Results 1  10
of
232
Noncommutative manifolds, the instanton algebra and isospectral deformations
 Comm. Math. Phys
"... We give new examples of noncommutative manifolds that are less standard than the NCtorus or Moyal deformations of R n. They arise naturally from basic considerations of noncommutative differential topology and have nontrivial global features. The new examples include the instanton algebra and the ..."
Abstract

Cited by 167 (29 self)
 Add to MetaCart
(Show Context)
We give new examples of noncommutative manifolds that are less standard than the NCtorus or Moyal deformations of R n. They arise naturally from basic considerations of noncommutative differential topology and have nontrivial global features. The new examples include the instanton algebra and the NC4spheres S4 θ. We construct the noncommutative algebras A = C ∞ (S4 θ) of functions on NCspheres as solutions to the vanishing, chj(e) = 0,j < 2, of the Chern character in the cyclic homology of A of an idempotent e ∈ M4(A), e2 = e, e = e ∗. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmanian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC3sphere intimately related to quantum group deformations SUq(2) of SU(2) but for unusual values (complex values of modulus one) of the parameter q of qanalogues, q = exp(2πiθ). We then construct the noncommutative geometry of S4 θ as given by a spectral triple (A, H,D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric gµν on S4 whose volume form √ g d4x is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation, e − 1
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
"... ..."
(Show Context)
Charge Deficiency, Charge Transport and Comparison of Dimensions
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1994
"... We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. ..."
Abstract

Cited by 41 (0 self)
 Add to MetaCart
We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.
A Proof of Tsygan’s formality conjecture for an arbitrary Smooth Manifold
, 2005
"... Proofs of Tsygan’s formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the AtiyahPatodiSinger index theorem and the RiemannRochHirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various ..."
Abstract

Cited by 36 (13 self)
 Add to MetaCart
Proofs of Tsygan’s formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the AtiyahPatodiSinger index theorem and the RiemannRochHirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various people the most general version of Tsygan’s formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevich’s formality quasiisomorphism for Hochschild cochains of R[[y 1,...,y d]] and Shoikhet’s formality quasiisomorphism for Hochschild chains of R[[y 1,...,y d]] I prove Tsygan’s formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold. The construction of the formality quasiisomorphism for Hochschild chains is manifestly functorial for isomorphisms of the pairs (M, ∇), where M is the manifold and ∇ is an affine connection on the
Positive representations of general commutation relations allowing wick ordering
 FUNCT ANAL
, 1995
"... 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 qcanonical commutation relations introduced by Greenberg, Bozejko, and Speicher, ..."
Abstract

Cited by 35 (7 self)
 Add to MetaCart
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 qcanonical 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 noncommutative 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.