Showing papers on "Lie group published in 1996"

Geometric control theory

Abstract: Introduction Acknowledgments Part I. Reachable Sets and Controllability: 1. Basic formalism and typical problems 2. Orbits of families of vector fields 3. Reachable sets of Lie-determined systems 4. Control affine systems 5. Linear and polynomial control systems 6. Systems on Lie groups and homogenous spaces Part II. Optimal Control Theory: 7. Linear systems with quadratic costs 8. The Riccati equation and quadratic systems 9. Singular linear quadratic problems 10. Time-optimal problems and Fuller's phenomenon 11. The maximum principle 12. Optimal problems on Lie groups 13. Symmetry, integrability and the Hamilton-Jacobi theory 14. Integrable Hamiltonian systems on Lie groups: the elastic problem, its non-Euclidean analogues and the rolling-sphere problem References Index.

Riemannian symmetric superspaces and their origin in random‐matrix theory

Abstract: Gaussian random‐matrix ensembles defined over the tangent spaces of the large families of Cartan’s symmetric spaces are considered. Such ensembles play a central role in mesoscopic physics, as they describe the universal ergodic limit of disordered and chaotic single‐particle systems. The generating function for the spectral correlations of each ensemble is reduced to an integral over a Riemannian symmetric superspace in the limit of large matrix dimension. Such a space is defined as a pair (G/H,M r ), where G/H is a complex‐analytic graded manifold homogeneous with respect to the action of a complex Lie supergroup G, and M r is a maximal Riemannian submanifold of the support of G/H.

Quantization of Lie bialgebras, II

Abstract: In the paper [Dr3] V. Drinfeld formulated a number of problems in quantum group theory. In particular, he raised the question about the existence of a quantization for Lie bialgebras, which arose from the problem of quantization of Poisson Lie groups. When the paper [KL] appeared Drinfeld asked whether the methods of [KL] could be useful for the problem of quantization of Lie bialgebras. This paper gives a positive answer to a number of Drinfeld's questions, using the methods and ideas of [KL]. In particular, we show the existence of a quantization for Lie bialgebras. The universality and functoriality properties of this quantization will be discussed in the second paper of this series. We plan to provide positive answers to most of the remaining questions in [Dr3] in the following papers of this series.

The Poisson boundary of the mapping class group

Abstract: A theory of random walks on the mapping class group and its non-elementary subgroups is developed. We prove convergence of sample paths in the Thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the Poisson boundary of random walks. The methods are based on an analysis of the asymptotic geometry of Teichmuller space and of the contraction properties of the action of the mapping class group on the Thurston boundary. We prove, in particular, that Teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees. Using our analysis of the mapping class group action on the Thurston boundary we prove that no non-elementary subgroup of the mapping class group can be a lattice in a higher rank semi-simple Lie group.

Barrier functions in interior point methods

Abstract: We show that the universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone. This interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone. This provides a simple method for calculating the universal barrier for homogeneous cones. We identify some known barriers as the universal barrier scaled by an appropriate constant. We also calculate some new universal barrier functions. Our results connect the field of interior point methods to several branches of mathematics such as Lie groups, Jordan algebras, Siegel domains, differential geometry, complex analysis of several variables, etc.

Introduction To Quantum Groups

Abstract: Part 1 Mathematical aspects of quantum groups theory and non-commutative geometry: Hopf algebra and Poisson structure of classical Lie groups and algebras quantum groups, algebras and their duality non-commutative spaces and quantum groups invariant differential calculi elements of quantum groups representations theory tensor products of representations q-tensors, q-vectors, q-scalars. Part 2 Deformation of harmonic oscillators: q-deformation of single-harmonic oscillator different forms of commutation relations representations q real and roots of unity cases algebraic maps from non-deformed to deformed oscillators path integral quantization of q-oscillator. Part 3 Q-deformation of space-time symmetries: classical relativistic space-time symmetries Poincare group as a "classical" deformation of Galilei group its representations multiparametric q-deformation of linear groups, twisted groups and algebras q-Poincare group as a q-subgroup of q-conformal one and its induced representations. Part 4 Non-commutative geometry and unification models: unified models, the problem of natural introduction of Higgs fields unified models of Higgs fields in the frame of non-commutative geometry and others.

