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Let G be a 2-group and B ( G ) × denote the group of units of the Burnside ring of G. For each subquotient H / K of G, there is a generalized induction map from B ( H / K ) × to B ( G ) × defined as the composition of inflation and multiplicative induction maps. We prove that the product of generalized induction maps ∏ B ( H / K ) × → B ( G ) × is surjective when the product is taken over the set of all subquotients that are isomorphic to the trivial group or a dihedral 2-group of order 2 n with n ⩾ 4 . As an application, we give an algebraic proof for a theorem by Tornehave [The unit group for the Burnside ring of a 2-group, Aarhus Universitet Preprint series 1983/84 41, May 1984] which states that tom Dieck's exponential map from the real representation ring of G to B ( G ) × is surjective. We also give a sufficient condition for the surjectivity of the exponential map from the Burnside ring of G to B ( G ) × .

The notion of group—‘concrete’ as a group of transformations of a set or system of differential equations – goes back to the eigthteenth century.

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

We define and study analogues of exponentials for functions on noncommutative two-tori that depend on a choice of a complex structure. The major difference with the commutative case is that our noncommutative exponentials can be defined only for sufficiently small functions. We show that this phenomenon is related to the existence of certain discriminant hypersurfaces in an irrational rotation algebra. As an application of our methods we give a very explicit characterization of connected components in the group of invertible elements of this algebra., Comment: AMSLatex, 23 pages, minor corrections

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.

Let E be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field K. The theory of complex multiplication associates E with a Hecke character ψ. The Hasse-Weil L-function of E equals the Hecke L-function of ψ, whose special value at s = 1 encodes important arithmetic information of E,as predicted by the Birch and Swinnerton-Dyer conjecture and verified by Rubin[Ru] when the special value is non-zero. For integers k, j, special values of the Hecke L-function associated to the Hecke character \({\psi ^k}{\bar \psi ^j}\) should encode arithmetic information of the Hecke character \({\psi ^k}{\bar \psi ^j}\) as predicted by the Bloch-Kato conjecture[B-K]. When p is a prime where E has good, ordinary reduction, the p-part of the conjecture has been verified when j = 0[Ha] and when j≠ 0, p > k[Gu]. To verify the conjecture in other cases, it is important to have an explicit description of the exponential map of Bloch and Kato. In this paper we provide such an explicit exponential map for the case j≠ 0, p ∤ k, by studying relation between the Hecke characters and formal group characters.

In this paper we characterize commutative Fréchet-Lie groups using the exponential map. In particular we prove that if a commutative Fréchet-Lie group G has an exponential map, which is a local diffeomorphism, then G is the limit of a projective system of Banach-Lie groups. © Akadémiai Kiadó.

For the quantum group $GL_{p,q}(2)$ and the corresponding quantum algebra $U_{p,q}(gl(2))$ Fronsdal and Galindo explicitly constructed the so-called universal $T$-matrix. In a previous paper we showed how this universal $T$-matrix can be used to exponentiate representations from the quantum algebra to get representations (left comodules) for the quantum group. Here, further properties of the universal $T$-matrix are illustrated. In particular, it is shown how to obtain comodules of the quantum algebra by exponentiating modules of the quantum group. Also the relation with the universal $R$-matrix is discussed., LaTeX-file, 7 pages. Submitted for the Proceedings of the 4th International Colloquium ``Quantum Groups and Integrable Systems,'' Prague, 22-24 June 1995

Unfortunately, it is not clear whether this applies to our cases. To solve this problem,we define a map sharing some of the properties of the exponential map, with the advan-tage of being defined whenever p is odd, and prove that some suitable sections of P arestrong subgroups.2. Preliminary results. Let G denote one of the above mentioned groups of Lie typeand set F = f

Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra $U_{p,q}(gl(2))$ to the quantum group $GL_{p,q}(2)$, we show how the $(2j+1)$-dimensional representations of $GL_{p,q}(2)$ can be obtained by `exponentiating' the well-known $(2j+1)$-dimensional representations of $U_{p,q}(gl(2))$ for $j$ $=$ $1,{3/2},... $; $j$ $=$ 1/2 corresponds to the defining 2-dimensional $T$-matrix. The earlier results on the finite-dimensional representations of $GL_q(2)$ and $SL_q(2)$ (or $SU_q(2)$) are obtained when $p$ $=$ $q$. Representations of $U_{\bar{q},q}(2)$ $(q$ $\in$ $\C \backslash \R$ and $U_q(2)$ $(q$ $\in$ $\R \backslash \{0\})$ are also considered. The structure of the Clebsch-Gordan matrix for $U_{p,q}(gl(2))$ is studied. The same Clebsch-Gordan coefficients are applicable in the reduction of the direct product representations of the quantum group $GL_{p,q}(2)$., 17 pages, LaTeX (latex twice), no figures. Changes consist of more general formula (4.13) for T-matrices, explicit Clebsch-Gordan coefficients, boson realization of group parameters, and typographical corrections

The nature of the visual system is discussed. It achieves “constancy” and shape recognition by means of an exact map. The ideals of this mapping are invariants of G V ,the Lie transformation group of the constancies and of shape recognition, thus generating “perception by exception”. The neuropsychological correlates, both neuronal and psychological, of the Lie transformation group are discussed at length in terms of the Bishop-Coombs-Henry model of the basic neocortical circuit and are shown to constitute a hyperbolic dynamical system. The local and global topologies of the latter are analysed. Shape recognition via annulment by the Lie derivatives of the constancies is discussed. Shape generation by means of the Lie transformation group of the constancies, using for this purpose the exponential map and “dragging the flow” along the group orbits, is then illustrated.

In this article we extend our previous results for the orthogonal group, $SO(2,4)$, to its homomorphic group $SU(2,2)$. Here we present a closed, finite formula for the exponential of a $4\times 4$ traceless matrix, which can be viewed as the generator (Lie algebra elements) of the $SL(4,C)$ group. We apply this result to the $SU(2,2)$ group, which Lie algebra can be represented by the Dirac matrices, and discuss how the exponential map for $SU(2,2)$ can be written by means of the Dirac matrices., Comment: 10 pages

We give a detailed discussion of the group ${\rm Sp}(2,\bold R)$, organized in such a way as to lead to explicit constructive descriptions of the metaplectic group ${\rm Mp}(2)$ and the universal covering group $\overline{{\rm Sp}(2,\bold R)}$. The aim is to make clear in easily visible fashion the global topological relationships among these groups of physical relevance, and to make practical calculations with them feasible. The properties of one-parameter subgroups and the exponential map, and of the Iwasawa decomposition, are also investigated in detail for these groups.

Let G be a connected reductive complex Lie group. Let EGbe the image of the exponential map of G and E'Gits complement in G. We give a purely algebraic characterization of the set EGand also describe an algorithm for finding all conjugacy classes of G in E'G. We are mainly interested in the case when the Lie algebra of G is simple and exceptional. Full details are provided for groups G of type G2, F4, and E6. If G is of type G2 then there are only two such conjugacy classes.

A convenient structure of Lie group to the entire group Aut P of G-automorphisms of a principal G-bundle without any assumption of compactness on the structure group G or on the base manifold. Its Lie algebra and the exponential map are illustrated. Some relevant principal bundles are discussed having Aut P or its subgroup Gau P of gauge transformations as structure group.

