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197
ON THE VASSILIEV KNOT INVARIANTS
, 1995
"... The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations coming from various quantum groups, and it is conjectured that these invariants are precisely as powerful a ..."
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Cited by 139 (0 self)
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The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations coming from various quantum groups, and it is conjectured that these invariants are precisely as powerful as those polynomials. As invariants of finite type are much easier to define and manipulate than the quantum group invariants, it is likely that in attempting to classify knots, invariants of finite type will play a more fundamental role than the various knot polynomials.
Twisted Ktheory of differentiable stacks
 ANN. SCI. ÉCOLE NORM. SUP
, 2004
"... In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framew ..."
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Cited by 42 (12 self)
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In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framework for studying various twisted Ktheories including the usual twisted Ktheory of topological spaces, twisted equivariant Ktheory, and the twisted Ktheory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted Kgroups can be expressed by socalled “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of Ktheory (KKtheory) of C ∗algebras.
The Ideal Structure of CuntzKrieger Algebras
 Ergod. Th. and Dyn. Sys
, 1996
"... We construct a universal CuntzKrieger algebra AO A , which is isomorphic to the usual CuntzKrieger algebra O A when A satises the condition (I) of Cuntz and Krieger. Cuntz's classication of ideals in O A when A satises condition (II) extends to a classication of the gauge invariant ideals in A ..."
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Cited by 39 (9 self)
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We construct a universal CuntzKrieger algebra AO A , which is isomorphic to the usual CuntzKrieger algebra O A when A satises the condition (I) of Cuntz and Krieger. Cuntz's classication of ideals in O A when A satises condition (II) extends to a classication of the gauge invariant ideals in AO A . We use this to describe the topology on the primitive ideal space of AO A . 1 Introduction In [4] Cuntz and Krieger studied C algebras generated by families of n nonzero partial isometries S i satisfying S i S i = n X i=1 A(i; j)S j S j and n X i=1 S i S i = 1; (1) where A is an nn matrix with entries in f0; 1g and no zero rows or columns. It was shown in [4, Theorem 2.13] that, if A satises a certain condition (I), then C (S i ) is unique up to canonical isomorphism (i.e., an isomorphism mapping generators to generators), so that O A := C (S i ) depends only on A. The algebra O A is simple if A is irreducible and not a permutation matrix [4, ...
Cluster algebra structures and semicanonical bases for unipotent groups
, 2008
"... Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQmodule M (these are certain preinjective CQmodules), we attach a natural subcategory CM of mod(Λ). We show that CM is a ..."
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Cited by 23 (1 self)
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Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQmodule M (these are certain preinjective CQmodules), we attach a natural subcategory CM of mod(Λ). We show that CM is a
The Århus Integral of Rational Homology 3Spheres I: A Highly Non Trivial Flat Connection on S³
, 2002
"... Path integrals do not really exist, but it is very useful to dream that they do and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly nontrivial mathematical theorems and theories. We argue that even though nontri ..."
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Cited by 20 (4 self)
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Path integrals do not really exist, but it is very useful to dream that they do and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly nontrivial mathematical theorems and theories. We argue that even though nontrivial at connections on S³ do not really exist, it is beneficial to dream that one exists (and, in fact, that it comes from the nonexistent ChernSimons path integral). Dreaming the right way, we are led to a rigorous construction of a universal finitetype invariant of rational homology spheres. We show that this invariant is equal (up to a normalization) to the LMO (LeMurakamiOhtsuki, [LMO]) invariant and that it recovers the Rozansky and Ohtsuki invariants. This is part I of a 4...
A short proof that Mn(A) is local if A is local and Fréchet
 Intl. Jour. Math
, 1992
"... We give a short and very general proof of the fact that the property of a dense Fréchet subalgebra of a Banach algebra being local, or closed under the holomorphic functional calculus in the Banach algebra, is preserved by tensoring with the n × n matrix algebra of the complex numbers. ..."
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Cited by 18 (2 self)
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We give a short and very general proof of the fact that the property of a dense Fréchet subalgebra of a Banach algebra being local, or closed under the holomorphic functional calculus in the Banach algebra, is preserved by tensoring with the n × n matrix algebra of the complex numbers.