Actions propres sur les espaces homogenes reductifs

Abstract: Let G be a linear semisimple real Lie group and H be a re- ductive subgroup of G. We give a necessary and sufficient condition for the existence of a nonabelian free discrete subgroup r of G acting properly on G/H. For instance, such a group r does exist for SL(2n, IR)/SL(2n - 1, IR) but does not for SL(2n + 1, R)/SL(2n, R) with n > 1.

Wavelets from square-integrable representations

Abstract: The continuous wavelet decompositions that arise from square-integrable representations of certain Lie groups on $L^2 (\mathbb{R}^n )$ are investigated. The groups are formed as the semidirect product of $\mathbb{R}^n $ with an n-dimensional subgroup H of $GL_n (\mathbb{R})$. There is a natural “translation and dilation” representation of such groups on $L^2 (\mathbb{R}^n )$. The basic formulas of Duflo and Moore, which lead to the resolution of the identity via a square-integrable representation, are given an elementary proof for this special case. Several two-dimensional examples are described. A method for discrete decompositions via frames is given using the representations under study.

Limit distributions of expanding translates of certain orbits on homogeneous spaces

Abstract: LetL be a Lie group and λ a lattice inL. SupposeG is a non-compact simple Lie group realized as a Lie subgroup ofL and $$\overline {GA} = L$$ . LetaeG be such that Ada is semisimple and not contained in a compact subgroup of Aut(Lie(G)). Consider the expanding horospherical subgroup ofG associated toa defined as U+ ={geG:a −n gan} →e as n → ∞. Let Ω be a non-empty open subset ofU + andn i → ∞ be any sequence. It is showed that $$\overline { \cup _{i = 1}^\infty a^n \Omega \Lambda } = L$$ . A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describeG-equivariant topological factors of L/gl × G/P, where the real rank ofG is greater than 1,P is a parabolic subgroup ofG andG acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.

Surfaces on Lie groups, on Lie algebras, and their integrability

Abstract: It is shown that the problem of the immersion of a 2-dimensional surface into a 3-dimensional Euclidean space, as well as then-dimensional generalization of this problem, is related to the problem of studying surfaces in Lie groups and surfaces in Lie algebras. A particular case of the general formalism presented here implies that any surface can be characterized in terms of 2×2 matrices using an arbitrary parametrization. It is also shown that this generality of parametrization is useful for studying integrable surfaces, i.e. surfaces described by integrable equations. In particular starting from a suitable Lax pair (i.e. a suitable integrable equation), it is possible to construct explicitly large classes of integrable surfaces.

On quaternionic discrete series representations, and their continuations.

Abstract: Among the discrete series representations of a real reductive group G, the simplest family to study are the holomorphic discrete series. These representations exist when the Symmetrie space G/ K has a G-invariant complex structure, and are admissible when restricted to a 1-dimensional torus S in the center of K. They can be constructed äs spaces of analytic sections of certain holomorphic vector bundles on G /K. In the category of (g, K)modules, holomorphic discrete series give rise to unitarizable highest weight modules. Other interesting unitarizable modules with a highest weight can be constructed by "analytic continuation" of the holomorphic discrete series (cf. [Wl]).

On Riemann-Roch Formulas for Multiplicities

Abstract: A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of the symplectically reduced space. Combined with the shifting-trick, this gives explicit formulas for the multiplicities of the various irreducible components. One of the assumptions of the Theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. In this paper, we prove a generalization of this result to the case where the reduced space may have orbifold singularities. Our proof uses localization techniques from equivariant cohomology, and relies in particular on recent work of Jeffrey-Kirwan and Guillemin. Since there are no complex geometry arguments involved, the result also extends to non Kaehlerian settings.

On special classes of n‐algebras

Abstract: We define n‐algebras as linear spaces on which the internal composition law involves n elements: m:V⊗n■V. It is known that such algebraic structures are interesting for their applications to problems of modern mathematical physics. Using the notion of a commutant of two subalgebras of an n‐algebra, we distinguish certain classes of n‐algebras with reasonable properties: semisimple, Abelian, nilpotent, solvable. We also consider a few examples of n‐algebras of different types, and show their properties.

