Showing papers on "Lie group published in 1987"

Quantization of Lie Groups and Lie Algebras

Abstract: Publisher Summary This chapter focuses on the quantization of lie groups and lie algebras. The Algebraic Bethe Ansatz—the quantum inverse scattering method—emerges as a natural development of the various directions in mathematical physics: the inverse scattering method for solving nonlinear equations of evolution, the quantum theory of magnets, the method of commuting transfer-matrices in classical statistical mechanics, and factorizable scattering theory. The chapter discusses quantum formal groups, a finite-dimensional example, an infinite-dimensional example, and reviews the deformation theory and quantum groups.

Local rigidity of discrete groups acting on complex hyperbolic space.

Abstract: The superrigidity theorem of Margulis, see Zimmer [17], classifies finite dimensional representations of lattices in semi-simple Lie groups of real rank strictly larger than 1. It is a fundamental problem to obtain the classification of finite dimensional representations of lattices in rank I semi-simple Lie groups. That this problem will be considerably harder than the previous one is suggested by the existence of continuous families of inequivalent representations or, in other words, the existence of non-trivial deformations. In Johnson-Millson [6] and Kourouniotis [8] such deformations were constructed for certain representations of lattices in SO(n, 1) based on a construction of Thurston called bending. The deformation space of the representation of a lattice F=SO(n, 1) obtained by restricting an inclusion SO(n, 1)-,SO(n+ l, 1) to F is of particular interest. If n > 2 the space of infinitesimal deformations is Hi(F, R "+1) where F acts on R "+1 by the restriction of the standard action of SO(n, 1). The space of infinitesimal deformations is non-zero for the standard arithmetic examples and the main point of the papers cited above was to establish that some of these infinitesimal deformations are integrable (in [6] it is also shown that some are not). In this paper, we study the complex analogue of the above example. We let F be a cocompact torsion free lattice in SU(n, 1) and consider the deformation space of the representation of F obtained by restricting an inclusion SU(n, 1)--*SU(n+I, 1). If n > l the space of infinitesimal deformations is H~(F, R)@H~(F, C,+ 1). In the first summand F acts trivially and the infinitesimal deformations are tangent to the obvious deformations obtained by deforming F in U(n, 1) by a curve of homomorphisms into the center of U(n, 1) (observe that the above inclusion factors as SU(n, 1)~U(n , 1)~SU(n+I, 1)). In the second summand F acts by the restriction of the standard action of SU(n, 1). This summand is non-zero for the standard arithmetic examples,

Analogues of the objects of lie group theory for nonlinear poisson brackets

Abstract: For general degenerate Poisson brackets, analogues are constructed of invariant vector fields, invariant forms, Haar measure and adjoint representation A pseudogroup operation is defined that corresponds to nonlinear Poisson brackets, and analogues are obtained for the three classical theorems of Lie The problem of constructing global pseudogroups is examined Bibliography: 49 titles

On the integrability of invariant hamiltonian systems with homogeneous configuration spaces

Abstract: All homogeneous spaces ( is a semisimple complex (compact) Lie group, a reductive subgroup) are enumerated for which arbitrary Hamiltonian flows on with -invariant Hamiltonians are integrable in the class of Noether integrals. It is proved that only for these spaces does the quasiregular representation of in the space of regular functions of the algebraic variety have a simple spectrum.Bibliography: 21 titles.

Trigonometric solutions of triangle equations. Simple lie superalgebras

Abstract: Trigonometric solutions of the graded triangle equation are constructed for the fundamental representations of all simple (nonexceptional) Lie superalgebras with nondegenerate metric. In Sec. 1, we introduce the concept of Z/sub 2/ graded spaces and give the basic definitions. In Sec. 2, we determine fundamental representations of the Lie superalgebras sl(m/vert bar/n) and osp(2r/vert bar/s) and give explicit realizations of the Coxeter automorphisms. In secs. 3 and 4, we give the trigonometric solutions of the graded triangle equation (quantum R matrices).

Lattices in Semisimple Groups and Invariant Geometric Structures on Compact Manifolds

Abstract: The aim of this paper is to describe a geometrization of the Mostow-Margulis theory of rigidity and representations of discrete subgroups of semisimple groups. More precisely let H be a connected semisimple Lie group and Γ ⊂ Halattice subgroup. Let G be another Lie group. The general problem considered by Mostow and Margulis was to study the homomorphisms π: Γ → G. The Mostow rigidity theorem of course deals with the case in which G is semisimple and π(Γ) is a lattice in G, and the Margulis superrigidity theorem deals with the more general case in which π(Γ) is merely assumed to be Zariski dense in G. While we shall recall the precise results later, we simply remark here that the ultimate conclusion is that one can essentially understand all such homomorphisms. Roughly speaking, π either extends to a smooth homomorphism of H or π(Γ) has compact closure (in which case one also has information on the closure), or is a combination of these cases. A geometric generalization of the notion of a homomorphism Γ → G is of course the notion of an action of Γ by automorphisms of a principal G-bundle.

Finite Presentability of S-Arithmetic Groups: Compact Presentability of Solvable Groups

Abstract: Compact presentability and contracting automorphisms.- Filtrations of Lie algebras and groups.- A necessary condition for compact presentability.- Implications of the necessary condition.- The second homology.- S-arithmetic groups.- S-arithmetic solvable groups.

