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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.

Dada uma aplicação f: B -> A entre duas C*-álgebras, o cone dessa aplicação, Cf, é o conjunto formado pelos pares (b,g) pertencentes à soma direta da C*-álgebra B com o cone CA tais que f(b) = g(0), sendo CA o cone de A. Neste trabalho estudamos o funtor determinado pela associação da sequência exata curta 0 -> SA -> Cf -> B -> 0 para cada *-homomorfismo f: B -> A, e demonstramos que esse funtor é exato. Caracterizamos as aplicações de conexão associadas à sequência exata 0 -> SA -> Cf -> B -> 0, mostrando que a aplicação do índice é dada por tAK1(f) e que a aplicação exponencial é dada por bAK0(f), sendo tA o isomorfismo entre K1(A) e K0(SA) e bA a aplicação de Bott. Por fim, usando que toda sequência exata curta de C*-álgebras pode ser vista na forma 0 -> Ker f -> B -> A -> 0, mostramos que as aplicações de conexão d1 e d0 associadas a cada sequência exata curta podem ser dadas por dn = Kn+1(j)-1 Kn+1(i) hn, em que j é a inclusão do núcleo de f em Cf, i é a inclusão da suspensão SA também em Cf, hn = bA e h1 = tA . If f: B A is a map between the C*-algebras A and B, the mapping cone is the set of pairs (b,g) in the direct sum of B and CA such that f(b) = g(0), where CA is the cone of A. In this work, we study the functor determined by the assignment of the exact sequence 0 SA Cf B 0 to each *-homomorphism f: B -> A, and we show that this functor is exact. We characterize the connecting maps associated with the short exact sequence 0 SA Cf B 0 and we prove that its index map is tA K1(f) and that its exponential map is bA K0(f), where tA is the isomorphism between K1(A) and K0(SA), and bA is the Bott map. Finally, using that every short exact sequence of C*-algebras can be seen as 0 Ker f B (f ) A 0, we prove that the connecting maps, d1 and d0, associated with a short exact sequence are given by dn = Kn+1(j)-1 Kn+1(i) hn, where j is the inclusion of f\'s kernel in Cf, i is the inclusion of the suspension SA in Cf, hn = bA e h1 = tA .

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.

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 paper we explore the relation between locally pseudocompact groups and b ƒ - spaces . Namely, we point out that b ƒ - continuous functions appear in a natural way when studying the exponential map. We introduce the concept of b ƒ - group and we prove that Σ-spaces of an arbitrary product of locally pseudocompact groups are b ƒ - groups . We show that the Dieudonne topological completion of an arbitrary product of locally pseudocompact groups agrees with its bilateral completion and then it is a topological group. We also study the distribution of the functor of the Dieudonne topological completion for coset spaces and characterize when the product of two b ƒ - groups is a b ƒ - group .

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 introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allow us to establish a Linearization Theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of the Weinstein-Zung Linearization Theorem for proper Lie groupoids. This new notion of metric has a simplicial nature which will be explored in future papers of this series., Comment: 29 pages; Final version accepted for publication in Journal f\"ur die reine und angewandte Mathematik (Crelle)

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.

We prove that the natural homomorphism from Kirchberg's ideal-related KK-theory, KKE(e, e'), with one specified ideal, into Hom_{\Lambda} (\underline{K}_{E} (e), \underline{K}_{E} (e')) is an isomorphism for all extensions e and e' of separable, nuclear C*-algebras in the bootstrap category N with the K-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the K_{1}-groups of the quotients being free abelian groups. This class includes all Cuntz-Krieger algebras with exactly one non-trivial ideal. Combining our results with the results of Kirchberg, we classify automorphisms of the stabilized purely infinite Cuntz-Krieger algebras with exactly one non-trivial ideal modulo asymptotically unitary equivalence. We also get a classification result modulo approximately unitary equivalence. The results in this paper also apply to certain graph algebras., Comment: 26 pages

Let $H$ be a linear algebraic group over an algebraically closed field of characteristic $p>0$. We prove that any "exponential map" for $H$ induces a bijection between the variety of $r$-tuples of commuting $[p]$-nilpotent elements in $Lie(H)$ and the variety of height $r$ infinitesimal one-parameter subgroups of $H$. In particular, we show that for a connected reductive group $G$ in pretty good characteristic, there is a canonical exponential map for $G$ and hence a canonical bijection between the aforementioned varieties, answering in this case questions raised both implicitly and explicitly by Suslin, Friedlander, and Bendel.

