Showing papers on "Lie group published in 1997"

Lie Group Valued Moment Maps

Abstract: We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg symplectic cross-section theorem and of convexity properties of the moment map. As an application we obtain moduli spaces of flat connections on an oriented compact 2-manifold with boundary as quasi-Hamiltonian quotients of the space G^2 x ... x G^2.

Harmonic Maps, Loop Groups, and Integrable Systems

Abstract: Preface Acknowledgements Part I. One-Dimensional Integrable Systems: 1. Lie groups 2. Lie algebras 3. Factorizations and homogeneous spaces 4. Hamilton's equations and Hamiltonian systems 5. Lax equations 6. Adler-Kostant-Symes 7. Adler-Kostant-Symes (continued) 8. Concluding remarks on one-dimensional Lax equations Part II. Two-Dimensional Integrable Systems: 9. Zero-curvature equations 10. Some solutions of zero-curvature equations 11. Loop groups and loop algebras 12. Factorizations and homogeneous spaces 13. The two-dimensional Toda lattice 14. T-functions and the Bruhat decomposition 15. Solutions of the two-dimensional Toda lattice 16. Harmonic maps from C to a Lie group G 17. Harmonic maps from C to a Lie group (continued) 18. Harmonic maps from C to a symmetric space 19. Harmonic maps from C to a symmetric space (continued) 20. Application: harmonic maps from S2 to CPn 21. Primitive maps 22. Weierstrass formulae for harmonic maps Part III. One-Dimensional and Two-Dimensional Integrable Systems: 23. From 2 Lax equations to 1 zero-curvature equation 24. Harmonic maps of finite type 25. Application: harmonic maps from T2 to S2 26. Epilogue References Index.

Ergodic theory on moduli spaces

Abstract: Let M be a compact surface with x(M) < 0 and let G be a compact Lie group whose Levi factor is a product of groups locally isomorphic to SU(2) (for example SU(2) itself). Then the mapping class group rM of M acts on the moduli space X(M) of flat G-bundles over M (possibly twisted by a fixed central element of G). When M is closed, then FM preserves a symplectic structure on X(M) which has finite total volume on M. More generally, the subspace of X(M) corresponding to flat bundles with fixed behavior over AM carries a rM-invariant symplectic structure. The main result is that rM acts ergodically on X(M) with respect to the measure induced by the symplectic structure.

Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction

Abstract: In this paper we establish necessary conditions for optimal control using the ideas of Lagrangian reduction in the sense of reduction under a symmetry group. The techniques developed here are designed for Lagrangian mechanical control systems with symmetry. The benefit of such an approach is that it makes use of the special structure of the system, especially its symmetry structure, and thus it leads rather directly to the desired conclusions for such systems. Lagrangian reduction can do in one step what one can alternatively do by applying the Pontryagin maximum principle followed by an application of Poisson reduction. The idea of using Lagrangian reduction in the sense of symmetry reduction was also obtained by Bloch and Crouch [Proc. 33rd CDC, IEEE, 1994, pp. 2584--2590] in a somewhat different context, and the general idea is closely related to those in Montgomery [Comm. Math. Phys., 128 (1990), pp. 565--592] and Vershik and Gershkovich [Dynamical Systems VII, V. Arnold and S. P. Novikov, eds., Springer-Verlag, 1994]. Here we develop this idea further and apply it to some known examples, such as optimal control on Lie groups and principal bundles (such as the ball and plate problem) and reorientation examples with zero angular momentum (such as the satellite with moveable masses). However, one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in the context of the work of Bloch, Krishnaprasad, Marsden, and Murray [Arch. Rational Mech. Anal., (1996), to appear]. The snakeboard is used to illustrate the method.

