Showing papers on "Lie group published in 2002"

Coherent states for arbitrary Lie group

Abstract: The concept of coherent states originally closely related to the nilpotent group of Weyl is generalized to arbitrary Lie group. For the simplest Lie groups the system of coherent states is constructed and its features are investigated.

Moment Maps, Cobordisms, and Hamiltonian Group Actions

Abstract: Introduction Part 1. Cobordism: Hamiltonian cobordism Abstract moment maps The linearization theorem Reduction and applications Part 2. Quantization: Geometric quantization The quantum version of the linearization theorem Quantization commutes with reduction Part 3. Appendices: Signs and normalization conventions Proper actions of Lie groups Equivariant cohomology Stable complex and Spin$^{\mathrm{c}}$structures Assignments and abstract moment maps Assignment cohomology Non-degenerate abstract moment maps Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization The Kawasaki Riemann-Roch formula Cobordism invariance of the index of a transversally elliptic operator Bibliography Index.

Introduction to Symmetry Analysis

Abstract: Preface 1. Introduction to symmetry 2. Dimensional analysis 3. Systems of ODE's, first order PDE's, state-space analysis 4. Classical dynamics 5. Introduction to one-parameter Lie groups 6. First order ordinary differential equations 7. Differential functions and notation 8. Ordinary differential equations 9. Partial differential equations 10. Laminar boundary layers 11. Incompressible flow 12. Compressible flow 13. Similarity rules for turbulent shear flows 14. Lie-Backlund transformations 15. Invariance condition for integrals, variational symmetries 16. Backlund transformations and non-local groups Appendix 1. Review of calculus and the theory of contact Appendix 2. Invariance of the contact conditions under Lie point transformation groups Appendix 3. Infinite-order structure of Lie-Backlund transformations Appendix 4. Symmetry analysis software.

D branes on group manifolds and fusion rings

Abstract: In this paper we compute the charge group for symmetry preserving D-branes on group manifolds for all simple, simply-connected, connected compact Lie groups G.

Newton's method on Riemannian manifolds and a geometric model for the human spine

Abstract: To study a geometric model of the human spine we are led to finding a constrained minimum of a real valued function defined on a product of special orthogonal groups. To take advantge of its Lie group structure we consider Newton's method on this manifold. Comparisons between measured spines and computed spines show the pertinence of this approach.

Lie Groups: An Introduction through Linear Groups

Abstract: Preface 1 The exponential map 2 Lie theory 3 The classical groups 4 Manifolds, homogeneous spaces, Lie groups 5 Integration 6 Representations Appendix: Analytic Functions and Inverse Function Theorem References Index

Unitary Representations of Uq(??}(2,ℝ)),¶the Modular Double and the Multiparticle q-Deformed¶Toda Chain

Abstract: The paper deals with the analytic theory of the quantum q-deformed Toda chains; the technique used combines the methods of representation theory and the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L. Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N-particle q-deformed open Toda chain are given as a multiple integral of the Mellin–Barnes type. For the periodic chain the two dual Baxter equations are derived.

On the structure theory of the Iwasawa algebra of a p-adic Lie group

Abstract: This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, Λ of a p-adic analytic group G. For G without any p-torsion element we prove that Λ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-nullΛ-module. This is classical when G=ℤ k p for some integer k≥1, but was previously unknown in the non-commutative case. Then the category of Λ-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the ℤ p -torsion part of a finitely generated Λ-module. We also prove a local duality theorem and a version of Auslander-Buchsbaum equality. The arithmetic applications to the Iwasawa theory of abelian varieties are published elsewhere.

