Quantization of Semi-Classical Twists and Noncommutative Geometry

A problem of defining the quantum analogues for semi-classical twists in U()[[t]] is considered. First, we study specialization at q = 1 of singular coboundary twists defined in Uq ())[[t]] for g being a nonexceptional Lie algebra, then we consider specialization of noncoboundary twists when = and obtain q-deformation of the semiclassical twist introduced by Connes and Moscovici in noncommutative geometry. Mathematics Subject Classification: 16W30, 17B37, 81R50     

Cyclic Cohomology and Hopf Algebra Symmetry

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.     

Differentials of Higher Order in Noncommutative Differential Geometry

In differential geometry, the notation dⁿ f along with the corresponding formalism has fallen into disuse since the birth of exterior calculus. However, differentials of higher-order are useful objects that can be interpreted in terms of functions on iterated tangent bundles (or in terms of jets). We generalize this notion to the case of noncommutative differential geometry. For an arbitrary algebra A, people already know how to define the differential algebra A of universal differential forms over A. We define Leibniz forms of order n (these are not forms of degree n, i.e. they are not elements of ⁿA) as particular elements of what we call the iterated frame algebra of order n, F_nA, which is itself defined as the2ⁿ tensor power of the algebra A. We give a system of generators for this iterated frame algebra and identify the A left-module of forms of order n as a particular vector subspace included in the space of universal 1-forms built over the iterated frame algebra of order n-1. We study the algebraic structure of these objects, recover the case of the commutative differential calculus of order n (Leibniz differentials) and give a few examples.     

Cyclic Cohomology and Hopf Algebras

We associate canonically a cyclic module to any Hopf algebra endowed with a modular pair in involution, consisting of a group-like element and a character. This provides the key construction for allowing the extension of cyclic cohomology to Hopf algebras in the nonunimodular case and, further, to developing a theory of characteristic classes for actions of Hopf algebras compatible not only with traces but also with the modular theory of weights. This applies to both ribbon and coribbon algebras as well as to quantum groups and their duals.     

Differential and Complex Geometry of Two-Dimensional Noncommutative Tori

We analyze in detail projective modules over two-dimensional noncommutative tori and complex structures on these modules. We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors generalizes the theory of theta-functions. The paper is self-contained; it can be used also as an introduction to the theory of noncommutative spaces with simplest space of this kind thoroughly analyzed as a basic example.     

Noncommutative Geometry and Integrable Models

A construction of conservation laws for -models in two dimensions is generalized in the framework of noncommutative geometry of commutative algebras. This is done by replacing theordinary calculus of differential forms with other differentialcalculi and introducing an analogue of the Hodge operator on thelatter. The general method is illustrated with several examples.     

Twists and Spectral Triples for Isospectral Deformations

We construct explicitly the symmetries of the isospectral deformations as twists of Lie algebras and demonstrate that they are isometries of the deformed spectral triples.     

Noncommutative Differential Calculus and Its Application on Discrete Spaces

We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincare lemma for the difference complex. Then the horizontal and vertical complexes are introduced with the total differential map and vertical exterior derivative. As the application of the differential calculus, we derive the schemes with the conservation of symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents. Then an Euler-Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.     

Lessons from Quantum Field Theory: Hopf Algebras and Spacetime Geometries

We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurrences of this, or closely related Hopf algebras, in other mathematical domains, such as foliations, Runge-Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.     

Noncommutative Geometry and Classical Orbits of Particles in a Central Force Potential

We investigate the effect of the noncommutative geometry on the classical orbits of particles in a central force potential The relation is implemented through the modified commutation relations [xi,xj] = iθij. Comparison with observations places severe constraints on the value of the noncommutativity parameter.     