Hypersymmetry: A Z3-graded generalization of supersymmetry

Abstract: We propose a generalization of non-commutative geometry and gauge theories based on ternary Z3-graded structures. In the new algebraic structures we define, all products of two entities are left free, the only constraining relations being imposed on ternary products. These relations reflect the action of the Z3-group, which may be either trivial, i.e., abc=bca=cab, generalizing the usual commutativity, or non-trivial, i.e., abc=jbca, with j=e(2πi)/3. The usual Z2-graded structures such as Grassmann, Lie, and Clifford algebras are generalized to the Z3-graded case. Certain suggestions concerning the eventual use of these new structures in physics of elementary particles and fields are exposed.

Regularity of quasi-linear equations in the Heisenberg group

Abstract: We prove the Cα regularity of the gradient of weak solutions of a class of quasi-linear equations in nilpotent stratified Lie groups of step two. As applications, we prove higher regularity theorems and a Liouville type theorem for 1-quasi-conformal mappings between domains of the Heisenberg group. © 1997 John Wiley & Sons, Inc.

Geometry of Lie groups

Abstract: Preface. 0. Structures of Geometry. I. Algebras and Lie Groups. II. Affine and Projective Geometries. III. Euclidean, Pseudo-Euclidean, Conformal and Pseudoconformal Geometries. IV. Elliptic, Hyperbolic, Pseudoelliptic, and Pseudohyperbolic Geometries. V. Quasielliptic, Quasihyperbolic, and Quasi-Euclidean Geometries. VI. Symplectic and Quasisymplectic Geometries. VII. Geometries of Exceptional Lie Groups. Metasymplectic Geometries. References. Index of Persons. Index of Subjects.

Quasi-flats and rigidity in higher rank symmetric spaces

Abstract: In this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces. Our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces, such as SL(n, 2), n > 3: Theorem 8.1 is a major step towards the proof of quasiisometric rigidity of such lattices ([E]). We also give a different, and effective, proof of the theorem of Kleiner-Leeb on the quasi-isometric rigidity of higher rank symmetric spaces ([KL]).

Lie group formalism for difference equations

Abstract: The methods of Lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations. Several examples are analysed, one of them being a nonlinear difference equation. For the linear equations the symmetry algebra of the discrete equation is found to be isomorphic to that of its continuous limit.

Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions

Abstract: We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism) This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the actions of cocompact lattices on Furstenberg boundaries, in particular, projective spaces The main new technical ingredient in the proofs is the use of a proper "non-stationary" generalization of the classical theory of normal forms for local contractions

Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions

Abstract: We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the actions of cocompact lattices on Furstenberg boundaries, in particular, projective spaces. The main new technical ingredient in the proofs is the use of a proper "non-stationary" generalization of the classical theory of normal forms for local contractions.

Lie bialgebras, Poisson Lie groups and dressing transformations

Abstract: In this course, we present an elementary introduction, including the proofs of the main theorems, to the theory of Lie bialgebras and Poisson Lie groups and its applications to the theory of integrable systems. We discuss r-matrices, the classical and modified Yang-Baxter equations, and the tensor notation. We study the dual and double of Poisson Lie groups, and the infinitesimal and global dressing transformations.

Infinite-dimensional Lie groups

Abstract: Introduction Infinite-dimensional calculus (Chapter I) Infinite-dimensional manifolds (Chapter II) Infinite-dimensional Lie groups (Chapter III) Geometric structures on orbits (Chapter IV) Fundamental theorems for differentiability (Chapter V) Groups of $C^\infty$ diffeomorphisms on compact manifolds (Chapter VI) Linear operators (Chapter VII) Several subgroups of ${\scr D}(M)$ (Chapter VIII) Smooth extension theorems (Chapter IX) The group of diffeomorphisms on cotangent bundles (Chapter X) Pseudodifferential operators on manifolds (Chapter XI) Lie algebra of vector fields (Chapter XII) Quantizations (Chapter XIII) Poisson manifolds and quantum groups (Chapter XIV) Weyl manifolds (Chapter XV) Infinite-dimensional Poisson manifolds (Chapter XVI) Appendix I Appendix II Appendix III References Index.

