Showing papers on "Operator algebra published in 1991"

A spatial operator algebra for manipulator modeling and control

Abstract: A recently developed spatial operator algebra for manipu lator modeling, control, and trajectory design is dis cussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high- level framework for describing the dynamic and kinematic behavior of a manipulator and for control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced concep tual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection. Thus the transition from an abstract problem formulation and solution to the detailed mechanizat...

Conformal field theories via Hamiltonian reduction

Abstract: Constraining theSL(3) WZW-model we construct a reduced theory which is invariant with respect to the new chiral algebraW 3 2 . This symmetry is generated by the stress-energy tensor, two bosonic currents with spins 3/2 and theU(1) current. We conjecture a Kac formula that describes the highly reducible representation for this algebra. We also discuss the quantum Hamiltonian reduction for the general type of constraints that leads to the new extended conformal algebras.

Operator algebras in dynamical systems

Abstract: This book is concerned with the theory of unbounded derivations in C*-algebras, a subject whose study was motivated by questions in quantum physics and statistical mechanics, and to which the author has made a considerable contribution. This is an active area of research, and one of the most ambitious aims of the theory is to develop quantum statistical mechanics within the framework of the C*-theory. The presentation, which is based on lectures given in Newcastle upon Tyne and Copenhagen, concentrates on topics involving quantum statistical mechanics and differentiations on manifolds. One of the goals is to formulate the absence theorem of phase transitions in its most general form within the C* setting. For the first time, he globally constructs, within that setting, derivations for a fairly wide class of interacting models, and presents a new axiomatic treatment of the construction of time evolutions and KMS states.

Differential Topology and Quantum Field Theory

Abstract: A Topological Preliminary. Elliptic Operators. Cohomology of Sheaves and Bundles. Index Theory for Elliptic Operators. Some Algebraic Geometry. Infinite Dimensional Groups. Morse Theory. Instantons and Monopoles. The Elliptic Geometry of Strings. Anomalies. Conformal Quantum Field Theories. Topological Quantum Field Theories. References.

( 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.

Braided groups and algebraic quantum field theories

Abstract: We introduce the notion of a braided group. This is analogous to a supergroup with Bose-Fermi statistics ±1 replaced by braid statistics. We show that every algebraic quantum field theory in two dimensions leads to a braided group of internal symmetries. Every quantum group can be viewed as a braided group.

Topological Mirrors and Quantum Rings

Abstract: Aspects of duality and mirror symmetry in string theory are discussed. We emphasize, through examples, the importance of loop spaces for a deeper understanding of the geometrical origin of dualities in string theory. Moreover we show that mirror symmetry can be reformulated in very simple terms as the statement of equivalence of two classes of topological theories: Topological sigma models and topological Landau-Ginzburg models. Some suggestions are made for generalization of the notion of mirror symmetry.

Spectral theory of families of self-adjoint operators

Abstract: Comments to the introduction.- I Families of Commuting Normal Operators.- 1. Spectral Analysis of Countable Families of Commuting Self-Adjoint Operators (CSO).- 2. Unitary Representations of Inductive Limits of Commutative Locally Compact Groups.- 3. Differential Operators With Constant Coefficients In Spaces of Functions of Infinitely Many Variables.- Inductive Limits of Finite-Dimensional Lie Algebras and Their Representations.- 4. Canonical Commutation Relations (CCR) of Systems with Countable Degrees of Freedom.- 5. Unitary Representations of The Group of Finite SU(2)-Currents on A Countable Set.- 6. Representations of The Group of Upper Triangular Matrices.- 7. A Class of Inductive Limits of Groups and Their Representations.- Collections of Unbounded Self-Adjoint operators Satisfying General Relations.- 8. Anticommuting Self-Adjoint Operators.- 9. Finite and Countable Collections of Gradedcommuting Self-Adjoint Operators (GCSO).- 10. Collections Of Unbounded CSO (Ak) And CSO (Bk) Satisfying General Commutation Relations.- Representations of Operator Algebras And Non-Commutative Random Sequences.- 11. C* -ALGEBRASU0? And Their Representations.- 12. Non-Commutative Random Sequences and Methods for Their Construction.

Geometric interpretation of vertex operator algebras.