A finite aspherical complex X is constructed whose group of homotopy classes of basepoint-preserving self-homotopy-equivalences is infinitely generated. 0. Introduction. Let EX denote the H-space of basepoint-preserving homotopy equivalences from X to X. The set 7ro(EX) becomes a group with multiplication induced by composition of representatives; it is denoted E (X) and called, by abuse of language, the group of homotopy equivalences of X. It was proved independently by D. Sullivan [Su] and C. Wilkerson [WI that & (X) is finitely presented when X is a simply-connected finite complex. In contrast, Frank and Kahn [F-K] showed that & (S' V SP V S2 41) is infinitely generated for p > 2. They also suggested the following alternative method for producing examples. Start with Lewin's [L] example of a finitely-presented group G with Aut(G) infinitely generated. It is well known (see e.g. [S, p. 427]) that if X is a K(XT, 1)-complex then ; (X) Aut(,f), and therefore any K(G, 1)-complex would have an infinitely-generated group of homotopy equivalences. In this note, I will construct a finite K(G, 1)-complex X, note some of its salient features, and raise some questions. 1. Construction of X. Let Gy be the group with presentation Py = and let GM be the group with presentation PM= . We will construct a K(Gy, 1)-complex Y and a K(GM, 1)-complex M. Let exp: C -+ C be the exponential map. Let T = S' X S' c C2 and let Pa Pb: I -> T be the loops pa(t) = (exp(27rit), 1) and pb(t) = (1, exp(27rit)). Then a = and b = generate ff1(T, (1, 1)) Z ED Z. If f: T-> T is the 4-fold covering map defined by f(exp(2'71itl), exp(2ffit2)) = (exp(4sfit,), exp(4sfit2)) Received by the editors September 7, 1979. AMS (MOS) subject classifications (1970). Primary 55D10, 55E05; Secondary 20E40, 20F55, 55D20, 57B 10.

Let G be a connected, complex, semi-simple Lie group Let g be an element in G. Let B be a Borel subgroup of G and g in B. Let m and n be the least positive integers such that the element gm lies on a one-parameter subgroup in G and the element gn lies on a one-parameter subgroup in B. We denote these integers by ind G (g) and ind B (g). In this note we prove the conjecture ind G (g) = ind B (g), if g is regular.

The group of gauge transformations of a smooth principal bundle P(M,G) over a not necessarily compact manifold M and with a not necessarily compact structure group G is proved to be a Schwartz–Lie group. Its Lie algebra and exponential map are discussed.

A new generalization of the rotation group involving a skew circulant matrix is given. Using the exponential map, a unified treatment is given to this generalization and to one due to Ungar. The special functions associated with the corresponding Lie groups are the trigonometric and hyperbolic functions of order n . Infinitesimal generators and invariants under the corresponding transformations are also obtained. A general theorem on linear transformations involving circulant and skew circulant matrices is also given.

In this paper, we establish a conjugacy theorem of homomorphisms of a locally compact connected semisimple group into a locally compact group.

We give a simple description of the interior, the exterior, and the boundary of the image of the exponential map for each of the following classes of complex Lie groups: GLfn C SLfn C Ofn C Spf2n C and also for each of their real forms. There is one exception: namely for special unitary groups we are not able to describe, in general, the interior and the boundary of the exponential image. In the cases GLfn R and SLfn R the results are due to M. Nishikawa, who has also handled the case of real orthogonal groups O(p,q) when 1⩽p⩽q⩽3.

In this paper we list all simple real Lie algebras g for which there exist connected Lie groups with dense images of the exponential function. We also describe the simple real Lie algebras for which the exponential functions of the associated simply connected Lie groups have dense images. Let us say that a Lie group is weakly exponential if the image of its exponential function is dense. Hofmann and Mukherjea (On the density of the image of the exponential function,Math. Ann.234(1978), 263–273) show how to reduce the problem of determining whetherGis weakly exponential to the semisimple case. We also give some methods which are useful in determining whether a reductive Lie group is weakly exponential or not. Our method is based on the fact that a maximal rank subgroup of a weakly exponential Lie group inherits the property of being weakly exponential. This finally permits us to characterize the reductive Lie algebras having a weakly exponential group of inner automorphisms as those where the centralizer of the compact part of a maximally non-compact Cartan subalgebra has a commutator algebra isomorphic to a product of s l (2, R )-factors. For the groups Sl(n, R ), Sp(n, R ), and SO(p, q)0, 2≤p,q,p,qeven, Hofmann and Mukherjea show that they are not weakly exponential. For the other classical groups the results of Đokovic (The interior and the exterior of the image of the exponential map in classical Lie groups,J. Algebra112(1988), 90–109) provide information as to whether they are weakly exponential or not. It is a classical results that complex and compact simple Lie groups are weakly exponential.

