Abstract not found

Abstract not found

Respectfully dedicate to Professor M. Sato on the occasion of his 70th birthday Abstract. Let X be a compact manifold with boundary. Suppose that the boundary is fibred, φ: ∂X − → Y, and let x ∈ C ∞ (X) be a boundary defining function. This data fixes the space of ‘fibred cusp ’ vector fields, consisting of those vector fields V on X satisfying V x = O(x 2) and which are tangent to the fibres of φ; it is a Lie algebra and C ∞ (X) module. This Lie algebra is quantized to the ‘small calculus ’ of pseudodifferential operators Ψ ∗ Φ (X). Mapping properties including boundedness, regularity, Fredholm condition and symbolic maps are discussed for this calculus. The spectrum of the Laplacian of an ‘exact fibred cusp ’ metric is analyzed as is the wavefront set associated to the calculus.

We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative three-dimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic K-theoretic equations. We find a 3-parameter family of deformations of the standard 3-sphere S 3 and a corresponding 3-parameter deformation of the 4-dimensional Euclidean space R 4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u are isomorphic to the algebras introduced by Sklyanin in connection with the Yang-Baxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θ-deformations, which we generalize in any dimension and various contexts, and study in some details. Here, and

It has been discussed earlier that ( weak quasi-) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra A of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey local braid relation. Contents 1 Introduction 1 2 The Notion of Quantum Symmetry 5 3 Algebraic Methods for Field Construction 9 3.1 Observables and superselection sectors in local quantum field theory . . . . 10 3.2 Localized endomorphisms and fusion structure . . . . . ....

We consider the problem of representing in Hilbert space commutation relations of the form aia ∗ j = δij1 + ∑ kℓ T kℓ ij a ∗ ℓ ak, where the T kℓ ij are essentially arbitrary scalar coefficients. Examples comprise the q-canonical commutation relations introduced by Greenberg, Bozejko, and Speicher, and the twisted canonical (anti-)commutation relations studied by Pusz and Woronowicz, as well as the quantum group SνU(2). Using these relations, any polynomial in the generators ai and their adjoints can uniquely be written in “Wick ordered form ” in which all starred generators are to the left of all unstarred ones. In this general framework we define the Fock representation, as well as coherent representations. We develop criteria for the natural scalar product in the associated representation spaces to be positive definite, and for the relations to have representations by bounded operators in a Hilbert space. We characterize the relations between the generators ai (not involving a ∗ i) which are compatible with the basic relations. The relations may also be interpreted as defining a non-commutative differential calculus. For generic coefficients T kℓ ij, however, all differential forms of degree 2 and higher vanish. We exhibit conditions for this not to be the case, and relate them to the ideal structure of the Wick algebra, and conditions of positivity. We show that the differential calculus is compatible with the involution iff the coefficients T define a representation of the braid group. This condition is also shown to imply improved bounds for the positivity of the Fock representation. Finally, we study the KMS states of the group of gauge transformations defined by aj ↦ → exp(it)aj.

We define a notion of a strong homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild cochains are formulated. We prove several results supporting these conjectures.

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of\Omega 1 . A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of\Omega 1 . The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois-Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of\Omega 1 . These constructions are illustrated with the example of the algebra of n \Theta n matrices. ? e-mail:MOURAD@celfi.univ-tours.fr 1. Introduction Non-commutative geometry [1,2] offers a novel and promising framework f...

We show that the isotropic harmonic oscillator in the ordinary euclidean space RN (N ≥ 3) admits a natural q-deformation into a new quantum mechanical model having a q-deformed symmetry (in the sense of quantum groups), SOq(N,R). The q-deformation is the consequence of replacing RN by RN q (the corresponding quantum space). This provides an example of quantum mechanics on a noncommutative geometrical space. To reach the goal, we also have to deal with a sensible definition of integration over RN q, which we use for the definition of the scalar product of states. 1

We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyclic homology. This completes results of Rickard’s [48] and Happel’s [18]. It also extends well known results on preservation of cyclic homology under Morita equivalence [10], [39], [25], [26], [41], [42]. We then show that under suitable flatness hypotheses, an exact sequence of derived categories of DG algebras yields a long exact sequence in cyclic homology. This may be viewed as an analogue of Thomason-Trobaugh’s [51] and Yao’s [58] localization theorems in K-theory (cf. also [55]).