Showing papers on "Lie group published in 1991"

Representation Theory: A First Course

Abstract: This volume represents a series of lectures which aims to introduce the beginner to the finite dimensional representations of Lie groups and Lie algebras. Following an introduction to representation theory of finite groups, the text explains how to work out the representations of classical groups.

Discrete subgroups of semisimple Lie groups

Abstract: 1. Statement of Main Results.- 2. Synopsis of the Chapters.- 3. Remarks on the Structure of the Book, References and Notation.- 1. Preliminaries.- 0. Notation, Terminology and Some Basic Facts.- 1. Algebraic Groups Over Arbitrary Fields.- 2. Algebraic Groups Over Local Fields.- 3. Arithmetic Groups.- 4. Measure Theory and Ergodic Theory.- 5. Unitary Representations and Amenable Groups.- II. Density and Ergodicity Theorems.- 1. Iterations of Linear Transformations.- 2. Density Theorems for Subgroups with Property (S)I.- 3. The Generalized Mautner Lemma and the Lebesgue Spectrum.- 4. Density Theorems for Subgroups with Property (S)II.- 5. Non-Discrete Closed Subgroups of Finite Covolume.- 6. Density of Projections and the Strong Approximation Theorem.- 7. Ergodicity of Actions on Quotient Spaces.- III. Property (T).- 1. Representations Which Are Isolated from the Trivial One-Dimensional Representation.- 2. Property (T) and Some of Its Consequences. Relationship Between Property (T) for Groups and for Their Subgroups.- 3. Property (T) and Decompositions of Groups into Amalgams.- 4. Property (R).- 5. Semisimple Groups with Property (T).- 6. Relationship Between the Structure of Closed Subgroups and Property (T) of Normal Subgroups.- IV. Factor Groups of Discrete Subgroups.- 1. b-metrics, Vitali's Covering Theorem and the Density Point Theorem.- 2. Invariant Algebras of Measurable Sets.- 3. Amenable Factor Groups of Lattices Lying in Direct Products.- 4. Finiteness of Factor Groups of Discrete Subgroups.- V. Characteristic Maps.- 1. Auxiliary Assertions.- 2. The Multiplicative Ergodic Theorem.- 3. Definition and Fundamental Properties of Characteristic Maps.- 4. Effective Pairs.- 5. Essential Pairs.- VI. Discrete Subgroups and Boundary Theory.- 1. Proximal G-Spaces and Boundaries.- 2. ?-Boundaries.- 3. Projective G-Spaces.- 4. Equivariant Measurable Maps to Algebraic Varieties.- VII. Rigidity.- 1. Auxiliary Assertions.- 2. Cocycles on G-Spaces.- 3. Finite-Dimensional Invariant Subspaces.- 4. Equivariant Measurable Maps and Continuous Extensions of Representations.- 5. Superrigidity (Continuous Extensions of Homomorphisms of Discrete Subgroups to Algebraic Groups Over Local Fields).- 6. Homomorphisms of Discrete Subgroups to Algebraic Groups Over Arbitrary Fields.- 7. Strong Rigidity (Continuous Extensions of Isomorphisms of Discrete Subgroups).- 8. Rigidity of Ergodic Actions of Semisimple Groups.- VIII. Normal Subgroups and "Abstract" Homomorphisms of Semisimple Algebraic Groups Over Global Fields.- 1. Some Properties of Fundamental Domains for S-Arithmetic Subgroups.- 2. Finiteness of Factor Groups of S-Arithmetic Subgroups.- 3. Homomorphisms of S-Arithmetic Subgroups to Algebraic Groups.- IX. Arithmeticity.- 1. Statement of the Arithmeticity Theorems.- 2. Proof of the Arithmeticity Theorems.- 3. Finite Generation of Lattices.- 4. Consequences of the Arithmeticity Theorems I.- 5. Consequences of the Arithmeticity Theorems II.- 6. Arithmeticity, Volume of Quotient Spaces, Finiteness of Factor Groups, and Superrigidity of Lattices in Semisimple Lie Groups.- 7. Applications to the Theory of Symmetric Spaces and Theory of Complex Manifolds.- Appendices.- A. Proof of the Multiplicative Ergodic Theorem.- B. Free Discrete Subgroups of Linear Groups.- C. Examples of Non-Arithmetic Lattices.- Historical and Bibliographical Notes.- References.