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

The dually conjugate Hopf superalgebras Funp,q(GL(1∣1)) and Up,q(gl(1∣1)) are studied using the Fronsdal-Galindo approach and the full Hopf structure of Up,q(gl(1∣1)) is extracted. A finite expression for the universal T-matrix, identified with the dual form and expressing the generalization of the exponential map of the classical groups, is obtained for Funp,q(GL(1∣1)). In a representation with a colour index, the T-matrix assumes a form that satisfies a coloured graded Yang-Baxter equation.

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 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.

The algebra of exponential fields and their extensions is developed. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, finitely presented extensions are defined, it is shown that finitely generated strong extensions are finitely presented, and these extensions are classified. An algebraic construction is given of Zilber's pseudo-exponential fields. As applications of the general results and methods of the paper, it is shown that Zilber's fields are not model-complete, answering a question of Macintyre, and a precise statement is given explaining how Schanuel's conjecture answers all transcendence questions about exponentials and logarithms. Connections with the Kontsevich-Zagier, Grothendieck, and Andr\'e transcendence conjectures on periods are discussed, and finally some open problems are suggested., Comment: 39 pages; v4 Minor changes and an improved discussion of the connections with transcendence theory

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.

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.

Let G be a simply connected solvable analytic group. We say that G is an exponential group if its exponential map is bijective. We say that G is an exponential group with real eigenvalues if Ad g has only real eigenvalues for every g in G , where Ad is the adjoint representation of G . By a lattice in G , we mean a discrete subgroup Γ of G such that G Γ is compact. Wedenote the collection of all lattices in G by L ( G ) with the Chabauty topology induced from limit of lattices. Let A ( G ) be the group of all topological automorphisms of G . Equipped with the compact-open topology, A ( G ) acts on L ( G ) continuously. We prove that for every exponential group G with real eigenvalues and for every lattice Γ in G , the orbit A ( G ) Γ is locally compact, and we give a counterexample to the case when G is merely an exponential group.

For a number field, we have a Tannaka category of mixed Tate motives at our disposal. We construct p-adic points of the associated Tannaka group by using p-adic Hodge theory. Extensions of two Tate objects yield functions on the Tannaka group, and we show that evaluation at our p-adic points is essentially given by the inverse of the Bloch-Kato exponential map., 24 pages

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.

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 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.

Let G be a Lie group. On the trivial principal G-bundle over the Lie algebra of G there is a natural connection whose curvature is the Lie bracket. The exponential map is given by parallel transport of this connection. If G is the diffeomorphism group of a manifold, the curvature of the natural connection is the Lie bracket of vectorfields on the manifold. The motion of a ball rolling on an oriented surface is the parallel transport of a similar connection on the trivial SO(3)-bundle over the surface. If the surface is a plane or a sphere, then the curvature of the connection is a scalar multiple of the Lie bracket in the Lie algebra of SO(3)., Comment: 13 pages, revised and expanded

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.

Identifying the algebra of exponential generating series with the shuffle algebra of formal power series, one can define an exponential map ${\mathop{exp}}_!:X\mathbb K[[X]]\longrightarrow 1+X\mathbb K[[X]]$ for the associated Lie group formed by exponential generating series with constant coefficient 1 over an arbitrary field $\mathbb K$. The main result of this paper states that the map ${\mathop{exp}}_!$ (and its inverse map ${\mathop{log}}_!$) induces a bijection between rational, respectively algebraic, series in $X\mathbb K [[X]]$ and $1+X\mathbb K[[X]]$ if the field $\mathbb K$ is a subfield of the algebraically closed field $\bar{\mathbb F}_p$ of characteristic $p$., Comment: 25 pages

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)