Infinite-dimensional Lie groups without completeness restrictions

Abstract: We describe a setting of infinite-dimensional smooth (resp., analytic) Lie groups modelled on arbitrary, not necessarily sequentially complete, locally convex spaces, generalizing the framework of Lie theory formulated in [R. Hamilton, The inverse function theorem of Nash and Moser , Bull. Amer. Math. Soc. 7 (1982), 65–222] for Fréchet modelling spaces and in [J. Milnor, Remarks on infinite-dimensional Lie groups, in: B. DeWitt and R. Stora (eds.), Relativity, Groups and Topology II, North-Holland, 1983] for sequentially complete modelling spaces. Our studies were dictated by the needs of infinite-dimensional Lie theory in the context of the existence problem of universal complexifications. We explain why satisfactory results in this area can only be obtained if the requirement of sequential completeness is abandoned. Introduction. The Fundamental Theorem of Calculus, asserting that

Finite Elastoplasticity Lie Groups and Geodesics on SL(d)

Abstract: The notions of nonlinear plasticity with finite deformations is interpreted in the sense of Lie groups. In particular, the plastic tensor P=F p −1 is considered as element of the Lie group SL(d). Moreover, the plastic dissipation defines a left-invariant Finsler metric on the tangent bundle of this Lie group. In the case of single crystal plasticity this metric is given interms of the different slip systems and is piecewise affine on each tangent space. For von Mises plasticity the metric is a left-invariant Riemannian metric. A main goal is to study the associated distance metric and the geodesic curves.

Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices

Abstract: The authors show that there is a generalization of Rodrigues’ formula for computing the exponential map exp: so(n) →SO(n) from skewsymmetric matrices to orthogonal matrices when n ≥ 4, and give a method for computing some determination of the (multivalued) function log: SO(n) → so(n). The key idea is the decomposition of a skew-symmetric n×n matrix B in terms of (unique) skew-symmetric matrices B1,...,Bp obtained from the diagonalization of B and satisfying some simple algebraic identities. A subproblem arising in computing logR, where R ∈SO(n), is the problem of finding a skewsymmetric matrix B, given the matrix B 2 , and knowing that B 2 has eigenvalues −1 and 0. The authors also consider the exponential map exp: se(n) →SE(n), where se(n) is the Lie algebra of the Lie group SE(n) of (affine) rigid motions. The authors show that there is a Rodrigues-like formula for computing this exponential map, and give a method for computing some determination of the (multivalued) function log: SE(n) → se(n). This yields a direct proof of the surjectivity of exp: se(n) →SE(n).

Group representations and the Euler characteristic of elliptically fibered Calabi–Yau threefolds

Abstract: To every elliptic Calabi-Yau threefold with a section X there can be associated a Lie group G and a representationof that group. The group is determined from the Weierstrass model, which has singularities that are generi- cally rational double points; these double points lead to local factors of G which are either the corresponding A-D-E groups or some associated non-simply laced groups. The representationis a sum of representations coming from the local factors of G, and of other representations which can be associated to the points at which the singularities are worse than generic. This construction first arose in physics, and the requirement of anomaly can- cellation in the associated physical theory makes some surprising predictions about the connection between X and �. In particular, an explicit formula (in terms of �) for the Euler characteristic of X is predicted. We give a purely math- ematical proof of that formula in this paper, introducing along the way a new invariant of elliptic Calabi-Yau threefolds. We also verify the other geometric predictions which are consequences of anomaly cancellation, under some (mild) hypotheses about the types of singularities which occur. As a byproduct we also discover a novel relation between the Coxeter number and the rank in the case of the simply laced groups in the "exceptional series" studied by Deligne.

Higher Yang-Mills Theory

Abstract: Electromagnetism can be generalized to Yang-Mills theory by replacing the group U(1)$ by a nonabelian Lie group This raises the question of whether one can similarly generalize 2-form electromagnetism to a kind of "higher-dimensional Yang-Mills theory" It turns out that to do this, one should replace the Lie group by a "Lie 2-group", which is a category C where the set of objects and the set of morphisms are Lie groups, and the source, target, identity and composition maps are homomorphisms We show that this is the same as a "Lie crossed module": a pair of Lie groups G,H with a homomorphism t: H -> G and an action of G on H satisfying two compatibility conditions Following Breen and Messing's ideas on the geometry of nonabelian gerbes, one can define "principal 2-bundles" for any Lie 2-group C and do gauge theory in this new context Here we only consider trivial 2-bundles, where a connection consists of a Lie(G)-valued 1-form together with an Lie(H)-valued 2-form, and its curvature consists of a Lie(G)-valued 2-form together with a Lie(H)-valued 3-form We generalize the Yang-Mills action for this sort of connection, and use this to derive "higher Yang-Mills equations" Finally, we show that in certain cases these equations admit self-dual solutions in five dimensions

