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Homology of pseudodifferential operators on manifolds with corners I. Manifolds with boundary
, 1996
"... 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, consi ..."
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Cited by 138 (26 self)
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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.
Noncommutative FiniteDimensional Manifolds  I. SPHERICAL MANIFOLDS AND RELATED EXAMPLES
, 2001
"... We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 d ..."
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Cited by 125 (15 self)
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We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic Ktheoretic equations. We find a 3parameter family of deformations of the standard 3sphere S 3 and a corresponding 3parameter deformation of the 4dimensional 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 YangBaxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θdeformations, which we generalize in any dimension and various contexts, and study in some details. Here, and
homotopy formulas and the GaussManin connection in cyclic homology
 in Quantum deformations of algebras and their representations, Israel Math. Conf. Proc. 7
, 1993
"... It is wellknown that the periodic cyclic homology HP•(A) of an algebra A is homotopy invariant (see Connes [3], Goodwillie [8] and Block [1]). Let A be an algebra over a field k and let Aν be a formal deformation of A, that is, an associative product mν ∈ Hom(A⊗2, A)[[ν1,..., νn]] such thatmν=0 is ..."
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Cited by 71 (1 self)
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It is wellknown that the periodic cyclic homology HP•(A) of an algebra A is homotopy invariant (see Connes [3], Goodwillie [8] and Block [1]). Let A be an algebra over a field k and let Aν be a formal deformation of A, that is, an associative product mν ∈ Hom(A⊗2, A)[[ν1,..., νn]] such thatmν=0 is the product on A. We will define a connection on the periodic cyclic bar complex of Aν for which the differential is covariant constant, thus inducing a connection on the periodic homology HP•(Aν), thought of as a module over k[[ν1,..., νn]]. This connection generalizes the classical GaussManin connection, and indeed we will prove that it has curvature chain homotopic to zero. The GaussManin connection is obtained by a generalization of Rinehart’s result: if D is a derivation on A, then the operator uL(D) on the cyclic bar complex C(A)[[u]] of A is chain homotopic to zero (see Rinehart [10], and also Goodwillie [8]). Inspired by the work of Nistor [9], we prove a more general result on the action of the cochains C•(A,A) on the cyclic bar complex C(A), where A is an A∞algebra. (Recall that A∞algebras are a generalization of differential graded algebras. It is shown in [6] that A∞algebras form a natural setting for the study of cyclic homology.) The author would like to thank J. Block, V. Nistor, D. Quillen and B. Tsygan for helpful conversations on the subject of this article; in particular, the term GaussManin connection is used at Quillen’s suggestion. Note that some similar formulas, in the setting of Hochschild homology, have been obtained by Gelfand, Daletskĭı and Tsygan [4]. 1. Hochschild cochains and A∞algebras In this paper, all vector spaces will be over a field k. If V and W are graded vector spaces, we denote by V ⊗W the graded tensor product: thus, if A ∈ End(V) and B ∈ End(W), then (A ⊗ B)(v ⊗ w) = (−1)B  v(Av) ⊗ (Bw). If V is a graded vector space, we denote by V (k) the tensor power V ⊗k. Let A be a graded vector space, and let sA be its suspension
Localization Bounds for an Electron Gas
, 1998
"... Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of a number of physical manifestations of localization, in regimes of strong disorder, extreme energies ..."
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Cited by 68 (9 self)
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Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of a number of physical manifestations of localization, in regimes of strong disorder, extreme energies, or weak disorder away from the unperturbed spectrum. The present work establishes on this basis exponential decay for the modulus of the two–point function, at all temperatures as well as in the ground state, for a Fermi gas within the one–particle approximation. Different implications, in particular for the Integral Quantum Hall Effect, are reviewed.
Construction of Field Algebras with Quantum Symmetry from Local Observables
, 1996
"... 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 reco ..."
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Cited by 61 (7 self)
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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 . . . . . ....
Noncommutative differential calculus, homotopy . . .
, 2000
"... 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. ..."
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Cited by 57 (1 self)
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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.
Invariance and localization for cyclic homology of DG algebras
 J. PURE APPL. ALGEBRA
, 1998
"... 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 cyc ..."
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Cited by 51 (6 self)
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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 ThomasonTrobaugh’s [51] and Yao’s [58] localization theorems in Ktheory (cf. also [55]).
Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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Cited by 49 (5 self)
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.