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1.
We show that the crossed modules and bicovariant differential calculi on two Hopf algebras related by a cocycle twist are
in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the
calculi are obtained as deformation-quantisations of the classical ones. As an application, we classify all bicovariant differential
calculi on the Planck scale Hopf algebra . This is a quantum group which has an limit as the functions on a classical but non-Abelian group and a limit as flat space quantum mechanics. We further study the noncommutative differential geometry and Fourier theory for this
Hopf algebra as a toy model for Planck scale physics. The Fourier theory implements a T-duality-like self-duality. The noncommutative
geometry turns out to be singular when and is therefore not visible in flat space quantum mechanics alone.
Received: 28 October 1998 / Accepted: 7 March 1999 相似文献
2.
In this paper, we establish the invariance of cyclic (co)homology of left Hopf algebroids under the change of Morita equivalent base algebras. The classical result on Morita invariance for cyclic homology of associative algebras appears as a special example of this theory. In our main application we consider the Morita equivalence between the algebra of complex-valued smooth functions on the classical 2-torus and the coordinate algebra of the noncommutative 2-torus with rational parameter. We then construct a Morita base change left Hopf algebroid over this noncommutative 2-torus and show that its cyclic (co)homology can be computed by means of the homology of the Lie algebroid of vector fields on the classical 2-torus. 相似文献
3.
Charles Radin 《Communications in Mathematical Physics》1973,33(4):283-292
Using noncommutative integration theory, we show why certain singular behavior has been appearing in the dynamics of large quantum mechanical systems, and how to isolate the singularities.Research supported by AFOSR under contract F 44620-71-C-0108. 相似文献
4.
5.
P. Aschieri 《Russian Journal of Mathematical Physics》2009,16(3):371-383
We consider noncommutative geometries obtained from a triangular Drinfeld twist. This allows us to construct and study a wide
class of noncommutative manifolds and their deformed Lie algebras of infinitesimal diffeomorphisms. Principles of symmetry
can be implemented in this way. Two examples are considered: (a) the general covariance in noncommutative spacetime, which
leads to a noncommutative theory of gravity, and b) symplectomorphims of the algebra of observables associated with noncommutative
configuration spaces, which leads to a geometric formulation of quantization on noncommutative spacetime (i.e., we establish
a noncommutative correspondence principle from ⋆-Poisson brackets to ⋆-commutators). 相似文献
6.
In this paper we construct noncommutative resolutions of a certain class of Calabi-Yau threefolds studied by Cachazo et al.
(Geometric transitions and N = 1 quiver theories. , 2001). The threefolds under consideration are fibered over a complex plane with the fibers being deformed Kleinian singularities.
The construction is in terms of a noncommutative algebra introduced by Ginzburg (Calabi-Yau algebras. , 2006) which we call the “N = 1 ADE quiver algebra”. 相似文献
7.
We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group
associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms
a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants
is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine
their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their
classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem
of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical
case with orthogonal quantum group symmetry. 相似文献
8.
The usual time-ordering operation and the corresponding time-ordered exponential play a fundamental role in physics and applied mathematics. In this work, we study a new approach to the understanding of time-ordering relying on recent progress made in the context of enveloping algebras of pre-Lie algebras. Various general formulas for pre-Lie and Rota–Baxter algebras are obtained in the process. Among others, we recover the noncommutative analog of the classical Bohnenblust–Spitzer formula, and get explicit formulae for operator products of time-ordered exponentials. 相似文献
9.
《Journal of Nonlinear Mathematical Physics》2013,20(1-2):62-68
Abstract The paper is devoted to application of the noncommutative integration method for linear partial differential equations. This method is based on the noncommutative integration theory for finite-dimensional Hamiltonian systems and is generalized for so-called functional algebras. 相似文献
10.
A. E. F. Djemai 《International Journal of Theoretical Physics》1995,34(6):801-887
We present a detailed review of the Dubois-Violette approach to noncommutative differential calculus. The noncommutative differential geometry of matrix algebras and the noncommutative Poisson structures are treated in some detail. We also present the analog of Maxwell's theory and new models of Yang-Mills-Higgs theories that can be constructed in this framework. In particular, some simple models are compared with the standard model. Finally, we discuss some perspectives and open questions. 相似文献
11.
This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres. We develop
the notion of central quadratic form for quadratic algebras, and a general theory which creates a bridge between noncommutative
differential geometry and its purely algebraic counterpart. It allows to construct a morphism from an involutive quadratic
algebra to a C*-algebra constructed from the characteristic variety and the hermitian line bundle associated to the central
quadratic form. We apply the general theory in the case of noncommutative 3-spheres and show that the above morphism corresponds
to a natural ramified covering by a noncommutative 3-dimensional nilmanifold. We then compute the Jacobian of the ramified
covering and obtain the answer as the product of a period (of an elliptic integral) by a rational function. We describe the
real and complex moduli spaces of noncommutative 3-spheres, relate the real one to root systems and the complex one to the
orbits of a birational cubic automorphism of three dimensional projective space. We classify the algebras and establish duality
relations between them. 相似文献
12.