Galois representations

Abstract: In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie groups In the second part we briefly review some limited recent progress on these conjectures

Geometric Quantization¶and the Generalized Segal--Bargmann Transform¶for Lie Groups of Compact Type

Abstract: Let K be a connected Lie group of compact type and let T *(K) be its cotangent bundle. This paper considers geometric quantization of T *(K), first using the vertical polarization and then using a natural Kahler polarization obtained by identifying T *(K) with the complexified group K ℂ. The first main result is that the Hilbert space obtained by using the Kahler polarization is naturally identifiable with the generalized Segal–Bargmann space introduced by the author from a different point of view, namely that of heat kernels. The second main result is that the pairing map of geometric quantization coincides with the generalized Segal–Bargmann transform introduced by the author. This means that the pairing map, in this case, is a constant multiple of a unitary map. For both results it is essential that the half-form correction be included when using the Kahler polarization. These results should be understood in the context of results of K. Wren and of the author with B. Driver concerning the quantization of (1+1)-dimensional Yang–Mills theory. Together with those results the present paper may be seen as an instance of “quantization commuting with reduction”.

Quantum superintegrability and exact solvability in n dimensions

Abstract: A family of maximally superintegrable systems containing the Coulomb atom as a special case is constructed in n-dimensional Euclidean space. Two different sets of n commuting second-order operators are found, overlapping in the Hamiltonian alone. The system is separable in several coordinate systems and is shown to be exactly solvable. It is solved in terms of classical orthogonal polynomials. The Hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine Lie algebra.

(Non-)abelian gauged supergravities in nine dimensions

Abstract: We construct five different two-parameter massive deformations of the unique nine-dimensional N = 2 supergravity. All of these deformations have a higher-dimensional origin via Scherk-Schwarz reduction and correspond to gauged supergravities. The gauge groups we encounter are SO(2), SO(1, 1)+, , + and the two-dimensional non-abelian Lie group A(1), which consists of scalings and translations in one dimension. We make a systematic search for half-supersymmetric domain walls and non-supersymmetric de Sitter space solutions. Furthermore, we discuss which of the supergravities can be considered as candidate low-energy limits of compactified superstring theory.

An SVD-based projection method for interpolation on SE(3)

Abstract: This paper develops a method for generating smooth trajectories for a moving rigid body with specified boundary conditions. Our method involves two key steps: 1) the generation of optimal trajectories in GA/sup +/ (n), a subgroup of the affine group in R/sup n/ and 2) the projection of the trajectories onto SE(3), the Lie group of rigid body displacements. The overall procedure is invariant with respect to both the local coordinates on the manifold and the choice of the inertial frame. The benefits of the method are threefold. First, it is possible to apply any of the variety of well-known efficient techniques to generate optimal curves on GA/sup +/ (n). Second, the method yields approximations to optimal solutions for general choices of Riemannian metrics on SE(3). Third, from a computational point of view, the method we propose is less expensive than traditional methods.

Minimal representations, spherical vectors and exceptional Theta series

Abstract: Theta series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group G is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of G, generalizing the Schrodinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the role of the summand in the automorphic theta series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the p-adic number fields provides the summation measure in the theta series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born–Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries.

On dense free subgroups of Lie groups

Abstract: We give a method for constructing dense and free subgroups in real Lie groups. In particular we show that any dense subgroup of a connected semisimple real Lie group G contains a free group on two generators which is still dense in G, and that any finitely generated dense subgroup in a connected non-solvable Lie group H contains a dense free subgroup of rank < 2 dim(H). This answer a question of Carriere and Ghys, and it gives an elementary proof of a conjecture of Connes and Sullivan on amenable actions, which was first proved by Zimmer.