An analogue of Hardy’s theorem for very rapidly decreasing functions on semi-simple Lie groups

Abstract: A celebrated theorem of L. Schwartz asserts that a function f on R is ‘rapidly decreasing’ (or in the ‘Schwartz class’) iff its Fourier transform is ‘rapidly decreasing’. Since this theorem is of fundamental importance in harmonic analysis, there is a whole body of literature devoted to generalizing this result to other Lie groups. (For example, see [18].) In sharp contrast to Schwartz’s theorem, is a result due to Hardy [5] which says that f and f cannot both be “very rapidly decreasing”. More precisely, if |f(x)| ≤ Ae−α|x|2 and |f(y)| ≤ Be−β|y| 2 and αβ > 1 4 , then f ≡ 0. (See [2], pp. 155-157.) However, as far as we are aware, until very recently no systematic attempt was made to generalize Hardy’s theorem to other Lie groups. In [12], [13], and [15], this result has been generalized to the Heisenberg groups Hn, the Euclidean motion groups M(n) and for certain eigenfunction expansions. In this paper we establish an analogue of Hardy’s theorem for a class of noncompact semisimple Lie groups and all symmetric spaces of the noncompact type. Hardy’s theorem can also be viewed as a sort of ‘Uncertainty Principle’. The results in [12] and [13] are presented from this point of view. (In [1], Cowling and Price have proved an “L − L” version of Hardy’s theorem on R. The theorem of Beurling in [9] is similar in spirit to Hardy’s theorem, although far more general, and indeed Hardy’s theorem, as well as the result of Cowling and Price, can be deduced from it as special cases.)

Harmonic two-spheres in compact symmetric spaces, revisited

Abstract: Uhlenbeck introduced an invariant, the (minimal) uniton number, of harmonic 2-spheres in a Lie group G and proved that when G=SU(n) the uniton number cannot exceed n-1. In this paper, using new methods inspired by Morse Theory, we explain this result and extend it to an arbitrary compact group G. The same methods also yield Weierstrass formulae for these harmonic maps and simple proofs of most of the known classification theorems for harmonic 2-spheres in symmetric spaces.

High powers of random elements of compact Lie groups

Abstract: If a random unitary matrix \(U\) is raised to a sufficiently high power, its eigenvalues are exactly distributed as independent, uniform phases. We prove this result, and apply it to give exact asymptotics of the variance of the number of eigenvalues of \(U\) falling in a given arc, as the dimension of \(U\) tends to infinity. The independence result, it turns out, extends to arbitrary representations of arbitrary compact Lie groups. We state and prove this more general theorem, paying special attention to the compact classical groups and to wreath products. This paper is excerpted from the author's doctoral thesis, [9].

Hypercomplex structures on four-dimensional Lie groups

Abstract: The purpose of this paper is to classify invariant hypercomplex structures on a 4-dimensional real Lie group G. It is shown that the 4dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group IHE of the quaternions, the multiplicative group H* of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, RH4 and CH2, respectively, and the semidirect product C i C. We show that the spaces CH2 and C x C possess an Rp2 of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian 4-manifolds are determined.

Complex Analysis and Geometry

Abstract: On the Limits of Manifolds with nef Canonical Bundles, M. Andreatta and T. Peternell On the Stability of the Restriction of TPn to Projective Curves, E. Ballico and B. Russo Theorie des (a,b)-Modules II. Extensions, D. Barlet Moduli of Reflexive K3 Surfaces, C. Bartocci, U. Bruzzo, and D. Hernandez Ruiperez New Examples of Domains with Non-Injective Proper Holomorphic Self-Maps, F. Berteloot and J. J. Loeb Q-Convexivity. A Survey, M. Coltoiu Commuting maps and Families of Hyperbolic Automorphisms, C. de Fabritiis An Alternative Proof of a Theorem of Boas-Straube-Yu, K. Diederich and G. Herbort Large Polynomial Hulls with No Analytic Structure, J. Duval and N. Levenberg Canonical Connections for Almost-Hypercomplex Structures, P. Gauduchon The Tangent Bundle of P2 Restricted to Plane Curves, G. Hein Quotients with Respects to Holomorphic Actions of Reductive Groups, P. Heinzner and L. Migliorini Adjunction Theory on Terminal Varieties, M. Mella Runge Theorem in Higher Dimensions, V. Vajaitu Only Countably Many Simply-Connected Lie Groups Admit Lattices, J. Winkelmann