Abstract: In this paper, Vafa's approach to the formulation of conformal field theories is combined with the formal calculus developed in Frenkel, Lepowsky, and Meurman's work on the vertex operator construction of the Monster to give a geometric definition of vertex operator algebras. The main result announced is the equivalence between this definition and the algebraic one in the sense that the categories determined by these definitions are isomorphic.

Explicit construction of universal operator algebras and applications to polynomial factorization

Abstract: Using the characterization of unital operator algebras developed in [6], we give explicit internal definitions of the free product and the maximal operator-algebra tensor product of operator algebras and of the group operator algebra OA(G) of a discrete semigroup G (if G is a discrete group, then OA(G) coincides with the group C*-algebra C*(G)). This approach leads to new factorization theorems for polynomials in one and two variables.

The Black hole horizon as a quantum surface

Abstract: In previous work it was argued why it is practically inevitable to assume the existence of an S-matrix that describes particle absorption and production by a black hole, and the relationship between this S-matrix and string theory was derived. The physical interpretation of the corresponding mathematical expressions however is quite different from string theory. We have an algebra of operators now defined on a two-dimensional Euclidean "world sheet". The algebra simplifies if one restricts it to the self dual projection of the fundamental surface elements Wμν. Our two dimensional functional integrals correspond to a rather unusual field theory. The long distance structure of this theory follows directly from the long distance structure of the standard model at the GeV scale. We emphasize the rather delicate physical interpretation of this approach.

Operator algebras in dynamical systems : the theory of unbounded derivations in C[*]-algebras

Abstract: Preface 1. Preliminaries 2. Bounded derivations 3. Unbounded derivations 4. C*-dynamical systems Index.

A FRACTIONAL KdV HIERARCHY

Abstract: We construct a new system of integrable nonlinear differential equations associated with. the operator algebra of Polyakov. Its members are fractional generalizations of KdV type flows corresponding to an alternative set of constraints on the 2-dim. SL(3) gauge connections. We obtain the first non-trivial examples by dimensional reductiion from self-dual Yang–Mills and then generate recursively the entire hierarchy and its conserved quantities using a bi-Hamiltonian structure. Certain relations with the Boussinesq equation are also discussed together with possible generalizations of the formalism to SL(N) gauge groups and operator algebras with arbitrary N and l.

Linear algebra and quantum chemistry

Abstract: (1991). Linear Algebra and Quantum Chemistry. The American Mathematical Monthly: Vol. 98, No. 8, pp. 710-718.

Recursive formulation of operational space control

Abstract: A spatial operator algebra approach to modeling and analysis of multibody robotic systems is discussed. It is shown that this modeling and analysis method can be used to develop O(n) recursive algorithms that compute the operational space mass matrix and the operational space coriolis/centrifugal and gravity terms of an n-link serial chain manipulator. It is also shown that, taken together, these algorithms enable an O(n) recursive implementation of operational space control. >

Spectral Theory in p-ADIC Quantum Mechanics, and Representation Theory

Abstract: Spectral theory is discussed for a harmonic oscillator in p-adic quantum mechanics. The problem of the decomposition into irreducible representations of the restriction of a projective representation of the symplectic group to a compact abelian subgroup is solved, the dimensions of the invariant subspaces are calculated, and the eigenfunctions are analyzed. Spectral problems are considered for p-adic pseudodifferential operators of Schrodinger type. A complete orthonormal system of eigenfunctions is constructed for the p-adic analog of the differentiation operator.

On certain properties of the relative entropy of states of operator algebras

Abstract: The aim of the present paper is to establish certain properties of the relative entropy functional in the general context of C*-algebras Most of these properties have been observed under more restrictive conditions in earlier papers (The celebrated strong subadditivity of the entropy gives such an example) The general approach is applied to prove the existence of the mean relative entropy for a noncommutative stationary Markov chain with conditional expectations

Paradigms of quantum algebras

Abstract: This is an informal overview of versions of quantum algebras which are currently finding applications in physics. Special attention is given to the quantum deformations of SU(2) and illustrations of general principles.

An example of a quantum exponential process

Abstract: In [3],R. L. Hudson andK. R. Parthasarathy showed that the Fock space based on the Heisenberg—Weyl algebra hosts Brownian motion and Poisson processes. In this paper we construct a quantum exponential process acting on the Fock space based on the finite-difference algebra ofP. J. Feinsilver ([2]).