Symmetries and integrability of difference equations

Abstract: The concept of integrability was introduced in classical mechanics in the 19th century for finite dimensional continuous Hamiltonian systems. It was extended to certain classes of nonlinear differential equations in the second half of the 20th century with the discovery of the inverse scattering transform and the birth of soliton theory. Also at the end of the 19th century Lie group theory was invented as a powerful tool for obtaining exact analytical solutions of large classes of differential equations. Together, Lie group theory and integrability theory in its most general sense provide the main tools for solving nonlinear differential equations. Like differential equations, difference equations play an important role in physics and other sciences. They occur very naturally in the description of phenomena that are genuinely discrete. Indeed, they may actually be more fundamental than differential equations if space-time is actually discrete at very short distances. On the other hand, even when treating continuous phenomena described by differential equations it is very often necessary to resort to numerical methods. This involves a discretization of the differential equation, i.e. a replacement of the differential equation by a difference one. Given the well developed and understood techniques of symmetry and integrability for differential equations a natural question to ask is whether it is possible to develop similar techniques for difference equations. The aim is, on one hand, to obtain powerful methods for solving `integrable' difference equations and to establish practical integrability criteria, telling us when the methods are applicable. On the other hand, Lie group methods can be adapted to solve difference equations analytically. Finally, integrability and symmetry methods can be combined with numerical methods to obtain improved numerical solutions of differential equations. The origin of the SIDE meetings goes back to the early 1990s and the first meeting with the name `Symmetries and Integrability of Discrete Equations (SIDE)' was held in Esterel, Quebec, Canada. This was organized by D Levi, P Winternitz and L Vinet. After the success of the first meeting the scientific community decided to hold bi-annual SIDE meetings. They were held in 1996 at the University of Kent (UK), 1998 in Sabaudia (Italy), 2000 at the University of Tokyo (Japan), 2002 in Giens (France), 2004 in Helsinki (Finland) and in 2006 at the University of Melbourne (Australia). In 2008 the SIDE 8 meeting was again organized near Montreal, in Ste-Adele, Quebec, Canada. The SIDE 8 International Advisory Committee (also the SIDE steering committee) consisted of Frank Nijhoff, Alexander Bobenko, Basil Grammaticos, Jarmo Hietarinta, Nalini Joshi, Decio Levi, Vassilis Papageorgiou, Junkichi Satsuma, Yuri Suris, Claude Vialet and Pavel Winternitz. The local organizing committee consisted of Pavel Winternitz, John Harnad, Veronique Hussin, Decio Levi, Peter Olver and Luc Vinet. Financial support came from the Centre de Recherches Mathematiques in Montreal and the National Science Foundation (through the University of Minnesota). Proceedings of the first three SIDE meetings were published in the LMS Lecture Note series. Since 2000 the emphasis has been on publishing selected refereed articles in response to a general call for papers issued after the conference. This allows for a wider author base, since the call for papers is not restricted to conference participants. The SIDE topics thus are represented in special issues of Journal of Physics A: Mathematical and General 34 (48) and Journal of Physics A: Mathematical and Theoretical, 40 (42) (SIDE 4 and SIDE 7, respectively), Journal of Nonlinear Mathematical Physics 10 (Suppl. 2) and 12 (Suppl. 2) (SIDE 5 and SIDE 6 respectively). The SIDE 8 meeting was organized around several topics and the contributions to this special issue reflect the diversity presented during the meeting. The papers presented at the SIDE 8 meeting were organized into the following special sessions: geometry of discrete and continuous Painleve equations; continuous symmetries of discrete equations—theory and computational applications; algebraic aspects of discrete equations; singularity confinement, algebraic entropy and Nevanlinna theory; discrete differential geometry; discrete integrable systems and isomonodromy transformations; special functions as solutions of difference and q-difference equations. This special issue of the journal is organized along similar lines. The first three articles are topical review articles appearing in alphabetical order (by first author). The article by Doliwa and Nieszporski describes the Darboux transformations in a discrete setting, namely for the discrete second order linear problem. The article by Grammaticos, Halburd, Ramani and Viallet concentrates on the integrability of the discrete systems, in particular they describe integrability tests for difference equations such as singularity confinement, algebraic entropy (growth and complexity), and analytic and arithmetic approaches. The topical review by Konopelchenko explores the relationship between the discrete integrable systems and deformations of associative algebras. All other articles are presented in alphabetical order (by first author). The contributions were solicited from all participants as well as from the general scientific community. The contributions published in this special issue can be loosely grouped into several overlapping topics, namely: •Geometry of discrete and continuous Painleve equations (articles by Spicer and Nijhoff and by Lobb and Nijhoff). •Continuous symmetries of discrete equations—theory and applications (articles by Dorodnitsyn and Kozlov; Levi, Petrera and Scimiterna; Scimiterna; Ste-Marie and Tremblay; Levi and Yamilov; Rebelo and Winternitz). •Yang--Baxter maps (article by Xenitidis and Papageorgiou). •Algebraic aspects of discrete equations (articles by Doliwa and Nieszporski; Konopelchenko; Tsarev and Wolf). •Singularity confinement, algebraic entropy and Nevanlinna theory (articles by Grammaticos, Halburd, Ramani and Viallet; Grammaticos, Ramani and Tamizhmani). •Discrete integrable systems and isomonodromy transformations (article by Dzhamay). •Special functions as solutions of difference and q-difference equations (articles by Atakishiyeva, Atakishiyev and Koornwinder; Bertola, Gekhtman and Szmigielski; Vinet and Zhedanov). •Other topics (articles by Atkinson; Grunbaum; Nagai, Kametaka and Watanabe; Nagiyev, Guliyeva and Jafarov; Sahadevan and Uma Maheswari; Svinin; Tian and Hu; Yao, Liu and Zeng). This issue is the result of the collaboration of many individuals. We would like to thank the authors who contributed and everyone else involved in the preparation of this special issue.