We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the D-module is holonomic, and hence, the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the D-module is the pushforward of the canonical D-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. We explain many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra. We compute the D-module in the case of surfaces with isolated singularities and show it is not always semisimple. We also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein–Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix we give a brief recollection of the theory of D-modules on singular varieties that we require. 相似文献
13.
Anatolij Dvurečenskij Thomas Vetterlein 《International Journal of Theoretical Physics》2001,40(3):685-701
As a noncommutative generalization of effect algebras, we introduce pseudoeffect algebras and list some of their basic properties. For the purpose of a structure theory, we further define several kinds of Riesz-like properties for pseudoeffect algebras and show how they are interrelated. 相似文献
14.
In a theory where the local observables are determined by local field algebras as the fixed points under a (a priori noncommutative) group of gauge transformations of the first kind, we show that, if the field algebras possess intermediate type I factors, we can construct observables having the meaning of local charge measurements, and local current algebras in the field algebras. 相似文献
15.
Julius Wess 《General Relativity and Gravitation》2007,39(8):1121-1134
We consider a formalism by which gauge theories can be constructed on noncommutative space time structures. The coordinates
are supposed to form an algebra, restricted by certain requirements that allow us to realise the algebra in terms of star
products. In this formulation it is useful to define derivatives and to extend the algebra of coordinates by these derivatives.
The elements of this extended algebra are deformed differential operators. We then show that there is a morphism between these
deformed differential operators and the usual higher order differential operators acting on functions of commuting coordinates.
In this way we obtain deformed gauge transformations and a deformed version of the algebra of diffeomorphisms. The deformation
of these algebras can be clearly seen in the category of Hopf algebras. The comultiplication will be twisted. These twisted
algebras can be realised on noncommutative spaces and allow the construction of deformed gauge theories and deformed gravity
theory.
Dedicated to the 60th birthday of Prof. Obregon. 相似文献
16.
《Nuclear Physics B》1995,447(1):35-61
We discuss the prepotential describing the effective field theory of N = 2 heterotic superstring models. At the one loop-level the prepotential develops logarithmic singularities due to the appearance of charged massless states at particular surfaces in the moduli space of vector multiplets. These singularities modify the classical duality symmetry group which now becomes a representation of the fundamental group of the moduli space minus the singular surfaces. For the simplest two-moduli case, this fundamental group turns out to be a certain braid group and we determine the resulting full duality transformations of the prepotential, which are exact in perturbation theory. 相似文献
17.
Cyclic cohomology has been recently adapted to the treatment of Hopf symmetry in noncommutative geometry. The resulting theory of characteristic classes for Hopf algebras and their actions on algebras allows the expansion of the range of applications of cyclic cohomology. It is the goal of this Letter to illustrate these recent developments, with special emphasis on the application to transverse index theory, and point towards future directions. In particular, we highlight the remarkable accord between our framework for cyclic cohomology of Hopf algebras on the one hand and both the algebraic as well as the analytic theory of quantum groups on the other, manifest in the construction of the modular square. 相似文献
18.
H. Fuchs U. Kasper D. -E. Liebscher V. Müller H. -J. Schmidt 《Fortschritte der Physik》1988,36(6):427-452
We review the present status of the singularity problem in gravitation theory. First we repeat the known results for Einstein's theory. After this, different other gravitation theories are considered under the following aspect: Which of the presumptions leading to singularities in Einstein's theory are not fulfilled, and can this prevent the singularities? In detail: scalar matter fields do not fulfil the positive energy condition, bimetric theories change the gravitational vacuum, theories with torsion and fourth order gravity add new geometric terms. It turns out that the answer is never simply no or yes, but requires new conditions, and new types of singularities occur. Otherwise there exist essential situations where the alternative theories satisfactorily prevent the singular behaviour known from Einstein's theory. Finally, we review on chaotic singularities. 相似文献
19.
The aim of this paper is to generalize the Hochschild-Kostant-Rosenberg theorem to the case of singular varieties, more precisely,
to manifolds with boundary and to varieties with isolated singularities. In these situations, we define suitable algebras
of functions and study the localization of the corresponding Hochschild homology. The tool we use is the Teleman localization
process. In the case of isolated singularities, the closed Hochschild homology corresponds to the intersection complex which
relates the objects defined here to intersection homology. 相似文献
20.
Heide Narnhofer 《Reports on Mathematical Physics》2005,55(1):93-107
In ergodic theory von Neumann emphasized the spectral analysis of the unitary implementor and the possibility to express point translations as automorphisms over abelian algebras. Replacing the abelian algebras by noncommutative algebras good ergodic behaviour asks for type II and III algebras. The possibility for existing K-systems and Anosov systems in this framework is discussed. Von Neumnns example of a type III algebra is examined from this viewpoint. 相似文献