Borcherds symmetries in M theory

Abstract: It is well known but rather mysterious that root spaces of the Ek Lie groups appear in the second integral cohomology of regular, complex, compact, del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms) of toroidal compactifications of M theory. Their Borel subgroups are actually subgroups of supergroups of finite dimension over the Grassmann algebra of differential forms on spacetime that have been shown to preserve the self-duality equation obeyed by all bosonic form-fields of the theory. We show here that the corresponding duality superalgebras are nothing but Borcherds superalgebras truncated by the above choice of Grassmann coefficients. The full Borcherds' root lattices are the second integral cohomology of the del Pezzo surfaces. Our choice of simple roots uses the anti-canonical form and its known orthogonal complement. Another result is the determination of del Pezzo surfaces associated to other string and field theory models. Dimensional reduction on Tk corresponds to blow-up of k points in general position with respect to each other. All theories of the Magic triangle that reduce to the En sigma model in three dimensions correspond to singular del Pezzo surfaces with A8?n (normal) singularity at a point. The case of type-I and heterotic theories if one drops their gauge sector corresponds to non-normal (singular along a curve) del Pezzo's. We comment on previous encounters with Borcherds algebras at the end of the paper.

Chapter 11 Dynamics of subgroup actions on homogeneous spaces of lie groups and applications to number theory

Abstract: Publisher Summary This chapter presents an exposition of homogeneous dynamics—that is, the dynamical and ergodic properties of actions on the homogeneous spaces of Lie groups. Many concepts of the modern theory of dynamical systems appeared in connection with the study of the geodesic flow on a compact surface of constant negative curvature. The algebraic nature of the phase space and the action itself allows obtaining much more advanced results as compared to the general theory of smooth dynamical systems. This can be seen in the example of smooth flows with polynomial divergence of trajectories. Homogeneous actions are discussed in the chapter and some basic examples include rectilinear flow on a torus, solvable flows on a three-dimensional locally Euclidean manifold, suspensions of toral automorphisms, nilflows on homogeneous spaces of the three-dimensional Heisenberg group, the geodesic and horocycle flows on a constant negative curvature surface, and geodesic flows on locally symmetric Riemannian spaces. The chapter presents the main link between homogeneous actions and number theory (Mahler's criterion and its consequences).

On the reduction methods for ordinary differential equations

Abstract: As tandard method based on the use of differential invariants of a Lie group, G ,e nables us to reduce any ordinary differential equation invariant under the action of G .W e show that this method is applicable to vector fields more general than those associated with Lie symmetries. We characterize all such vector fields and study their relationship with nonlocal symmetries and λsymmetries (Govinder K S and Leach P G L 1995 J. Phys. A: Math. Gen. 28 5349–59, Muriel C and Romero L 2001 IMA J. Appl. Math. 66 111–25).

Conformal field theory in four and six dimensions

Abstract: Introduction In this paper, I will be considering conformal field theory (CFT) mainly in four and six dimensions, occasionally recalling facts about two dimensions. The notion of conformal field theory is familiar to physicists. From a mathematical point of view, we can keep in mind Graeme Segal's definition of conformal field theory. Instead of just summarizing the definition here, I will review how physicists actually study examples of quantum field theory, as this will make clear the motivation for the definition. When possible (and we will later consider examples in which this is not possible), physicists make models of quantum field theory using path integrals. This means first of all that, for any n -manifold M n , we are given a space of fields on M n ; let us call the fields Φ. The fields might be, for example, real-valued functions, or gauge fields (connections on a G -bundle over M n for some fixed Lie group G ), or p -forms on M n for some fixed p , or they might be maps Φ : M n → W for some fixed manifold W . Then we are given a local action functional I (Φ). ‘Local’ means that the Euler–Lagrange equations for a critical point of I are partial differential equations. If we are constructing a quantum field theory that is not required to be conformally invariant, I may be defined using a metric on M n . For conformal field theory, I should be defined using only a conformal structure.

Fixed point ratios in actions of finite exceptional groups of lie type.