Representation of Lie groups and special functions

Abstract: At first only elementary functions were studied in mathematical analysis. Then new functions were introduced to evaluate integrals. They were named special functions: integral sine, logarithms, the exponential function, the probability integral and so on. Elliptic integrals proved to be the most important. They are connected with rectification of arcs of certain curves. The remarkable idea of Abel to replace these integrals by the corresponding inverse functions led to the creation of the theory of elliptic functions.

On Raghunathan’s measure conjecture

Abstract: This paper represents the last in our three-part series on Raghunathan's measure conjecture (see [R1], [R2] for Parts I and II). More specifically, let G be a real Lie group (all groups in this paper are assumed to be second countable) with the Lie algebra (, r a discrete subgroup of G and ir: G-> r \ G the projection lwg) = rg. The group G acts by right translations on r \ G, x ->x g, x E r \ G, g E G. Let gt be a Borel probability measure on r \ G. Define

Boson realizations of Lie algebras with applications to nuclear physics

Abstract: The concept of boson realization (or mapping) of Lie algebras appeared first in nuclear physics in 1962 as the idea of expanding bilinear forms in fermion creation and annihilation operators in Taylor series of boson operators, with the object of converting the study of nuclear vibrational motion into a problem of coupled oscillators. The physical situations of interest are quite diverse, depending, for instance, on whether excitations for fixed- or variable-particle number are being studied, on how total angular momentum is decomposed into orbital and spin parts, and on whether isotopic spin and other intrinsic degrees of freedom enter. As a consequence, all of the semisimple algebras other than the exceptional ones have proved to be of interest at one time or another, and all are studied in this review. Though the salient historical facts are presented in the introduction, in the body of the review the progression is (generally) from the simplest algebras to the more complex ones. With a sufficiently broad view of the physics requirements, the mathematical problem is the realization of an arbitrary representation of a Lie algebra in a subspace of a suitably chosen Hilbert space of bosons (Heisenberg-Weyl algebra). Indeed, if one includes the study of odd nuclei, one is forced to consider the mappings to spaces that are direct-product spaces of bosons and (quasi)fermions. Though all the methods that have been used for these problems are reviewed, emphasis is placed on a relatively new algebraic method that has emerged over the past decade. Many of the classic results are rederived, and some new results are obtained for odd systems. The major application of these ideas is to the derivation, starting from the shell model, of the phenomenological models of nuclear collective motion, in particular, the geometric model of Bohr and Mottelson and the more recently developed interacting boson model of Arima and Iachello. A critical discussion of those applications is interwoven with the theoretical developments on which they are based; many other applications are included, some of practical interest, some simply to illustrate the concepts, and some to suggest new lines of inquiry.

Examples of braided groups and braided matrices

Abstract: Matrix braided groups are developed as an analog of the ‘‘coordinate functions’’ on a group or supergroup. The ±1 in the super case is replaced by braid statistics. There are braided group analogs of all the classical simple Lie groups as well as braided matrix groups and braided matrices B(R) for every regular solution R of the quantum Yang–Baxter equations. A direct verification of B(R) is provided and some of the simplest examples are computed in detail.

Associated Varieties and Unipotent Representations

Abstract: Suppose G ℝ is a semisimple Lie group. The philosophy of coadjoint orbits, as propounded by Kirillov and Kostant, suggests that unitary rep-resentations of G ℝ are closely related to the orbits of G ℝ on the dual \( \mathfrak{g}_{\mathbb{R}}^{ * } \) of the Lie algebra g ℝ of G ℝ. One knows how to attach representations to semisimple orbits, but the methods used (which rely on the existence of nice “polarizing subalgebras” of g) cannot be applied to most nilpotent orbits.

Elliptic Operators and Lie Groups

Abstract: Introduction Elliptic operators Analytic elements Semigroup kernels Second-order operators Elliptic operators with variable coefficients Appendices.

Computer calculation of Witten's $3$-manifold invariant

Abstract: Witten's 2+1 dimensional Chern-Simons theory is exactly solvable. We compute the partition function, a topological invariant of 3-manifolds, on generalized Seifert spaces. Thus we test the path integral using the theory of 3-manifolds. In particular, we compare the exact solution with the asymptotic formula predicted by perturbation theory. We conclude that this path integral works as advertised and gives an effective topological invariant.

