Results 1  10
of
235
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 ..."
Abstract

Cited by 144 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 50 (12 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 41 (11 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 24 (2 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 24 (4 self)
 Add to MetaCart
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 S&sup3; 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...
Category O: Quivers and Endomorphism Rings of Projectives
, 2003
"... We describe an algorithm for computing quivers of category O of a finite dimensional semisimple Lie algebra. The main tool for this is Soergel’s description of the endomorphism ring of the antidominant indecomposable projective module of a regular block as an algebra of coinvariants. We give explic ..."
Abstract

Cited by 21 (7 self)
 Add to MetaCart
We describe an algorithm for computing quivers of category O of a finite dimensional semisimple Lie algebra. The main tool for this is Soergel’s description of the endomorphism ring of the antidominant indecomposable projective module of a regular block as an algebra of coinvariants. We give explicit calculations for root systems of rank 1 and 2 for regular and singular blocks and also quivers for regular blocks for type A3. The main result in this paper is a necessary and sufficient condition for an endomorphism ring of an indecomposable projective object of O to be commutative. We give also an explicit formula for the socle of a projective object with a short proof using Soergel’s functor V and finish with a generalization of this functor to HarishChandra bimodules and parabolic versions of category O.
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. ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
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.
Between classical and quantum
, 2008
"... The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, inclu ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg’s ‘quantumtheoretical Umdeutung (reinterpretation) of classical observables’, which lies at the basis of quantization theory. Similarly, Bohr’s correspondence principle (in somewhat revised form) and Schrödinger’s wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely
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 ..."
Abstract

Cited by 18 (6 self)
 Add to MetaCart
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...
A counterexample to Hilbert’s Fourteenth Problem in dimension six. Transformation Groups 5
, 2000
"... Abstract. We survey counterexamples to Hilbert’s Fourteenth Problem, beginning with those of Nagata in the late 1950s, and including recent counterexamples in low dimension constructed with locally nilpotent derivations. Historical framework and pertinent references are provided. We also include 8 ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We survey counterexamples to Hilbert’s Fourteenth Problem, beginning with those of Nagata in the late 1950s, and including recent counterexamples in low dimension constructed with locally nilpotent derivations. Historical framework and pertinent references are provided. We also include 8 important open questions. 1. Introduction. One