Differential and holomorphic differential operators on noncommutative algebras

This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of as category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considered, though without the quotient to ensure zero curvature.Comment: 31 pages. Comments welcome on the contents or on references that should be added for a possible revised version. Please send any comments to the autho

Pointwise bounded asymptotic morphisms and Thomsen's non-stable k-theory

In this paper I show that pointwise bounded asymptotic morphisms between separable metrisable locally convex *-algebras induce continuous maps between the quasi-unitary groups of the algebras, provided that the algebras support a certain amount of functional calculus. This links the asymptotic morphisms directly to Thomsen's non-stable definition of k-theory in the C* algebra case. A result on composition of asymptotic morphisms is also given.Comment: LaTex, approx 14page

Noncommutative geodesics and the KSGNS construction

We study geodesics in noncommutative geometry by means of bimodule connections and completely positive maps using the Kasparov, Stinespring, Gel'fand, Naimark & Segal (KSGNS) construction. This is motivated from classical geometry, and we also consider examples on the algebras M_2(C) and C(Z_n), though restricting to classical real time. On the way we have to consider the reality of a noncommutative vector field, and for this we propose a definition depending on a state on the algebra.Comment: Version 2: Major revision - the main change is the inclusion of a reality condition on vector fields which enforces normalisation of states. The examples have been recomputed to include thi

Two-forms and Noncommutative Hamiltonian dynamics

In this paper we extend the standard differential geometric theory of Hamiltonian dynamics to noncommutative spaces, beginning with symplectic forms. Derivations on the algebra are used instead of vector fields, and interior products and Lie derivatives with respect to derivations are discussed. Then the Poisson bracket of certain algebra elements can be defined by a choice of closed 2-form. Examples are given using the noncommutative torus, the Cuntz algebra, the algebra of matrices, and the algebra of matrix valued functions on $\Bbb R^2$.Comment: 10 pages LaTe

The Majid-Ruegg model and the Planck scales

A novel differential calculus with central inner product is introduced for kappa-Minkowski space. The `bad' behaviour of this differential calculus is discussed with reference to symplectic quantisation and A-infinity algebras. Using this calculus in the Schrodinger equation gives two values which can be compared with the Planck mass and length. This comparison gives an approximate numerical value for the deformation parameter in kappa-Minkowski space. We present numerical evidence that there is a potentially observable variation of propagation speed in the Klein-Gordon equation. The modified equations of electrodynamics (without a spinor field) are derived from noncommutative covariant derivatives. We note that these equations suggest that the speed of light is independent of frequency, in contrast to the KG results (with the caveat that zero current is not the same as in vacuum). We end with some philosophical comments on measurement related to quantum theory and gravity (not necessarily quantum gravity) and noncommutative geometry.Comment: The authors would be happy to receive all comments or additional references. Version 2 - additional references adde

A Leray spectral sequence for noncommutative differential fibrations

This paper describes the Leray spectral sequence associated to a differential fibration. The differential fibration is described by base and total differential graded algebras. The cohomology used is noncommutative differential sheaf cohomology. For this purpose, a sheaf over an algebra is a left module with zero curvature covariant derivative. As a special case, we can recover the Serre spectral sequence for a noncommutative fibration.Comment: The authors would be grateful for any comments or additional reference

Quantum Bianchi identities and characteristic classes via DG categories

We show how DG categories arise naturally in noncommutative differential geometry and use them to derive noncommutative analogues of the Bianchi identities for the curvature of a connection. We also give a derivation of formulae for characteristic classes in noncommutative geometry following Chern's original derivation, rather than using cyclic cohomology. We show that a related DG category for extendable bimodule connections is a monoidal tensor category and in the metric compatible case give an analogue of a classical antisymmetry of the Riemann tensor. The monoidal structure implies the existence of a cup product on noncommutative sheaf cohomology. Another application is to prove that the curvature of a line module reduces to a 2-form on the base algebra. We also extend our geometric approach to Dirac operators. We illustrate the theory on the q-sphere, the permutation group S_3 and the bicrossproduct model quantum spacetime with algebra [r,t]=\lambda r.Comment: 37 pages, added a small result about the square of the Dirac o

Semiquantisation Functor and Poisson-Riemannian Geometry, I

We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter $\lambda$, using a functorial approach. The data for quantisation of the cotangent bundle is known to be a Poisson structure and Poisson preconnection and we now show that this data defines to a functor $Q$ from the monoidal category of classical vector bundles equipped with connections to the monodial category of bimodules equipped with bimodule connections over the quantised algebra. We adapt this functor to quantise the wedge product of the exterior algebra and in the Riemannian case, the metric and the Levi-Civita connection. Full metric compatibility requires vanishing of an obstruction in the classical data, expressed in terms of a generalised Ricci 2-form, without which our quantum Levi-Civita connection is still the best possible. We apply the theory to the Schwarzschild black-hole and to Riemann surfaces as examples, as well as verifying our results on the 2D bicrossproduct model quantum spacetime. The quantized Schwarzschild black-hole in particular has features similar to those encountered in $q$-deformed models, notably the necessity of nonassociativity of any rotationally invariant quantum differential calculus of classical dimensions.Comment: 57 pages AMS LATEX, no figure

Noncommutative complex differential geometry

This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a differential structure on a noncommutative algebra defined in terms of a differential graded algebra. This is compared to current ideas on noncommutative algebraic geometry.Comment: 42 pages. A few small changes and corrections to the previous version after being refereed. To appear in the special issue of the Journal for Geometry and Physics for the meeting "Noncommutative Algebraic Geometry and its Applications to Physics" 19-23 March 2012, Leiden N

Analogue-digital systems with modes of physical behaviour

Complex environments, processes and systems may exhibit several distinct modes of physical behaviour or operation. Thus, for example, in their design, a set of mathematical models may be needed, each model having its own domain of application and representing a particular mode of behaviour or operation of physical reality. The models may be of disparate kinds { discrete or continuous in data, time and space. Furthermore, some physical modes may not have a reliable model. Physical measurements determine modes of operation. We explore the question: What is a mode of behaviour? How do we specify algorithms and software that monitor or govern a complex physical situation with many modes? How do we specify a portfolio of modes, and the computational problem of transitioning from using one mode to another mode as physical modes change? We propose a general definition of an analogue-digital system with modes. We show how any diverse set of modes { with or without models { can be bound together, and how the transitions between modes can be determined, by constructing a topological data type based upon a simplicial complex. We illustrate the ideas of physical modes and our theory by reflecting on simple examples, including driverless racing cars.Comment: Ver 3: references and background material adde