D-Branes on Group Manifolds

Abstract: Possible Dirichlet boundary states for WZW models with untwisted affine super Kac-Moody symmetry are classified for all compact simple Lie groups. They are obtained by inner- and outer-automorphism of the group. D-brane world-volume turns out to be a group manifold of a symmetric subgroup, so that the moduli space of D-brane is a irreducible Riemannian symmetric space. It is also clarified how these D-branes are transformed to each other under abelian T-duality of WZW model. Our result implies, for example, there is no D-particle on the compact simple group manifold. When the D-brane world-volume contains $S^1$ factor, the D-brane moduli space becomes hermitian symmetric space and the open string world-sheet instantons are allowed.

The complete Kepler group can be derived by Lie group analysis

Abstract: It is shown that the complete symmetry group for the Kepler problem, as introduced by Krause, can be derived by Lie group analysis. The same result is true for any autonomous system.

Affine structures on nilmanifolds

Abstract: We investigate the existence of affine structures on nilmanifolds Γ\G in the case where the Lie algebra g of the Lie group G is filiform nilpotent of dimension less or equal to 11. Here we obtain examples of nilmanifolds without any affine structure in dimensions 10, 11. These are new counterexamples to the Milnor conjecture. So far examples in dimension 11 were known where the proof is complicated, see [5] and [4]. Using certain 2-cocycles we realize the filiform Lie algebras as deformation algebras from a standard graded filiform algebra. Thus we study the affine algebraic variety of complex filiform nilpotent Lie algebra structures of a given dimension ≤11. This approach simplifies the calculations, and the counterexamples in dimension 10 are less complicated than the known ones. We also obtain results for the minimal dimension µ(g) of a faithful g-module for these filiform Lie algebras g.

The Chern character of a transversally elliptic symbol and the equivariant index

Abstract: Let G be a compact Lie group acting on a compact manifold M. In this article, we associate to a G-transversally elliptic symbol on M a G- invariant generalized function on G, constructed in terms of equivariant closed dierential forms on the cotangent bundle TM.

Hall’s transform and the Segal-Bargmann map

Abstract: It is shown how Hall’s transform for a compact Lie group can be derived from the infinite dimensional Segal-Bargmann transform by means of stochastic analysis.

Four-dimensional BF theory as a topological quantum field theory

Abstract: Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that four-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G=GL(4,ℝ) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of four-manifolds.

Left-invariant lorentzian metrics on 3-dimensional lie groups

Abstract: We find the Riemann curvature tensors of all leftinvariant Lorentzian metrics on 3-dimensional Lie groups. MSC(1991): Primary 53C50; Secondary 53B30, 53C30. −−−−−−−−−−−−−−−−−−−−−−−−−−→Υ· ∞·←−−−−−−−−−−−−−−−−−−−−−−−−−− Supported by Project XUGA8050189, Xunta de Galicia, Spain. On leave from Math. Dept., Wichita State Univ., Wichita KS 67260, U.S.A., pparker@twsuvm.uc.twsu.edu Partially supported by DGICYT-Spain.