Edge current channels and Chern numbers in the integer quantum Hall effect
, 2000
"... A quantization theorem for the edge currents is proven for discrete magnetic halfplane operators. Hence the edge channel number is a valid concept also in presence of a disordered potential. Under a gap condition on the corresponding planar model, this quantum number is shown to be equal to the quan ..."
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Cited by 18 (6 self)
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A quantization theorem for the edge currents is proven for discrete magnetic halfplane operators. Hence the edge channel number is a valid concept also in presence of a disordered potential. Under a gap condition on the corresponding planar model, this quantum number is shown to be equal to the quantized Hall conductivity as given by the KuboChern formula. For the proof of this equality, we consider an exact sequence of C algebras (the Toeplitz extension) linking the halfplane and the planar problem, and use a duality theorem for the pairings of Kgroups with cyclic cohomology. 1 Introduction In quantum Hall effect (QHE) experiments, one observes the quantization of the Hall conductance of an effectively twodimensional semiconductor in units of the universal constant e 2 =h [35, 45]. As the Hall conductance is a macroscopic quantity, this effect is of completely different nature than any quantization in atomic physics resulting from BohrSommerfeld rules. Although also a pur...
Random operators and crossed products
 Mathematical Physics, Analysis and Geometry
, 1999
"... Abstract. This article is concerned with crossed products and their applications to random operators. We study the von Neumann algebra of a dynamical system using the underlying Hilbert algebra structure. This gives a particularly easy way to introduce a trace on this von Neumann algebra. We review ..."
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Cited by 15 (13 self)
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Abstract. This article is concerned with crossed products and their applications to random operators. We study the von Neumann algebra of a dynamical system using the underlying Hilbert algebra structure. This gives a particularly easy way to introduce a trace on this von Neumann algebra. We review several formulas for this trace, show how it comes as an application of Connes ’ non commutative integration theory and discuss Shubin’s trace formula. We then restrict ourselves to the case of an action of a group on a group and include new proofs for some theorems of Bellissard and Testard on an analogue of the classical Plancherel Theorem. We show that the integrated density of states is a spectral measure in the periodic case, therby generalizing a result of Kaminker and Xia. Finally, we discuss duality results and apply a method of Gordon et al. to establish a duality result for crossed products by Z. Families of random operators arise in the study of disordered media. More precisely, one is given a topological space X and a family of operators (Hx)x∈X on L 2 (G).
The index of projective families of elliptic operators
 Geom. Topol. 9 (2005) 341–373, math.DG/0206002, MR 2140985, Zbl 1083.58021
"... Abstract. An index theory for projective families of elliptic pseudodifferential operators is developed. The topological and the analytic index of such a family both take values in twisted Ktheory of the parametrizing space, X. The main result is the equality of these two notions of index when the ..."
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Cited by 15 (5 self)
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Abstract. An index theory for projective families of elliptic pseudodifferential operators is developed. The topological and the analytic index of such a family both take values in twisted Ktheory of the parametrizing space, X. The main result is the equality of these two notions of index when the twisting class is in the torsion subgroup tor(H 3 (X; Z)). The Chern character of the index class is then computed.
Duality of compact groups and Hilbert C*systems for C*algebras with a nontrivial center
, 2004
"... In the present paper we prove a duality theory for compact groups in the case when the C*algebra A, the fixed point algebra of the corresponding Hilbert C*system (F, G), has a nontrivial center Z ⊃ C and the relative commutant satisfies the minimality condition A ′ ∩ F = Z, as well as a technical ..."
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Cited by 15 (2 self)
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In the present paper we prove a duality theory for compact groups in the case when the C*algebra A, the fixed point algebra of the corresponding Hilbert C*system (F, G), has a nontrivial center Z ⊃ C and the relative commutant satisfies the minimality condition A ′ ∩ F = Z, as well as a technical condition called regularity. The abstract characterization of the mentioned Hilbert C*system is expressed by means of an inclusion of C*categories TC < T, where TC is a suitable DRcategory and T a full subcategory of the category of endomorphisms of A. Both categories have the same objects and the arrows of T can be generated from the arrows of TC and the center Z. A crucial new element that appears in the present analysis is an abelian group C(G), which we call the chain group of G, and that can be constructed from certain equivalence relation defined on ̂ G, the dual object of G. The chain group, which is isomorphic to the character group of the center of G, determines the action of irreducible endomorphisms of A when restricted to Z. Moreover, C(G) encodes the possibility of defining a symmetry ǫ also for the