Discontinuous Groups and Clifford—Klein Forms of Pseudo-Riemannian Homogeneous Manifolds

Abstract: Let G be a Lie group and H a subgroup. A Clifford-Klein form of the homogeneous manifold G/H is a double coset space / G/H , where is a subgroup of G acting properly discontinuously and freely on G/H . For example, any closed Riemann surface M with genus ≥2 is biholomorphic to a compact Clifford-Klein form of the Poincare plane G/H =PSL(2,ℝ葷)/SO(2). On the other hand, there is no compact Clifford-Klein form of the hyperboloid of one sheet G/H =PSL(2,ℝ葷)/SO(1,1). Even more, there is no infinite discrete subgroup of G that acts properly discontinuously on G/H (the Calabi-Markus phenomenon). We discuss recent developments in the theory of discontinuous groups acting on G/H where G is a real reductive Lie group and H a noncompact reductive subgroup. Geometric ideas of various methods together with a number of examples are presented regarding the fundamental problems: Which homogeneous manifolds G/H admit properly discontinuous actions of infinite discrete subgroups of G? Which homogenous manifolds admit compact Clifford-Klein forms?

A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps

Abstract: The classical Hilbert 5th problem [14] asks whether every ( nite-dimensional) locally Euclidean topological group is necessarily a Lie group. It was solved, in the a rmative, by von Neumann [23] for compact groups in 1933, and by Gleason [11] and by Montgomery and Zippin [20] for locally compact groups in 1952. A more general version of the Hilbert 5th problem, called the Hilbert-Smith Conjecture, asserts that among all locally compact groups only Lie groups G can act e ectively on ( nite-dimensional) manifolds M (i.e. each g ∈ G\{e} moves at least one point of M) [28]. It follows from the work of Newman [24] and Smith [29] that this conjecture is equivalent to its special case when the acting group G is the group of p-adic integers Ap. In 1946 Bochner and Montgomery [3] proved the Hilbert-Smith Conjecture for groups G acting e ectively on a manifold M by di eomorphisms. A simpler, geometrical proof was obtained by Skopenkov and the authors [25] using the idea of smooth homogeneity: a compact subset K ⊂ M of a smooth manifold M is said to be smoothly ambiently homogeneous, i.e. for each x; y ∈ K there exists a di eomorphism h : (M;K; x)→ (M;K; y). It was shown that this property implies that K is a smooth submanifold of M (therefore G ∼= K is a Lie group). The proof reveals a close relationship between homogeneity and taming theory for compact subsets of Rn, which are pinched by tangent balls (the latter problem was investigated in the past by various authors [6,10,12,16,17]). See also a very interesting paper by Hahn [13]. An interesting approach to the Hilbert-Smith conjecture is via wild Cantor sets in Rn with strong homogeneity properties. Note that the Antoine necklace

Robertson intelligent states

Abstract: The diagonalization of the uncertainty matrix and the minimization of Robertson inequality for n observables are considered. It is proved that for even n this relation is minimized in states which are eigenstates of n/2 independent complex linear combinations of the observables. In the case of canonical observables, this eigenvalue condition is also necessary. Such minimizing states are called Robertson intelligent states (RIS). The group-related coherent states (CS) with maximal symmetry (for semisimple Lie groups) are a particular case of RIS for the quadratures of Weyl generators. Explicit constructions of RIS are considered for operators of su(1,1), su(2), and sp(N,R) algebras. Unlike the group-related CS, RIS can exhibit strong squeezing of group generators. Multimode squared amplitude squeezed states are naturally introduced as sp(N,R) RIS. It is shown that the uncertainty matrices for quadratures of q-deformed boson operators (q>0) and of any k power of are positive definite and can be diagonalized by symplectic linear transformations.