Some Quantum-like Hopf Algebras which Remain Noncommutative when q = 1

Starting with only three of the six relations defining the standard (Manin) GL _q (2), we try to construct a quantum group. The antipode condition requires some new relations, but the process stops at a Hopf algebra with a Birkhoff–Witt basis of irreducible monomials. The quantum determinant is group-like but not central, even when q = 1. So, the two Hopf algebras constructed in this way are not isomorphic to the Manin GL _q (2), all of whose group-like elements are central. Analogous constructions can be made starting with the Dipper–Donkin version of GL _q (2), but these turn out to be included in the two classes of Hopf algebras described above.     

Towards a $$Z\prime $$ Gauge Boson in Noncommutative Geometry

We study all possible U(1)-extensions of the standard model within the framework of noncommutative geometry with the algebra . Comparison to experimental data about the mass of a hypothetical gauge boson leads to the necessity of introducing at least one new family of heavy fermions.     

Vacuum Solutions of Classical Gravity on Cyclic Groups from Noncommutative Geometry

Based on the observation that the moduli of a link variable on a cyclic group modify Connes‘ distance on this group,we construct several action functionals for this link variable within the framework of noncommutative geometry.After solving the equations of motion,we find that one type of action gives nontrivial vacuum solution for gravity on this cyclic group in a broad range of coupling constants and that such a solution can be expressed with Chebyshev‘s polynomials.     

More Noncommutative 4-Spheres

New examples of noncommutative 4-spheres are introduced.     

On Spin Structures and Dirac Operators on the Noncommutative Torus

We find and classify possible equivariant spin structures with Dirac operators on the noncommutative torus, proving that, similarly as in the classical case, the spectrum of the Dirac operator depends on the spin structure.     

DKP Oscillator in a Noncommutative Space

We present the DKP oscillator model of spins 0 and 1, in a noncommutative space. In the case of spin 0, the equation is reduced to Klein Gordon oscillator type, the wave functions are then deduced and compared with the DKP spinless particle subjected to the interaction of a constant magnetic field. For the case of spin 1, the problem is equivalent with the behavior of the DKP equation of spin 1 in a commutative space describing the movement of a vectorial boson subjected to the action of a constant magnetic field with additional correction which depends on the noncommutativity parameter.     

Noncommutative Differential Calculus Approach to Discrete Symplectic Schemes on Regular Lattice

By means of a noncommutative differential calculus on function space of discrete Abelian groups and that of the regular lattice with equal spacing as well as the discrete symplectic geometry and a kind of classical mechanical systems with separable Hamiltonian of the type H(p, q) = T(p) + V(q) on regular lattice, we introduce the discrete symplectic algorithm, i.e., the phase-space discrete counterpart of the symplectic algorithm including original symplectic schemes and the jet-symplectic schemes in terms of the discrete time jet bundle formalism, on the regular lattice. We show some numerical calculation examples and compare the results of different schemes.     

DKP Oscillator in a Noncommutative Space

We present the DKP oscillator model of spins 0 and 1, in a noncommutative space. In the case of spin 0, the equation is reduced to Klein-Gordon oscillator type, the wave functions are then deduced and compared with the DKP spinless particle subjected to the interaction of a constant magnetic field. For the case of spin 1, the problem is equivalent with the behavior of the DKP equation of spin 1 in a commutative space describing the movement of a vectorial boson subjected to the action of a constant magnetic field with additional correction which depends on the noncommutativity parameter.     

Quantum Observables Algebras and Abstract Differential Geometry: The Topos-Theoretic Dynamics of Diagrams of Commutative Algebraic Localizations

We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), à la Mallios, in a topos-theoretic environment, and hence, the extension of the “mechanism of differentials” in the quantum regime. The process of gluing information, within diagrams of commutative algebraic localizations, generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection.     

A Noncommutative Generalization of the Free-Field Yang–Mills Equations

The purpose of this paper is to propose a noncommutative generalization of a gauge connection and the free-field Yang–Mills equations. The paper draws upon the techniques proposed by Heller et al. for the noncommutative generalization of the Einstein field equations.