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It is known that the Lyapunov exponent for multifrequency analytic cocycles is weak-Hölder continuous in cocycle for certain Diophantine frequencies, and that this implies certain regularity of the integrated density of states in energy for Jacobi operators. In this paper, we establish the pointwise modulus of continuity in both cocycle and frequency and obtain analogous regularity of the integrated density of states in energy, potential, and frequency. [ABSTRACT FROM AUTHOR]

Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example, square roots) although the final result may be expressed without algebraic extensions. In this article, I present an improvement of this algorithm, which avoids algebraic extensions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the locally uniform convergence from a family of pullback random attractors to a deterministic attractor. We establish criteria by using joint-pathwise convergence of the cocycles, collective locally uniform compactness, and eventually deterministic of the random attractors. As an application of the abstract result, it is shown that the family of random attractors for the stochastic g-Navier–Stokes equation is local uniformly upper semi-continuous as the density of random noise tends to zero. [ABSTRACT FROM AUTHOR]

The limiting dynamics of stochastic 2D nonautonomous g-Navier-Stokes equations defined on a sequence of expanding domains are investigated, where the limiting domain is unbounded. By generalizing the energy-equation method, we show that the sequence of expanding cocycles is weakly equicontinuous and strongly equiasymptotically compact, which lead to both existence and upper semicontinuity of the null-expansion of the corresponding random attractor when the bounded domain approaches to the unbounded domain. [ABSTRACT FROM AUTHOR]

Motivated by the study of area laws for the entanglement entropy of gapped ground states of quantum spin systems and their stability, we prove that the unitary cocycle generated by a local time-dependent Hamiltonian can be approximated, for any finite set X, by a tensor product of the corresponding unitaries in X and its complement, multiplied by a dynamics strictly supported in the neighbourhood of the surface ∂X. The error decays almost exponentially in the size of the neighbourhood and grows with the square of the area ∣∂X∣2. [ABSTRACT FROM AUTHOR]

To a weighted graph can be associated a bipartite graph planar algebra P. We construct and study the symmetric enveloping inclusion of P. We show that this construction is equivariant with respect to the automorphism group of P.We consider subgroups G of the automorphism of P such that the G-fixed point space PG is a subfactor planar algebra. As an application we show that if G is amenable, then PG is amenable as a subfactor planar algebra. We define the notions of a cocycle action of a Hecke pair on a tracial von Neumann algebra and the corresponding crossed product. We show that a large class of symmetric enveloping inclusions of subfactor planar algebras can be described by such a crossed product. [ABSTRACT FROM AUTHOR]

We construct a Mickelsson-Rajeev-type cocycle corresponding to five space dimensions. In the three dimensional case, this cocycle is shown to be equivalent to the original cocycle proposed by Mickelsson and Rajeev. Furthermore, we construct a local representative for this cocycle and evaluate it explicitly on a five dimensional torus. [ABSTRACT FROM AUTHOR]

In this paper we discuss reproducing kernels whose ranges are contained in a C*-algebra or a Hilbert C*-module. Using the construction of a reproducing Hilbert C*-module associated with a reproducing kernel, we show how such a reproducing kernel can naturally be expressed in terms of operators on a Hilbert C*-module using representations on Hilbert C*-modules. We prove that for each positive definite G-kernel associated with a cocycle there is a representation associated with an operator-valued cocycle on the corresponding Hilbert C*-module. Finally, some examples will be considered. [ABSTRACT FROM AUTHOR]

We use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a Drinfeld-type cochain twist. These are the so-called quasialgebras and include the octonions as braided-commutative but nonassociative coordinate rings, as well as quasialgebra versions Cq(G) of the standard q-deformation quantum groups. We introduce the notion of ribbon algebras in the category, which are algebras equipped with a suitable generalized automorphism σ, and obtain the required generalization of cyclic cohomology. We show that this braided cyclic cohomology is invariant under a cochain twist. We also extend to our generalization the relation between cyclic cohomology and differential calculus on the ribbon quasialgebra. The paper includes differential calculus and cyclic cocycles on the octonions as a finite nonassociative geometry, as well as the algebraic noncommutative torus as an associative example. [ABSTRACT FROM AUTHOR]

Develops the twisting cocycles in fundamental representation and triangular bi-crossproduct Hopf algebras. Implications of classical universal enveloping Lie algebras for twisting 2-cocycles; Role of the tensor bialgebra in twist-equivalences; Relationship of the inhomogeneous universal enveloping Lie algebras with associative rings.

Introduces a class of 2-cocycles defined explicitly on the generators of certain multiparameter standard quantum groups. Generalizations of both the Cremmer-Gervais deformation of SL(3) and the esoteric quantum groups of Fronsdal and Galindo; Twisting of the familiar standard quantum groups to generate examples of non-standard quantum groups.