Algebras of functions on compact quantum groups, Schubert cells and quantum tori

Abstract: The structures of Poisson Lie groups on a simple compact group are parametrized by pairs (a, u), wherea∈R,\(u \in \Lambda ^2 \mathfrak{h}_R\), and\(\mathfrak{h}_R\) is a real Cartan subalgebra of complexification of Lie algebra of the group in question. In the present article the description of the symplectic leaves for all pairs (a,u) is given. Also, the corresponding quantized algebras of functions are constructed and their irreducible representations are described. In the course of investigation Schubert cells and quantum tori appear. At the end of the article the quantum analog of the Weyl group is constructed and some of its applications, among them the formula for the universalR-matrix, are given.

bounds for spectral multipliers on nilpotent groups

Abstract: A criterion is given for the LP boundedness of a class of spec- tral multiplier operators associated to left-invariant, homogeneous subelliptic second-order differential operators on nilpotent Lie groups, generalizing a theo- rem of Hormander for radial Fourier multipliers on Euclidean space. The order of differentiability required is half the homogeneous dimension of the group, improving previous results in the same direction.

Differential calculus on quantized simple lie groups

Abstract: Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q ∈ ℝ are also discussed.

S-matrix approach to massless higher spins theory ii: the case of internal symmetry

Abstract: The system of higher massive spins s = 0, 1, 2,… (every spin lies in o(n)-matrix algebra) is studied in the framework of the light-cone formalism. It is shown that the conditions of closure of the Poincare-Lorentz (PL) algebra for a four-legged diagram allow one to express all the cubic interaction constants in terms of some unique universal constant. We also shown that the systems in which even (odd) spins are realized as n × n (anti) symmetric matrices admit the consistent formulation. A complete list of four-legged vertices of the theory is given.

( T * G ) t : A toy model for conformal field theory

Abstract: We study a chiral operator algebra of conformal field theory and quantum deformation of the finite-dimensional Lie group to obtain the definition of (T * G) t and its representation. The closeness of the Kac-Moody algebras, constituting the chiral operator algebra of a typical (and generic) conformal field theory model, namely the WZNW model, and quantum deformation of corresponding finite-dimensional Lie groupG has become more and more evident in recent years [1–5]. This in particular prompts further investigation of the differential geometry of such deformations. The notion of tangent and cotangent bundles is basic in classical differential geometry. It is only natural that the quantum deformations ofTG andT * G are to be introduced alongside those forG itself. Physical ideas could be useful for this goal. Indeed, theT * G can be interpreted as a phase space for a kind of a top, generalizing the usual top associated withG=SO(3). The classical mechanics is a natural language to describe differential geometry, whereas the usual quantization is nothing but the representation theory. In this paper we put corresponding formulas in such a fashion that their deformation becomes almost evident, given the experience in this domain. As a result we get the definition of (T * G) t and its representation (t is the deformation parameter). To make the exposition most simple and formulas transparent we shall work on an example ofG=sl(2) and present results in such a way that the generalizations become evident. We shall stick to generic complex versions, real and especially compact forms requiring some additional consideration, not all of which are self-evident.

Classical $W$-algebras

Abstract: We reconsider the relation between classicalW-algebras and deformations of differential operators, emphasizing the consistency with diffeomorphisms. Generators of theW-algebra that arek-differentials are constructed by a systematic procedure. The method extends, following Drinfeld and Sokolov, toW-algebras based on arbitrary simple Lie algebras.

Application of Lie groups and differentiable manifolds to general methods for simplifying systems of partial differential equations

Abstract: General techniques are developed to obtain: (1) the completion of a sys- temof nonlinear first-order partid differential equations (PDES) which is an indepem dent set of further PDES derivable from the system by differentiation and elimination; and (2) simplifications of the system by choosing appropriate new independent and dependent variables using a result from Lie group theory The number of dependent and independent variables is reduced to the minimum. The theory specializes to the clasricd theory of a single nonlinear PDE with one unknown and can be combined with the methods of Olver, Edelen and Estabmok and Wahlquist. Most of the meth- ods appear to be sufficiently well defined for automation as are the techniques in Olvcr. A second-order nonlinear equation in n dimensions is given which is related to a fuoctional differential equation in statistical mechanics. It is reducible to two dimensions for any value of n 2 2.