The x-ray transform: singular value decomposition and resolution

Abstract: The x-ray transform maps a compactly supported function in Rn to its integrals over all the straight lines in Rn. A singular value decomposition (SVD) for this operator is given in arbitrary dimensions. The proof uses results from the representation theory of Lie groups. This paper addresses questions concerning the stability of the inversion problem. The SVD shows which parts of a reconstructed function are affected by data errors and by how much. The resolution in the reconstruction is determined if only a finite set of data is available.

Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula

Abstract: Let G be a connected real semisimple Lie group with finite centre, and let z be an involutive automorphism of G. Put G'={x~G: z(x)=x}, and let H be a closed subgroup of G with (G'),cHcG'; here (G'), denotes the identity component of GL In this paper we investigate some properties of the algebra D(X) of invariant differential operators on the semisimple symmetric space X=G/H. Our main results are that the action of D(X) diagonalizes over the discrete part of L2(X) (Theorem 1.5), and that the irreducible constituents of an abstract Plancherel formula for X occur with finite multiplicities (Theorem 3.1). Both results are proved by using techniques of Harish--Chandra adapted to the situation at hand.

Lie groupoids and Lie algebroids in Differential Geometry: Lie groupoids and Lie algebroids

Abstract: 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.

N=2, d=2 supergeometry

Abstract: N=2, d=2 supergeometry is formulated in superspaces with U(1)*U(1) and U(1) internal symmetry factors in the tangent space group. The N=2 geometry is compared with the N=1 and (p, q)=(1, 0) geometries, and the superconformal transformations are derived from the superconformal Killing equation.

A Cancellation Theorem for Modules Over the Group C*-Algebras of Certain Nilpotent Lie Groups

Abstract: In recent years, there has been a rapid growth of the K-theory of C*-algebras. From a certain point of view, C*-algebras can be treated as “non-commutative topological spaces”, while finitely generated projective modules over them can be thought of as “non-commutative vector bundles”. The K-theory of C*-algebras [30] then generalizes the classical K-theory of topological spaces [1]. In particular, the K 0-group of a unital C*-algebra A is the group “generated” by (or more precisely, the Grothendieck group of) the commutative semigroup of stable isomorphism classes of finitely generated projective modules over A with direct summation as the binary operation. The semigroup gives an order structure on K 0(A) and is usually called the positive cone in K 0(A). Around 1980, the work of Pimsner and Voiculescu [18] and of A. Connes [4] provided effective ways to compute the K-groups of C*-algebras. Then the classification of finitely generated projective modules over certain unital C*-algebras up to stable isomorphism could be done by computing their K 0-groups as ordered groups. Later on, inspired by A. Connes's development of non-commutative differential geometry on finitely generated projective modules [2], the deeper question of classifying such modules up to isomorphism and hence the so-called cancellation question were raised (cf. [21] ).

Intrinsic Nilpotent Approximation.

Abstract: : This report is a preliminary version of work on an intrinsic approximation process arising in the context of a non-isotropic perturbation theory for certain classes of linear differential and pseudodifferential operators P on a minifold M. A basic issue is that the structure of P itself determines the minimal information that the initial approximation must contain. This may vary from point to point, and requires corresponding approximate state spaces or phase spaces. This approximation process is most naturally viewed from a seemingly abstract algebraic context, namely the approximation of certain infinite dimensional filtered Lie algebras L by (finite-dimensional) graded nilpotent Lie algebras.

Maximal subsemigroups of lie groups that are total

Abstract: The major problem with which this paper is concerned is determining criteria that allow one to decide whether the subsemigroup generated by a subset B of a group G is all of G. Motivations for considering this problem arise from at least two sources.

Superposition formulas for rectangular matrix Riccati equations

Abstract: A system of nonlinear ordinary differential equations allowing a superposition formula can be associated with every Lie group–subgroup pair G⊇G0. We consider the case when G=SL(n+k,C) and G0=P(k) is a maximal parabolic subgroup of G, leaving a k‐dimensional vector space invariant (1≤k≤n). The nonlinear ordinary differential equations (ODE’s) in this case are rectangular matrix Riccati equations for a matrix W(t)∈Cn×k. The special case n=rk (n,r,k∈N) is considered and a superposition formula is obtained, expressing the general solution in terms of r+3 particular solutions for r≥2, k≥2. For r=1 (square matrix Riccati equations) five solutions are needed, for r=n (projective Riccati equations) the required number is n+2.

The Batalin, Fradkin, and Vilkovisky Formalism for Higher Order Theories

Abstract: The generalized Batalin, Fradkin, and Vilkovisky (BFV) formalism was developed as a method for determining the ghost structure for theories, such as gravity and supergravity, whose Hamiltonian formalism has constraints not related to a Lie algebra action. Previously, the classical dynamical content of the BFV description of Yang–Mills theory was investigated. There it was found that this approach had a homological interpretation, derived from the Lie algebra cohomology of the gauge group, which allowed one to understand the construction in terms of the Dirac approach to constrained systems. In this paper the dynamical consequences of the generalized BFV formalism are investigated. It is found that even though one no longer has a Lie algebra structure associated with the constraints, one can still develop a homology theory that reproduces the Dirac analysis and from which the generalized BFV formalism can be derived. Some of the consequences of this approach are discussed.

Super poincaré groups and division algebras

Abstract: The Super Poincare Groups in d = 3, 4, 6 and 10 are represented by graded matrices over the four division algebras. It is shown that classical superstring actions can be written in the same dimensions in terms of the supergroup elements. Possible applications are discussed.