Studies gyrogroups that are equipped with a cocycle form and their resulting Lorentz groups. Definition of gyrogroups; Description of the cocycle form needed for the extension of gyrogroups into Lorentz groups; Extension of gyrogroups into real and complex space-times; Action of Lorentz transformations on abstract space-times.

In the framework of operator algebraic quantum statistical mechanics, the global nonequilibrium dynamics for a general class of interactions of infinite mean field quantum lattice systems with the boson field is investigated. The associated interaction operators consist of arbitrary powers of the collective density operators of the mean field system and are linear with respect to the bosonic field operators. Instead of the usual perturbation expansions, here the interacting dynamics are studied by means of cocycle techniques. The cocycle methods are more appropriate to the considered class of interactions, and lead to explicit closed expressions for the dynamical automorphism groups. The cocycle equations connect the classical, collective dynamical behavior of the mean field system with the one-boson dynamics. In physical applications such systems are due to collectively ordered finite-level atoms or the Josephson junction in the thermodynamic limit weakly interacting with the electromagnetic field. [ABSTRACT FROM AUTHOR]

Presents the intrinsic structure of the space of Lie algebra two-cocycles of an affine transformation group. Equivariant morphism from the space into a space of quadratic maps on the affine space; Two-cocycles of the Galilei Lie algebra that correspond to certain affine tensor fields on Galilean space-time.

Investigates the occurrence of a three-cocycle in quantum mechanics or quantum field theory as a violation of the Jacobi identity. Prevention of the existence of a certain extension of one Lie algebra; Derivation of observables of quantum fields; Formulation of the canonical commutation relations.

The relation between complex line bundles and certain group cocycles is explored in general to obtain explicit formulae for the transition functions and curvature of the determinant line bundle DET of a family of Dirac operators coupled to Yang–Mills fields. A covariant derivative on sections of DET is constructed which realizes the curvature and ‘‘minimally couples’’ to the integrated anomaly which thus appears as a ‘‘functional magnetic field’’ on gauge orbit space. The transcription of group cohomological (cocycles) into geometrical (line bundles) information is refined in such a way that the relevant cohomology groups can be computed in many cases, giving insight into the classification of lifts of principal group actions. [ABSTRACT FROM AUTHOR]

It is shown that the cocycles of extension of one massless representation of the Poincaré group P3 in 2+1 dimensions by the tensor product of n massless representations are coboundaries when the space of the representation is the space of the C∞ functions on R2 rapidly decreasing at infinity and at the origin. As a direct consequence, the equivalence classes of formal nonlinear representations of P3 with an irreducible physical representation as linear term are isomorphic to the classes of extension of the linear term by its symmetrical tensor product of order 2. [ABSTRACT FROM AUTHOR]

The symplectic two-cocycle on the semidirect product Lie algebra g(W+V*+V) is shown to be canonically related to the dual spaces of the Lie algebras (a) g(W+(gV)) and (b) g(W+(gV*)). This fact (a) explains the second Poisson bracket for irrotational 4He and (b) leads to a derivation of a new nonlinear Poisson bracket for rotating 4He. [ABSTRACT FROM AUTHOR]

It is shown that the two-cocycle involved in the Hamiltonian description of the superfluid [sup 4]He, both nonrotating and rotating, is a particular case of generalized symplectic two-cocycles on semidirect product Lie algebras. [ABSTRACT FROM AUTHOR]

Non-Abelian Toda lattices with a finite number of "above-diagonal" variables are related to appropriate Lie algebras of operators and a certain two-cocycle. [ABSTRACT FROM AUTHOR]

The theory of α*-cohomology is studied thoroughly and it is shown that in each cohomology class there exists a unique 2-cocycle, the harmonic form, which generates a particular Groenewold-Moyal star product. This leads to an algebraic classification of translation-invariant non-commutative structures and shows that any general translation-invariant non-commutative quantum field theory is physically equivalent to a Groenewold-Moyal non-commutative quantum field theory. [ABSTRACT FROM AUTHOR]

We study the quantization of chiral fermions coupled to generalized Dirac operators arising in NCG Yang–Mills theory. The cocycles describing chiral symmetry breaking are calculated. In particular, we introduce a generalized locality principle for the cocycles. Local cocycles are by definition expressions which can be written as generalized traces of operator commutators. In the case of pseudodifferential operators, these traces lead in fact to integrals of ordinary local de Rham forms. As an application of the general ideas we discuss the case of noncommutative tori. We also develop a gerbe theoretic approach to the chiral anomaly in the Hamiltonian quantization of NCG field theory. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]