Bicovariant differential calculus on quantum groups SU q (N) and SO q (N)

Abstract: Following Woronowicz's proposal the bicovariant differential calculus on the quantum groupsSU q (N) andSO q (N) is constructed. A systematic construction of bicovariant bimodules by using the $$\hat R_q $$ matrix is presented. The relation between the Hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules, and theq-deformed universal enveloping algebras is given. Imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined. As an application the Maurer-Cartan equations and theq-analogue of the structure constants are formulated.

Non-Compact WZW Conformal Field Theories

Abstract: We discuss non-compact WZW sigma models, especially the ones with symmetric space $H^{\bf C}/H$ as the target, for $H$ a compact Lie group. They offer examples of non-rational conformal field theories. We remind their relation to the compact WZW models but stress their distinctive features like the continuous spectrum of conformal weights, diverging partition functions and the presence of two types of operators analogous to the local and non-local insertions recently discussed in the Liouville theory. Gauging non-compact abelian subgroups of $H^{\bf C}$ leads to non-rational coset theories. In particular, gauging one-parameter boosts in the $SL(2,\bC)/SU(2)$ model gives an alternative, explicitly stable construction of a conformal sigma model with the euclidean 2D black hole target. We compute the (regularized) toroidal partition function and discuss the spectrum of the theory. A comparison is made with more standard approach based on the $U(1)$ coset of the $SU(1,1)$ WZW theory where stability is not evident but where unitarity becomes more transparent.

Lattices in rank one Lie groups over local fields

Abstract: We prove that if $$G = \underline G (K)$$ is theK-rational points of aK-rank one semisimple group $$\underline G $$ over a non archimedean local fieldK, thenG has cocompact non-arithmetic lattices and if char(K)>0 also non-uniform ones. We also give a general structure theorem for lattices inG, from which we confirm Serre's conjecture that such arithmetic lattices do not satisfy the congruence subgroup property.

The three‐dimensional real Lie algebras and their contractions

Abstract: In this paper, the general contractions of the three‐dimensional real Lie algebras are determined. A short summary will be given of the history, definitions, properties, examples, and applications of contractions of Lie algebras. So far, mostly simple or generalized Inonu–Wigner contractions have been used. The three‐dimensional real Lie algebras are classified in such a way that their general contractions can be easily read off. All these contractions can be realized by generalized Inonu–Wigner contractions, which supports the argument that these are the interesting objects to study.

Recognition and semi-differential invariants

Abstract: Semidifferential invariants, combining coordinates in different points together with their derivatives, are used for the description of planar contours. Their use can be seen as a tradeoff between two extreme strategies currently used in shape recognition: (invariant) feature extraction methods, involving high-order derivatives, and invariant coordinate descriptions, leading to the correspondence problem of reference points. The method for the derivation of such invariants, based on Lie group theory and applicable to a wide spectrum of transformation groups, is described. As an example, invariant curve parameterizations are developed for affine and projective transformations. The usefulness of the approach is illustrated with two examples: (1) recognition of a test set of 12 planar objects viewed under conditions allowing affine approximations, and (2) the detection of symmetry in perspective projections of curves. >

Hochschild and cyclic homology of quantum groups

Abstract: For an arbitrary complex linear semisimple Lie groupG, we consider Hopf algebras of the deformations of the formal and algebraic functions onG. The Hochschild and cyclic homology of these Hopf algebras are computed when the value of the deformation parameter is generic.

Geometry and the Dynamic Interpolation Problem

Abstract: In this paper we consider the dynamic interpolation problem for control systems in which certain dynamic variables of state trajectories are forced to pass through specific points by suitable choices of controls. This problem can be viewed as an extension of the spline problem. Following Noakes, Heinzinger and Paden [16], we give a derivation of suitable interpolating cubic splines on a Riemannian manifold extending the variational approach in Milnor [15]. For the special case of compact Lie groups, the relation with optimal control problems and singular Riemannian Geometry is spelled out in detail.