Let G \mathfrak {G} be a connected (real or complex) Lie group with Lie algebra G. Define a conjugate point g of G \mathfrak {G} as a point g = exp ⁡ x g = \exp x for some x ∈ G x \in G and d exp x d{\exp _x} is a noninvertible linear map. We prove that g ∈ G g \in \mathfrak {G} is a conjugate point if and only if g = exp ⁡ x λ g = \exp {x_\lambda } for at least a (complex parameter) family of elements x λ ( λ ∈ C ) {x_\lambda }(\lambda \in {\mathbf {C}}) in G.

This section contains a brief introduction to matrix Lie groups. The focus is on developing tools for the construction of factorizations of a group with respect to some subgroup. In this connection the Lie algebra and the exponential map are the central concepts. It may be noted that all the groups occurring in the examples considered in these notes are matrix Lie groups.

We consider the Heisenberg group N as being the simply connected Lie group of Lie algebra η. Via the exponential map exp, N is identified with the 2n + 1 vector space V + ℝE with the multiplication law: $$ \exp \left( {v + tE} \right) \cdot \exp \left( {v' + t'E} \right) = \exp \left( {v + v' + \left( {t + t' + \frac{{B\left( {v,v'} \right)}}{2}} \right)E} \right) $$ where v,v′ Є V, t,t′ Є ℝ

The two main results in this chapter concern a class of Lie groups which contains all compact connected Lie groups, namely the class of complete connected Lie groups which possess invariant Riemannian metrics. For groups in this class it is first shown that geodesies and translates of 1-parameter subgroups are essentially the same (Theorem 4. 3. 3) and then, as a corollary, that the associated exponential map is surjective (Corollary 4. 3. 5). These results are presented solely for their own interest and are not needed to establish the main structure Theorems described in Chapter 6. The first section of this chapter contains a brief discussion of Riemannian manifolds (with some statements being given without proof) while the second section is concerned primarily with the establishment of a necessary and sufficient condition for a Lie group to possess an invariant Riemannian metric. In the third section these preliminaries culminate with the proofs of the main results described above. Riemannian manifolds Throughout this section we will suppose that M is an analytic manifold. By selecting p in M and imposing a norm on T p (M) we can arrive at a notion of distance in an ‘infinitesimal’ neighbourhood of p in the following way. Temporarily ignoring all the usual caveats against talking about infinitesimals, let q be a point in this neighbourhood of p and let ξ be an analytic curve on M with ξ(0) = p and ξ(δt)=q.

The philosophy behind this chapter is that Lie groupoids and Lie algebroids are much like Lie groups and Lie algebras, even with respect to those phenomena – such as connection theory – which have no parallel in the case of Lie groups and Lie algebras. We begin therefore with an introductory section, §1, which treats the differentiable versions of the theory of topological groupoids, as developed in Chapter II, §§1-6. Note that a Lie groupoid is a differentiable groupoid which is locally trivial. Most care has to be paid to the question of the submanifold structure on the transitivity components, and on subgroupoids. §2 introduces Lie algebroids, as briefly as is possible preparatory to the construction in §3 of the Lie algebroid of a differentiable groupoid. The construction given in §3 is presented so as to emphasize that it is a natural generalization of the construction of the Lie algebra of a Lie group. One difference that might appear arbitrary is that we use right-invariant vector fields to define the Lie algebroid bracket, rather than the left-invariant fields which are standard in Lie group theory. This is for compatibility with principal bundle theory, where it is universal to take the group action to be a right action. In §4 we construct the exponential map of a differentiable groupoid, and give the groupoid version of the standard formulas relating the adjoint maps and the exponential.