Abstract: Let G be a finite exceptional group of Lie type acting transitively on a set O. For x in G, the fixed point ratio of x is the proportion of elements of O which are fixed by x. We obtain new bounds for such fixed point ratios. When a point-stabilizer is parabolic we use character theory; and in other cases, we use results on an analogous problem for algebraic groups in Lawther, Liebeck & Seitz, 2002. These give dimension bounds on fixed point spaces of elements of exceptional algebraic groups, which we apply by passing to finite groups via a Frobenius morphism.

A Family of Novel Orientational 3-DOF Parallel Robots

Abstract: This paper presents a family of novel orientational 3-dof parallel robots, which are not overconstrained whereas most of the previously described mechanisms are. Using equivalencies that stem from the algebraic structure of Lie group of the set of Euclidean displacements, we have found many new tripod limbs and some of them are truly amazing by including prismatic pairs.

Reduction and reconstruction for self-similar dynamical systems

Abstract: We present a general method for analysing and numerically solving partial differential equations with self-similar solutions. The method employs ideas from symmetry reduction in geometric mechanics, and involves separating the dynamics on the shape space (which determines the overall shape of the solution) from those on the group space (which determines the size and scale of the solution). The method is computationally tractable as well, allowing one to compute self-similar solutions by evolving a dynamical system to a steady state, in a scaled reference frame where the self-similarity has been factored out. More generally, bifurcation techniques can be used to find self-similar solutions, and determine their behaviour as parameters in the equations are varied. The method is given for an arbitrary Lie group, providing equations for the dynamics on the reduced space, for reconstructing the full dynamics and for determining the resulting scaling laws for self-similar solutions. We illustrate the technique with a numerical example, computing self-similar solutions of the Burgers equation.

sh-Lie algebras induced by gauge transformations

Abstract: Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, as Lie algebra actions. A significant generalization is required when “gauge parameters” act in a field dependent way. Such symmetries appear in several field theories, most notably in a “Poisson induced” class due to Schaller and Strobl [SS94] and to Ikeda [Ike94], and employed by Cattaneo and Felder [CF99] to implement Kontsevich's deformation quantization [Kon97]. Consideration of “particles of spin > 2” led Berends, Burgers and van Dam [Bur85,BBvD84,BBvD85] to study “field dependent parameters” in a setting permitting an analysis in terms of smooth functions. Having recognized the resulting structure as that of an sh-Lie algebra (L∞-algebra), we have now formulated such structures entirely algebraically and applied it to a more general class of theories with field dependent symmetries.

Dynamics of Subgroup Actions on Homogeneous Spaces of Lie Groups and Applications to Number Theory

Abstract: Publisher Summary This chapter presents an exposition of homogeneous dynamics—that is, the dynamical and ergodic properties of actions on the homogeneous spaces of Lie groups. Many concepts of the modern theory of dynamical systems appeared in connection with the study of the geodesic flow on a compact surface of constant negative curvature. The algebraic nature of the phase space and the action itself allows obtaining much more advanced results as compared to the general theory of smooth dynamical systems. This can be seen in the example of smooth flows with polynomial divergence of trajectories. Homogeneous actions are discussed in the chapter and some basic examples include rectilinear flow on a torus, solvable flows on a three-dimensional locally Euclidean manifold, suspensions of toral automorphisms, nilflows on homogeneous spaces of the three-dimensional Heisenberg group, the geodesic and horocycle flows on a constant negative curvature surface, and geodesic flows on locally symmetric Riemannian spaces. The chapter presents the main link between homogeneous actions and number theory (Mahler's criterion and its consequences).

On the K + P Problem for a Three-Level Quantum System: Optimality Implies Resonance

Abstract: We apply techniques of subriemannian geometry on Lie groups to laser-induced population transfer in a three-level quantum system. The aim is to induce transitions by two laser pulses, of arbitrary shape and frequency, minimizing the pulse energy. We prove that the Hamiltonian system given by the Pontryagin maximum principle is completely integrable, since this problem can be stated as a “k ⊕ p problem” on a simple Lie group. Optimal trajectories and controls are exhausted. The main result is that optimal controls correspond to lasers that are “in resonance”.