mathbb{Z}$ to itself, locally like an exponential map, yet satisfying a recurrence property., 27 pages. Revised version. We address results by Glebsky and Kassabov-Kuperberg-Riley that go against the heuristic, allowing for the construction of functions that behave like $x\to m^x$, $m\ne 2$ and satisfy a recurrence property

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.

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

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.

Based on the previously developed technique using the Laplace transformation of distributions with compact support, we calculate the Hochschild cohomologies of the algebra of smooth functions on a finite-dimensional real vector space with coefficients in the adjoint representation.

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

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.

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.

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.