This survey of the algebraic structure of Lie groups and Lie algebras (mainly semi simple) is a considerably expanded version of the oral lectures at the symposium. It is limited to what is necessary for representation theory, which is another way of saying that very little has been left out. In spite of its length, it contains few proofs or even indications of proofs, nor have I given chapter and verse for each of the multitude of unproved assertions throughout the text. Instead, I have appended references to each section, from which the diligent reader should have no difficulty in tracking down the proofs. Lie Groups and Lie Algebras Vector fields Let M be a smooth (C ∞ ) manifold, and for each point x ∈ M let T x (M) denote the vector space of tangent vectors to M at x. The union of all the T x (M) is the tangent bundle T(M) of M, Locally, if U if a coordinate neighbourhood in M, the restriction of T(M) to U is just U × R n , where n is the dimension of M. Each smooth map ϕ: M → N, where N is another smooth manifold, gives rise to a tangent map T(ϕ) : T(M) → T(N), whose restriction T x (M) to the tangent space T x (M) is a linear mapping of T x (M) into T ϕ(x) , (N). In terms of local coordinates in M and N, T x (ϕ) is given by the x Jacobian matrix.

Let $\mathscr{S}_{r}$, be the real topological vector space of real-valued rapidly decreasing functions and let $\mathcal{O}(\mathscr{S}_{r})$ be the group of rotations of $\mathscr{S}_{r}$. Then every one-parameter subgroup of $\mathcal{O}(\mathscr{S}_{r})$ induces a flow in $\mathscr{S}_{r}^{*}$ the conjugate space of $\mathscr{S}_{r}$ with the Gaussian White Noise as an invariant measure. ¶ The author constructed a group of functions which is isomorphic to a subgroup of $\mathcal{O}(\mathscr{S}_{r})$ and some of its one-parameter subgroups. ¶ But the problem whether it contains sufficiently many one-parameter subgroups has been a problem. In Part I of the present paper, we answer this problem affirmatively by constructing two classes of one-parameter subgroups in a concrete way. ¶ In Part II, we construct an infinite dimensional Lie subgroup of $\mathcal{O}(\mathscr{S}_{r})$ and the corresponding Lie algebra. Namely, we construct a topological subgroup $\mathfrak{G}$ of $\mathcal{O}(\mathscr{S}_{r})$ which is coordinated by the nuclear space $\mathscr{S}_{r}$ and the algebra $\mathfrak{a}$ of generators of one-parameter subgroups of $\mathfrak{G}$ which is closed under the commutation. Furthermore, we establish the exponential map from $\mathfrak{a}$ into $\mathfrak{G}$ and prove continuity.

International audience; We address here the construction of wrapped probability densities on Lie groups and quotient of Lie groups using the exponential map. The paper starts by briefly reviewing the different approaches to build densities on a manifold and shows the interest of wrapped distributions. We then construct wrapped densities on SE(n) and discuss their statistical estimation. We conclude by an opening to the case of symmetric spaces.

This paper reviews the Euler–Rodrigues formula in the axis–angle representation of rotations, studies its variations and derivations in different mathematical forms as vectors, quaternions and Lie groups and investigates their intrinsic connections. The Euler–Rodrigues formula in the Taylor series expansion is presented and its use as an exponential map of Lie algebras is discussed particularly with a non-normalized vector. The connection between Euler–Rodrigues parameters and the Euler–Rodrigues formula is then demonstrated through quaternion conjugation and the equivalence between quaternion conjugation and an adjoint action of the Lie group is subsequently presented. The paper provides a rich reference for the Euler–Rodrigues formula, the variations and their connections and for their use in rigid body kinematics, dynamics and computer graphics.

Introduction. Suppose that G is an analytic subgroup and S is a subgroup of G. S is called a uniform subgroup of G if G/S is compact. S is called a full subgroup of G if the only analytic subgroup of G that contains S is G itself. The notion of full subgroup is first brought up by G. D. Mostow (131). Full subgroups of exponential groups (a simply connected solvable analytic group is said to be exponential if its exponential map is bijective) have been studied by the authors ([2]). It is the aim of this paper to study full subgroups of simply connected solvable analytic groups in genleral. In view of a result of G. D. Mostow ([3], Theorem on P. 12), in simply conniected solvable analytic groups, every closed uniform subgroup is a full subgroup. In section one, we give a condition when a full subgroup is a uniformll subgroup. We have the following theorem.